28th Biennial Conferenceon

Numerical Analysis

25 – 28 June, 2019

Contents

Introduction 1Invited Speakers 3Abstracts of Invited Talks 4Abstracts of Minisymposia 8Abstracts of Contributed Talks 36

Nothing in here

Introduction

Dear Participant,

On behalf of the Strathclyde Numerical Analysis and Scientific Computing Group, it is our plea-sure to welcome you to the 28th Biennial Numerical Analysis conference. This is the sixth timethe meeting has been held at Strathclyde, continuing the long series of conferences originallyhosted in Dundee. We are delighted to have over 180 registered participants from all over theworld, some who are familiar faces at this series of meetings, but also many who are attendingfor the first time.

The conference is rather unusual in the sense that it seeks to encompass all areas of numericalanalysis, and the list of invited speakers reflects this aim. We have once again been extremelyfortunate in securing a top line-up of plenary speakers, and we very much hope that you enjoysampling the wide range of interesting topics which their presentations will cover. The spe-cialised minisymposia and sessions of contributed talks scheduled cover a wealth of additionalareas, confirming the continued strength of numerical analysis as an active field of research al-most 55 years since the first meeting in the series.

The meeting is funded almost entirely from the registration fees of the participants. However,we are very pleased to be able to subsidise the student registration fee and provide additional fi-nancial support for some overseas participants thanks to the Dundee Numerical Analysis Fund,started by Professor Gene Golub from Stanford University in 2007. We are also thankful tohave again received sponsorship from the UKIE section of the Society for Industrial and AppliedMathematics for three prizes for the best student talks which, if past meetings are anything togo by, will be a very hard-fought competition!

Outwith the scientific content, we hope you will also enjoy meeting new people and socialisingwith the other participants. We are indebted to the City of Glasgow for once again generouslysponsoring a wine reception in the spectacular City Chambers building: this will take place onTuesday evening. Another social highlight will be our conference dinner on Thursday evening,which will be held in the Trades Hall, another of Glasgow’s impressive historic buildings.

Welcome to Glasgow, and enjoy the meeting!

Philip KnightJohn MackenzieAlison Ramage

Conference Organising Committee

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Information for participants

• General. There will be a registration desk in the foyer of the John Anderson building.The organisers can be contacted there during tea and coffee breaks.

• Accommodation. All rooms are in the Campus Village. Check-out time is 10:00 on dayof departure. On Friday morning, luggage may be left in room JA327.

• Meals. Breakfast (Tue-Fri) is available from 07:30 until 09:00 in the Aroma Dining Roomin the Lord Todd building. The times of lunches and dinners are as indicated in theconference programme. Dinner (Tue, Wed) will also be served in Aroma. Buffet lunches(Tue-Thu) will be served in the Urban Bean Java Cafe and (Fri) in the foyer outside JA325.Coffee and tea will be provided at the advertised times in the foyer outside JA325.

• Lecture rooms. These are all in the John Anderson building. The main auditorium(JA325) is down one floor from the main entrance, along with rooms JA314 and JA317.The additional rooms for parallel sessions are JA505, JA506 and JA507 (on level 5 of theJohn Anderson building).

• Chairing sessions. It is hoped that if you are listed as chairing a session, you will bewilling to help in this way. Minisymposium organisers should organise chairpeople for theirown sessions (including any contributed talks which follow) as appropriate. A break of5 minutes has been allowed for moving between rooms. Please keep speakers to thetimetable!

• UKIE SIAM student prizes. If you are chairing a session, please rate any student talksusing the forms provided in the lecture rooms.

• Book displays. There will be books on display for the duration of the conference in roomJA326. Room JA327 is also available for conference participants.

• Reception. A reception for all participants hosted by Glasgow City Council will be held inthe City Chambers on Tuesday 23rd June from 20.00 to 21.00. Entry to the City Chambersis from George Square.

• Conference dinner. The conference dinner will be held in the Trades Hall of Glasgowon Thursday 29th June at 19:30 (for 20:00 dinner). The venue is located at 84 GlassfordStreet, Glasgow G1 1UH, which is 10 minutes walk from the conference venue. The guestspeaker will be Professor Jeremy Levesley, University of Leicester.

• Internet Access. Complimentary WiFi is available throughout the campus from the‘WifiGuest’ network, which should appear on the list of available networks on your portabledevice. This network uses the same authentication system as ‘The Cloud’ network foundin public places across the UK. You can log in to ‘WifiGuest’ using existing ‘The Cloud’credentials, or set up a new account which you can then use wherever ‘The Cloud’ isavailable. More comprehensive internet access is available to eduroam users. Computerterminals will be available from 09:00-17:00 Tuesday-Thursday in room JA512 of the JohnAnderson Building. If you require access to a fixed terminal, please contact the organisersto obtain a username/password.

• Bar. There is a bar in the Lord Todd building next to the Aroma Dining Room.

• Sports facilities. Delegates can access facilities at the University Sports Centre for a feeof £5 per visit.

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Invited Speakers

Raymond Chan City University of Hong Kong rchan.sci@cityu.edu.hk

Paul Constantine University of Colorado paul.constantine@colorado.edu

Alistair Forbes National Physical Laboratory alistair.forbes@npl.co.uk

Vivette Girault Sorbonne Universite girault@ann.jussieu.fr

Des Higham University of Edinburgh d.j.higham@ed.ac.uk

Natalia Kopteva University of Limerick natalia.kopteva@ul.ie

Gunilla Kreiss Uppsala University gunilla.kreiss@it.uu.se

Frances Kuo University of NSW f.kuo@unsw.edu.au

Ulrich Rude FAU Erlangen-Nurnberg ulrich.ruede@fau.de

Carola-Bibiane Schonlieb Univ of Cambridge cbs31@cam.ac.uk

Holger Wendland Universitat Bayreuth holger.wendland@uni-bayreuth.de

Margaret Wright Courant Institute mhw@cs.nyu.edu

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Abstracts of Invited Talks

Flexible methodology for image segmentation

Raymond H. Chan (City University of Hong Kong)

In this talk, we introduce a SaT (Smoothing andThresholding) method for multiphase segmentation ofimages corrupted with different degradations: noise,information loss and blur. At the first stage, a con-vex variant of the Mumford-Shah model is applied toobtain a smooth image. We show that the model hasunique solution under different degradations. In thesecond stage, we apply clustering and thresholdingtechniques to find the segmentation. The number ofphases is only required in the last stage, so users canmodify it without the need of repeating the first stageagain. The methodology can be applied to variouskinds of segmentation problems, including color im-age segmentation, hyper-spectral image classification,and point cloud segmentation. Experiments demon-strate that our SaT method gives excellent results interms of segmentation quality and CPU time in com-parison with other state-of-the-art methods.

Joint work with: X.H. Cai (UCL) and T.Y. Zeng(CUHK)

The uncertainty contribution of numerical soft-ware

Alistair Forbes (National Physical Laboratory)

The National Physical Laboratory (NPL) is the UK’sNational Metrology Institute, responsible for definingand dissemination standard units of measurement, themetre, the kilogram, etc. Two key concepts in metrol-ogy are those of traceability and uncertainty. Mea-surements results need to be traceable back to relevantstandard units though a chain of calibrations. Eachmeasurement result must be accompanied by an un-certainty statement that summarises the factors thatinfluence the accuracy of the measurement and quan-tifies their uncertainty contribution. Almost all mea-surement results involve computation, often signifi-cantly, and it is therefore necessary to assess the un-certainty contribution of the numerical software per-forming the computation. There are many ways inwhich numerical software can make a significant un-certainty contribution. There are issues relating tofinite precision, numerical stability of algorithms andill-conditioning. But there are also contributions re-

lated to model approximations and discretisation, al-gorithms that address a different computational aim,iterative schemes involving convergence tolerances andtuning parameters. Finally, the software implement-ing the algorithms might not have been coded cor-rectly.

We can regard numerical software as fit for purpose ifits uncertainty contribution is small compared to otheruncertainty sources such as those associated with theinput data. A dependable uncertainty quantificationscheme also opens up the possibility of implementingsimpler, quicker algorithms, perhaps in reduced pre-cision, that still meet the requirements of the user interms of accuracy.

In this talk, I will summarise some of the work wehave undertaken at NPL relating to assessing the un-certainty contribution of numerical software with ap-plications to the analysis of metrology data. I will alsotake the opportunity to discuss the redefinition of theSI units in terms of fundamental physical constants.

Discretization of some elastic models with im-plicit non linear constitutive relations

V. Girault (Sorbonne Universite)

There are situations where linearization of classicalCauchy elasticity is not applicable. This occurs forexample in the vicinity of crack tips and notches ina brittle material. In this case, the generalization ofCauchy elasticity proposed by K.R. Rajagopal, withimplicit non linear constitutive relation, leads to agood description of the phenomenon. One of its ad-vantages is that it permits the strain (i.e., the sym-metric gradient of the displacement) to stay bounded,even when the stress is very large. This approach hasnot been much explored and is an interesting new lineof research.

With this motivation, I shall discuss some finite ele-ment discretizations of a family of such models, wherethe strain is a non linear monotone function of thestress, but without coercivity. The problem consistsof a system of two equations, the balance of linearmomentum and the constitutive equation, in two un-knowns, the displacement and the stress. The theo-retical analysis is done by stabilizing the constitutiveequation by the adjunction of a small stabilizing co-ercive term, and by proving convergence of the stabi-lized solution. Therefore, I shall discretize the stabi-lized form and prove convergence of its finite elementsolution. Also, as the resulting non linear system islarge, I shall use an alternating-direction algorithm to

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decouple the computation of the two equations, andpresent a numerical experiment. The monotone char-acter of the operator plays a fundamental part in thenumerical (and theoretical) analysis, as well as in thealgorithm’s convergence.

If there is time, I shall briefly present other applica-tions to similar models.

This is joint work with A. Bonito, D. Guignard, K.Rajagopal, Department of Mathematics, TAMU, E.Suli, Mathematical Institute, University of Oxford

Walk this way

Desmond J. Higham (University of Edinburgh)

The notion of a walk around a graph is both natu-ral and useful. The walker may follow any availableroute, with nodes and edges being revisited at anystage. However, in some settings, walks that back-track—leave a node and then return to it on the nextstep—are best avoided. The concept of restrictingattention to nonbacktracking walks has arisen, essen-tially independently, in a wide range of seemingly un-related fields, including spectral graph theory, num-ber theory, discrete mathematics, quantum chaos, ran-dom matrix theory, stochastic analysis, applied linearalgebra and computer science. Nonbacktracking hasmore recently been considered in the context of ma-trix computation, where it has been shown to formthe basis of effective algorithms in network sciencefor community detection, centrality measurement andgraph alignment. I will discuss some recent theoret-ical and practical advances that combine ideas frommatrix polynomial theory, spectral analysis, mathe-matical physics and what Herbert S. Wilf called “gen-eratingfunctionology.” Topics will include

• localization effects,

• graph pruning for efficiency,

• alternative interpretations of nonbacktracking al-gorithms,

• a nonbacktracking version of Google’s PageR-ank,

• extensions to nontriangulating, nonsquaring and,generally, non-k-cycling walks.

Results will be illustrated on some large scale networksarising in the social sciences. My nonbacktracking col-laborators are Francesca Arrigo (Strathclyde), PeterGrindrod (Oxford) and Vanni Noferini (Aalto).

A posteriori error estimation on anisotropic

meshes

Natalia Kopteva (University of Limerick)

Solutions of partial differential equations frequentlyexhibit corner singularities and/or sharp boundary andinterior layers. To obtain reliable numerical approxi-mations of such solutions in an efficient way, one maywant to use meshes that are adapted to solution sin-gularities. Such meshes can be constructed using apriori information on the exact solutions, however itis rarely available in real-life applications. Thereforethe automated mesh construction by adaptive tech-niques is frequently employed. This approach requiresno initial asymptotic understanding of the nature ofthe solutions and the solution singularity locations.Note that reliable adaptive algorithms are based on aposteriori error estimates, i.e. estimates of the errorin terms of values obtained in the computation pro-cess: computed solution and current mesh. Such aposteriori error estimates for elliptic partial differen-tial equations will be the subject of this talk.

Our main goal in this talk is to present residual-typea posteriori error estimates in the maximum norm, aswell as in the energy norm, on anisotropic meshes, i.e.we allow mesh elements to have extremely high aspectratios. The error constants in these estimates are in-dependent of the diameters and the aspect ratios ofmesh elements. Note also that, in contrast to some aposteriori error estimates on anisotropic meshes in theliterature, our error constants do not involve so-calledmatching functions (that depend on the unknown er-ror and, in general, may be as large as mesh aspectratios).

To deal with anisotropic elements, a number of tech-nical issues have been addressed. For example, an in-spection of standard proofs for shape-regular meshesreveals that one obstacle in extending them to aniso-tropic meshes lies in the application of a scaled tracedtheorem when estimating the jump residual terms (thiscauses the mesh aspect ratios to appear in the es-timator). For the estimation in the energy norm, aspecial quasi-interpolation operator is constructed onanisotropic meshes, which may be of independent in-terest.

In the second part of the talk, the focus will shiftto the efficiency of energy-norm error estimators onanisotropic meshes.

A simple numerical example will be given that clearlydemonstrates that the standard bubble function ap-proach does not yield sharp lower error bounds. To

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be more precise, short-edge jump residual terms insuch bounds are not sharp. To remedy this, we shallpresent a new upper bound for the short-edge jumpresidual terms of the estimator.

We shall also touch on that certain perceptions needto be adjusted for the case of anisotropic meshes. Inparticular, it is not always the case that the computed-solution error in the maximum norm is closely relatedto the corresponding interpolation error.

Stability and accuracy for initial-boundaryvalue problems revisited

Gunilla Kreiss (Uppsala University)

Stability and accuracy for a numerical method approx-imating an initial boundary value problem are inher-ently linked together. Stability means that perturba-tions of data have a bounded effect on the discretesolution, and is usually characterized by a precise es-timate, which links the perturbations of data to per-turbations of the solution. Such an estimate can be di-rectly used to quantify the accuracy of the method. Avery convenient and common way to investigate stabil-ity, and hence accuracy, is to use the energy method.If this approach fails one may instead attempt to getresults by Laplace transforming in time. Such analy-sis is usually more involved, but sharper results mayfollow. In this talk we will discuss the two basic ap-proaches, both in continuous and discrete settings. Wewill also show two examples where, even though theenergy method is applicable, it is rewarding to con-sider the problem in the Laplace domain. In the firstcase we get sharper accuracy results, and in the secondcase we get sharper temporal bounds.

High dimensional integration and approxima-tion: the Quasi-Monte Carlo (QMC) way

Frances Y. Kuo (UNSW Sydney)

High dimensional computation – that is, numericalcomputation in which there are very many or even in-finitely many continuous variables – is a new frontierin scientific computing. Often the high dimensionalitycomes from uncertainty or randomness in the model ordata (e.g., in groundwater flow it can arise from mod-eling the permeability field that is rapidly varying anduncertain). High dimensional problems present majorchallenges to computational resources, and require se-rious theoretical numerical analysis in devising newand effective methods.

This talk will begin with a contemporary review ofQuasi-Monte Carlo (QMC) methods, which offer tai-lored point constructions for solving high dimensionalintegration and approximation problems by sampling.By exploiting the smoothness properties of the un-derlying mathematical functions, QMC methods areproven to achieve higher order convergence rates, beat-ing standard Monte Carlo sampling. Moreover, QMCerror bounds can be independent of the dimension un-der appropriate theoretical function space settings.

In recent years the modern QMC theory has been suc-cessfully applied to a number of applications in uncer-tainty quantification. This talk will showcase someongoing works where we take QMC methods to newterritories including neutron transport as a high di-mensional PDE eigenvalue problem, high frequencywave scattering in random media, optimal control con-strained by PDE with random coefficients, and a rev-olutionary approach to model random fields using pe-riodic random variables. The common and essentialtheme among these collaborations with various inter-national teams is that we are not just applying anoff-the-shelf method; rather, we provide rigorous errorand cost analysis to design QMC methods tailored tothe features of the underlying mathematical functionsin these applications.

Parallel multigrid for systems with a trillionunknowns and beyond

Ulrich Rude (Friedrich-Alexander-UniversitatErlangen-Nurnberg and CERFACS, Toulouse)

For discretized elliptic PDE, multigrid methods canprovide asymptotically optimal solvers so that thenumber of operations grows only proportional to thenumber of unknowns. Such algorithms can then beused to develop scalable parallel solvers, i.e. concurrentalgorithms that can solve twice the size of a system inthe same time when it is given twice the number ofprocessors. This is known as weak scaling. We willalso discuss strong scaling, i.e. solving the same sys-tem in half the time when given twice as many pro-cessors, since this is significantly harder to achieve.However, multigrid methods do not only exhibit fa-vorable asymptotic behavior. They are also efficientin an absolute sense, since they often beat other meth-ods in terms of total number of flops and time spentfor computing a sufficiently accurate solution for agiven system size. To reach fast compute times, it isalso essential that the methods are implemented withcare and being aware of the architecture of modernCPUs, exploiting multicore architectures, instruction

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level parallelism, and vector units. We will also discusshow parallel multigrid methods can be excellent com-panions for Multilevel Monte Carlo methods, wheremany problems of varying size must be solved.

Finally, we will present techniques that are necessaryto realize multigrid methods that can exploit super-computers with up to a million parallel processes. Thelargest example problem will be a Stokes system thatis discretized with finite elements and that results inan indefinite linear system with more than 1013 un-knowns. This corresponds to a solution vector thatrequires 80 TByte of memory. For solving these prob-lems of this size, matrix free techniques have beendeveloped that do not need to store the stiffness ormass matrices. The talk will conclude with examplesfrom Earth mantle convection that are grand challengeproblems in geophysics.

This is joint work with Dominik Bartuschat, SimonBauer, Hans-Peter Bunge, Daniel Drzisga, BjornGmeiner, Markus Huber, Lorenz John, Nils Kohl,Philippe Leleux, Marcus Mohr, Holger Stengel, Do-minik Thonnes, Christian Waluga, Jens Weismuller,and Barbara Wohlmuth in the project TerraNeo withinthe DFG priority program Software for Exascale Com-puting.

Variational models and partial differential equa-tions for mathematical imaging

Carola-Bibiane Schonlieb (Cambridge University)

Images are a rich source of beautiful mathematical for-malism and analysis. Associated mathematical prob-lems arise in functional and non-smooth analysis, thetheory and numerical analysis of partial differentialequations, harmonic, stochastic and statistical analy-sis, and optimisation. Starting with a discussion onthe intrinsic structure of images and their mathemati-cal representation, in this talk we will learn about vari-ational models for image analysis and their connectionto partial differential equations, and go all the way tothe challenges of their mathematical analysis as wellas the hurdles for solving these - typically non-smooth- models computationally. The talk is furnished withapplications of the introduced models to digital artrestoration, cancer imaging and forest conservation.If time allows I might say a few words about usingmachine learning to learn better variational modelsfor imaging.

Kernel-based reconstructions for uncertainty quan-tification

Holger Wendland (University of Bayreuth)

In uncertainty quantification, an unknown quantityhas to be reconstructed which depends typically onthe solution of a partial differential equation. Thispartial differential equation itself depends on parame-ters, some of them are deterministic and some are ran-dom. To approximate the unknown quantity one thushas to solve the partial differential equation (usuallynumerically) for several instances of the parametersand then reconstruct the quantity from these simu-lations. As the number of parameters may be large,this becomes a high-dimensional reconstruction prob-lem. In this talk, I will address the topic of recon-structing such unknown quantities using kernel-basedreconstruction methods on sparse grids. I will give anintroduction to the topic. After that, I will explaindifferent reconstruction processes using kernel-basedmethods such as support vector machines and multi-scale radial basis functions and the so-called Smolyakalgorithm. I will discuss techniques for deriving errorestimates and show convergence results. If time per-mits, I will also give numerical examples.

This talk is based upon joint work with R. Kempf(Bayreuth) and C. Rieger (Aachen/Bonn).

Optimization and machine learning: what arethe connections?

Margaret Wright (Courant Institute)

To answer the question in the title, one might turn tothe hundreds of millions of relevant websites producedby a favorite search engine. Unfortunately, the an-swers are highly inconsistent, nor are they necessarilytrustworthy. (Some offer to charge a large fee so thatyou can become a data scientist in 6 weeks.) There areexperts who claim that there is very little differencebetween machine learning and optimization; others as-sert that machine learning is merely a consumer ofoptimization; and still others state emphatically thatoptimization is a strict subset of machine learning.The speaker (an optimizer) will discuss the increas-ing number of interactions between machine learningand “classical” optimization, emphasizing recent andcontinuing substantive changes.

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Minisymposia abstracts

Minisymposium M1

Fractional-derivative problemsOrganisers

Martin Stynes and Yubin Yan

Error analysis of the Grunwald–Letnikovscheme for fractional initial-value problemswith weakly singular solutions

Hu Chen & Finbarr Holland & Martin Stynes(Beijing Computational Science Research Center)

A convergence analysis is given for the Grunwald-Letnikov discretisation of a Riemann-Liouville frac-tional initial-value problem on a uniform mesh tm =mτ with m = 0, 1, . . . ,M . For given smooth data, theunknown solution of the problem will usually have aweak singularity at the initial time t = 0. Our anal-ysis is the first to prove a convergence result for thismethod while assuming such non-smooth behaviour inthe unknown solution. In part our study imitates pre-vious analyses of the L1 discretisation of such prob-lems, but the introduction of some additional ideasenables exact formulas for the stability multipliers inthe Grunwald-Letnikov analysis to be obtained (theearlier L1 analyses yielded only estimates of their sta-bility multipliers). Armed with this information, it isshown that the solution computed by the Grunwald-Letnikov scheme is O(τtα−1

m ) at each mesh point tm;hence the scheme is globally only O(τα) accurate, butit isO(τ) accurate for mesh points tm that are boundedaway from t = 0. Numerical results for a test exampleshow that these theoretical results are sharp.

Mittag-Leffler Euler integrator for solving semi-linear time fractional stochastic partial differ-ential equations driven by fractionally inte-grated additive noise

Ye Hu (LvLiang University, University of Chester)& Changpin Li (Shanghai University) & Yubin Yan(University of Chester)

In this paper, we introduce a spectral element methodin space and an exponential time discretization schemein time to approximate a semilinear stochastic partialdifferential equation driven by fractionally integrated

additive noise, which is the continuation of the workin Jin et al. where the numerical methods for thelinear stochastic partial differential equations drivenby fractionally integrated additive noise are consid-ered. The solution of the problem involves three dif-ferent Mittag-Leffler functions and the regularities ofthe solution are studied based on the properties of theMittag-Leffler functions. The optimal convergence er-ror estimates are obtained which precisely reflect howthe fractional orders α, γ affect the convergence rates.Numerical experiments show that the numerical re-sults are consistent with the theoretical findings.

Sharp spatial H1-norm analysis of a finiteelement method for a time-fractional initial-boundary value problem

Chaobao Huang & Martin Stynes (Beijing Compu-tational Science Research Center)

A time-fractional initial-boundary value problem Dαt u

−∆u = f , where Dαt is a Caputo fractional derivative

of order α ∈ (0, 1), is considered on the space-timedomain Ω× [0, T ], where Ω ⊂ Rd (d ≥ 1) is a boundedLipschitz domain. Typical solutions u(x, t) of suchproblems have components that behave like a multipleof tα as t→ 0+, so the integer-order temporal deriva-tives of u blow up at t = 0. The numerical methodof the paper uses a standard finite element method inspace on a quasiuniform mesh and considers both theL1 discretisation and Alikhanov’s L2-1σ discretisationof the Caputo derivative on suitably graded temporalmeshes. Optimal error bounds in H1(Ω) are proved;no previous analysis of a discretisation of this problemusing finite elements in space has established such abound. Furthermore, the optimal grading of the tem-poral mesh can be deduced from our analysis. Numer-ical experiments show that our error bounds are sharp.

References

[1] Chaobao Huang and Martin Stynes, Optimal spa-tial H1-norm analysis of a finite element method fora time-fractional diffusion equation. (Submitted forpublication).

Mittag-Leffler Euler integrator for a stochasticfractional order equation with additive noise

Mihaly Kovacs & Stig Larsson & Fardin Saedpanah(PPKE ITK)

Motivated by fractional derivative models in viscoelas-ticity, a class of semilinear stochastic Volterra integro-

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differential equations, and their deterministic counter-part, are considered. A generalized exponential Eulermethod, named here as the Mittag-Leffler Euler inte-grator, is used for the temporal discretization whilethe spatial discretization is performed by the spectralGalerkin method. The temporal rate of strong conver-gence is found to be (almost) twice compared to whenthe backward Euler method is used together with aconvolution quadrature for time discretization.

Subdiffusion with a time-dependent coefficient:analysis and numerical solution

Bangti Jin & Buyang Li & Zhi Zhou (The Polytech-nic University of Hong Kong)

In this work, a complete error analysis is presentedfor fully discrete solutions of the subdiffusion equationwith a time-dependent diffusion coefficient, obtainedby the Galerkin finite element method with conform-ing piecewise linear finite elements in space and back-ward Euler convolution quadrature in time. The reg-ularity of the solutions of the subdiffusion model isproved for both nonsmooth initial data and incom-patible source term. Optimal-order convergence of thenumerical solutions is established using the proven so-lution regularity and a novel perturbation argumentvia freezing the diffusion coefficient at a fixed time.The analysis is supported by numerical experiments.

Error analysis for a fractional-derivative para-bolic problem on quasi-graded meshes usingbarrier functions

Natalia Kopteva (University of Limerick) &Xiangyun Meng (Beijing Computational Science Re-search Center)

An initial-boundary value problem with a Caputo timederivative of fractional order α ∈ (0, 1) is considered,solutions of which typically exhibit a singularbehaviour at an initial time. For this problem, we givea simple and general numerical-stability analysis usingbarrier functions, which yields sharp pointwise-in-timeerror bounds on quasi-graded temporal meshes witharbitrary degree of grading. L1-type and Alikhanov-type discretization in time are considered. In par-ticular, those results imply that milder (compared tothe optimal) grading yields optimal convergence ratesin positive time. Semi-discretizations in time and fulldiscretizations are addressed. The theoretical findings

are illustrated by numerical experiments.

Blow-up in time-fractional initial-boundaryvalue problems

Martin Stynes & Hu Chen (Beijing ComputationalScience Research Center)

Time-fractional initial-boundary value problems of theform Dα

t u− p∂2u/∂x2 + cu = f are considered, whereDαt u is a Caputo fractional derivative of order α ∈

(0, 1). As α → 1−, we prove that the solution u con-verges, uniformly on the space-time domain, to the so-lution of the classical parabolic initial-boundary valueproblem where Dα

t u is replaced by ∂u/∂t. Neverthe-less, most of the rigorous analyses of numerical meth-ods for this time-fractional problem have error boundsthat blow up as α→ 1−, as we demonstrate. We showthat in some cases these analyses can be modified toobtain robust error bounds that do not blow up asα→ 1−.

Convergence of L1-Galerkin FEMs for nonlin-ear time-fractional Schrodinger equations

Jilu Wang (Beijing Computational Science ResearchCenter) & Dongfang Li (Huazhong University of Sci-ence and Technology) & Jiwei Zhang (Wuhan Univer-sity)

In this work, a linearized L1-Galerkin finite elementmethod is proposed to solve the multi-dimensionalnonlinear time-fractional Schrodinger equation. Interms of a temporal-spatial error splitting argument,we prove that the finite element approximations in L2-norm and L∞-norm are bounded without any timestepsize conditions. More importantly, by using a dis-crete fractional Gronwall type inequality, optimal er-ror estimates of the numerical schemes are obtainedunconditionally, while the classical analysis for multi-dimensional nonlinear fractional problems always re-quired certain time-step restrictions dependent on thespatial mesh size. Numerical examples are given toillustrate our theoretical results.

Laplace transform method for solving the frac-tional cable equation with nonsmooth data

Yanyong Wang & Yubin Yan & Amiya K. Pani

(Lvliang University, China; University of Chester;Indian Institute of Technology Bombay)

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We introduce two time discretization schemes for solv-ing the time fractional cable equation. The time deriva-tive is approximated by using the backward Eulermethod and the second order backward difference for-mula, respectively. The Riemann-Liouville fractionalderivatives are approximated by using the backwardEuler convolution quadrature method and the sec-ond order backward difference convolution quadraturemethod, respectively. The nonsmooth data error es-timates with the convergence orders O(k) and O(k2)are proved in detail. Instead of using the discretizedoperational calculus approach to prove the nonsmoothdata error estimates of the time discretization schemesfor solving time fractional cable equation as used inliterature, we directly bound the kernel function inthe resolvent and obtain the nonsmooth data errorestimates. To the best of our knowledge, this is thefirst work to consider the nonsmooth data error esti-mates for solving the time fractional cable equationby directly bounding the kernel function in the resol-vent. This argument may be applied to consider thenonsmooth data error estimates for solving the timefractional cable equation where the Riemann-Liouvillefractional derivatives are approximated by using otherschemes, for example, L1 scheme.

Minisymposium M2

Recent Advances in the NumericalSolution of Inverse Problems in Imaging

OrganiserSilvia Gazzola

Adaptive regularization parameter choicerules for large-scale problems

Malena Sabate Landman & Silvia Gazzola(University of Bath)

This talk introduces a new class of adaptive regular-ization parameter choice strategies that can be effi-ciently applied when regularizing large-scale linear in-verse problems using a combination of projection ontoKrylov subspaces and Tikhonov regularization, andthat can be regarded as special instances of bilevel op-timization methods. The links between Gauss quadra-ture and Golub-Kahan bidiagonalization are exploitedto prove convergence results for some of the consideredapproaches, and numerical tests are shown to give in-

sight.

Learning a sampling pattern for MRI

Ferdia Sherry & Matthias J. Ehrhardt & Carola-Bibiane Schonlieb (University of Cambridge)

The discovery of the theory of compressed sensingbrought the realisation that many inverse problemscan be solved satisfactorily even when measurementsare incomplete. This is particularly interesting in animaging modality such as magnetic resonance imaging(MRI), where long acquisition times are problematic.The measurements taken in MRI are often modelledas samples of the Fourier transform of the signal to berecovered. In this work, we consider the problem oflearning a sparse sampling pattern that can be used toobtain high quality reconstructions for images that aresimilar to the images used in training, while only hav-ing to sample along the learned sampling pattern. Theapproach that is taken is that of bilevel learning: aminimisation problem is solved, which has constraintsthat are formulated as the solution to variational reg-ularisation problems (the problems that are solved toreconstruct a signal from measurements). The frame-work that we set up for learning a sampling patternallows a sampling pattern of scattered points or ofCartesian lines in k-space to be learned and can easilybe extended to other parametrisations of the samplingpattern.

Some augmented methods for treatment of ill-posed problems

Kirk M Soodhalter (Trinity College Dublin)

In this talk, we discuss some strategies for treatingdiscretize ill-posed problems of the form,

Ax = b with A ∈ Rn×n and b ∈ Rn,

which inherits a discrete version of ill conditioningfrom the continuum problem. We propose methodsbuilt on top of the Arnoldi-Tikhonov method of Lewisand Reichel, whereby one builds the Krylov subspace

Kj(A,w) = spanw,Aw,A2w, . . . ,Aj−1w

where

w ∈ b,Aband solves the discretized Tikhonov minimization prob-lem projected onto that subspace. We propose to ex-tend this strategy to setting of augmented Krylov sub-space methods. Thus we project onto a sum of sub-spaces of the form U +Kj where U is a fixed subspace

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and Kj is a Krylov subspace. There are a number ofapproaches one can take to combine these strategies.We discuss them and demonstrate how they work withsome example problems.

Variational models for image registration prob-lems

Anis Theljani & Ke Chen (University of Liverpool)

Image registration is a core problem in Medical Imag-ing. It aims to align two or more images so that in-formation can be compared and/or fused to highlightdifferences and complement information. Mathemati-cally the task is to find a suitable transform that mapspatterns and shapes of one image to those of another.The different imaging modalities lead to images withdifferent properties and often complementary infor-mation which makes the task particularly useful. Al-though most methods work very well on mono-modalimages, there is a high demand for improved mod-els for registration of multi-modality images. In thistalk, we present some new works on improved varia-tional models for multi-modal images. Experimentsand comparisons with competitive models show thegood performance of the proposed models in gener-ating an accurate diffeomorphic transformation. Wealso show how these models can be in used in a deeplearning framework to order to get an accurate solu-tion in fast speed for real time applications

Blind image deconvolution using a non-separable point spread function

Joab Winkler (University of Sheffield)

This paper considers the problem of the removal ofblur from an image that is degraded by a non-separablepoint spread function (PSF), when information on thePSF is not known. The non-separable nature of thePSF implies that two blurred images, taken by thesame system such that the PSF can be assumed to bethe same for both images, are required to determinethe PSF. The most difficult part of the computation isthe determination of the size of the PSF because thisproblem reduces to the determination of the rank oftwo matrices (one matrix for the horizontal componentof the PSF and one matrix for the vertical componentof the PSF). It is shown that this computation requiresthe determination of the greatest common divisor oftwo polynomials, after they have been transformed tothe Fourier domain. The Sylvester resultant matrixand its subresultant matrices are used for this compu-

tation. A structure-preserving matrix method is usedto perform each deconvolution, and thereby computedeblurred forms of the given blurred images becausethis method preserves the Tœplitz structure of the co-efficient matrix in the linear algebraic equation.

The presentation includes examples that demonstratethe method.

Expectation propagation for Poisson data

Chen Zhang & Simon Arridge & Bangti Jin(University College London)

The Poisson distribution arises naturally when deal-ing with data involving counts, and it has found manyapplications in inverse problems and imaging. In thiswork, we develop an approximate Bayesian inferencetechnique based on expectation propagation for ap-proximating the posterior distribution formed fromthe Poisson likelihood function and a Laplace typeprior distribution, e.g., the anisotropic total varia-tion prior. The approach iteratively yields a Gaus-sian approximation, and at each iteration, it updatesthe Gaussian approximation to one factor of the pos-terior distribution by moment matching. We deriveexplicit update formulas in terms of one-dimensionalintegrals, and also discuss stable and efficient quadra-ture rules for evaluating these integrals. The methodis showcased on two-dimensional PET images.

Minisymposium M3

Trends in numerical methods for partialdifferential equations

OrganisersA M Portillo, M J Moreta and N

Reguera

Transparency: boundary conditions that imi-tate the Cauchy problem for differential andfinite-difference equations and systems

Vladimir A. Gordin & Alexander A. Shemendyuk(National Research University “Higher School of Eco-nomics”) & Hydrometeorological Centre of Russia)

In many practical problems of mathematical physicsthe necessary complete set of physically adequateboundary conditions is absent. For example, in theproblem of weather forecasting for a limited area V ,

11

we can set Dirichlet boundary conditions at the bor-der ∂V , where the right hand side of the conditions istaken from a larger scale (global) forecasting model.We can consider the difference between meteorologicalfields in these two numerical models. The dynamicsof this difference show that waves coming out of thecomputational area are reflected from the boundary∂V . It is not a physical effect, which worsens the re-gional model’s forecasts.

We construct the special kind of boundary conditions(BC) that do not reflect outgoing waves both for dif-ferential and finite-difference problems. The mixedinitial-boundary problem under such BC give the so-lution, which is identical to the solution of the cor-responding Cauchy problem, i.e. the BC imitate theCauchy problem.

The ICP BC may be constructed for a wide class ofPDE. As a rule, such BC are not local: Au = B(dxu)for differential problems and Au(x = 0) = Bu(x = h)for finite-difference ones. Here A, B are convolutionoperators (continuous or discrete) with respect to thevariables that tangent to ∂V , and the variable x isdirected along the normal to it. The coefficients ofthe convolution may be interpreted as the coefficientsof the Pade – Hermite rational approximation. Thebigger number of the ICP BC are necessary for higherorder (with respect to x) of PDE (e.g. for the equa-tion of rod transverse vibrations). We use here a moresophisticated kind of rational approximation. We con-firm the ICP-property by numerical experiments forvarious difference schemes for various PDE.

The work was prepared within the framework of theAcademic Fund Program at the National ResearchUniversity Higher School of Economics (HSE) in 2018– 2019 (grant No 18-05-0011) and by the Russian Aca-demic Excellence Project 5-100.

References

[1] V.A.Gordin. “About Mixed Boundary Problemthat Imitates Cauchy’s Problem,” Uspekhi Matem-aticheskikh Nauk, vol. 33, no. 5, pp. 181-182, 1978.

[2] V.A.Gordin. Mathematical Problems and Meth-ods in Hydrodynamical Weather Forecasting. Gordon& Breach, 2000.

[3] V.A.Gordin. Mathematics, Computer, WeatherForecast and Other Scenaria of Mathematical Physics,Moscow: FIZMATLIT, 2010, 2013 (in Russian).

[4] V.A. Gordin, A.A.Shemendyuk. “Transparent”Boundary Conditions for the Equation of Rod Trans-verse Vibrations. Preprint https://arxiv.org/submit/

2447606/view 2018.

A Schwarz method for a Rayleigh-Benard prob-lem

Henar Herrero & Francisco Pla (Universidad deCastilla-La Mancha)

A study of a Schwarz domain decomposition numeri-cal method for a Rayleigh-Benard convection problemis presented. The model equations are incompress-ible Navier-Stokes coupled with a heat equation underBoussinesq approximation. The problem is defined ina rectangular domain. The nonlinear stationary prob-lem is dealt with by a Newton method. Each step inthe Newton method is solved with a Schwarz domaindecomposition method with the domain partitionedinto two subdomains with appropriate interface condi-tions. The convergence properties of a linear case arestudied theoretically in a simplified domain includingtwo artificial parameters in the equations. The numer-ical resolution of the problem confirms the theoreticalresults. The convergence rate is less than one whenoverlap is considered. Convergence is optimal for somevalues of the parameters.

Solving efficiently time dependent singularlyperturbed convection-diffusion systems

J.C. Jorge & C. Clavero (Public University ofNavarra)

In this talk we deal with the efficient numerical res-olution of one dimensional parabolic singularly per-turbed systems of convection-diffusion-reaction typewhich are coupled by the reaction terms. We assumethat the convection matrix is diagonal while we permitthat the diffusion parameters can be different at eachequation and even they can have different orders ofmagnitude. The numerical method used to solve thecontinuous problem combines a standard upwind fi-nite difference scheme, defined on a special mesh ofShishkin type, to discretize in space and the frac-tional implicit Euler method together with an appro-priate splitting by components to integrate in time.It is proven that the resulting numerical algorithmis uniformly convergent of first order in time and ofalmost first order in space. Besides, the splitting-by-components technique used here makes that onlysmall tridiagonal linear systems must be solved to ad-vance in time; because of this, the proposed algorithmis more efficient than other classical implicit methodsused to solve the same type of problems. Some nu-

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merical experiments are shown which corroborate inpractice both the uniform convergence and the effi-ciency of the algorithm.

Exponential quadrature rules without order re-duction for integrating linear initial boundaryvalue problems

M. J. Moreta & B. Cano (Universidad Complutensede Madrid)

In this talk a technique is suggested to completelyavoid the order reduction that it is observed when ex-ponential quadrature rules are used to integrate linearinitial boundary value problems as

u′(t) = Au(t) + f(t), 0 ≤ t ≤ T,u(0) = u0,

∂u(t) = g(t),

When the s nodes of the quadrature rule are chosen ina suitable way, we can obtain order 2s in time, whilewith the classical approach the maximum order thatis reached is s.

A thorough error analysis is given for both the classi-cal approach of integrating the problem firstly in spaceand then in time and of doing it in the reverse order ina suitable manner in order to avoid the order reduc-tion. Time-dependent boundary conditions are con-sidered with both approaches and full discretizationformulas are given to implement the methods oncethe quadrature nodes have been chosen for the timeintegration and a particular (although very general)scheme is selected for the space discretization. Numer-ical experiments are shown which corroborate that,for example, with the suggested technique, order 2s isobtained when choosing the s nodes of the Gaussianquadrature rule.

References

[1] B. Cano, M. J. Moreta, Exponential quadraturerules without order reduction for integrating linear ini-tial boundary value problems, SIAM J. NUMER. ANAL.Vol. 56, No. 3 (2018), pp. 1187–1209.

This work was funded by Ministerio de Economıa y Com-

petitividad, Junta de Castilla y Leon and FEDER through

projects MTM 2015- 66837-P and VA024P17.

High-order full discretization for two-dimensional coupled seismic wave equations

A.M. Portillo (University of Valladolid)

Two-dimensional coupled seismic waves on a rectan-gular domain with initial–periodic boundary condi-tions are considered. Second order spatial derivatives,in the x direction, in the y direction and mixed deriva-tive, are approximated by second and fourth order fi-nite differences on a uniform grid. If the initial con-ditions have compact support contained in an openset included in the domain, the discrete energies con-sidered are approximations of the continuous energyof second and fourth order respectively. The ordinarysecond order in time system obtained is transformedinto a first order in time system in the displacementand velocity vectors. For the time integration of thissystem, second order and fourth order exponentialsplitting methods, which are geometric integrators,are proposed. These explicit splitting methods arenot unconditionally stable and the stability conditionfor time step and space step ratio is deduced. In thenumerical experiments are displayed energy errors forthe second and fourth order finite differences at timet = 0 as well as the good behavior in the long timeintegration of the splitting methods.

Avoiding order reduction when integrating non-linear problems with Strang splitting

N. Reguera & I. Alonso-Mallo & B. Cano (Univer-sidad de Burgos)

In this work we focus on the problem of avoidingorder reduction when using Strang splitting to inte-grate nonlinear initial boundary value problems withtime-dependent boundary conditions. In the liter-ature there are several references dealing with thisproblem [1], [3].

We are going to present the technique proposed in [1],along with the results obtained for the local and globalerrors. On the other hand, we will show a numericalcomparison in terms of computational efficiency [2]between the techniques [1] and [3]. We will see that itis important to consider an exponential method, thatavoids also order reduction, for the integration of thelinear nonhomogeneous and stiff part in [3] so that itis similar in efficiency to the technique proposed in [1].

References

[1] I. Alonso-Mallo, B. Cano and N. Reguera,Avoiding order reduction when integrating reaction-diffusion boundary value problems with exponentialsplitting methods, J. Computational and Applied Math-

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ematics 357 (2019) 228-250.

[2] I. Alonso-Mallo, B. Cano and N. Reguera,Comparison of efficiency among different techniquesto avoid order reduction with Strang splitting, submit-ted for publication.

[3] L. Einkemmer and A. Ostermann, Overcomingorder reduction in diffusion-reaction splitting. Part 2:Oblique boundary conditions, SIAM J. Sci. Comput.38 (2016) A3741-A3757.

Minisymposium M4

Preconditioning and iterative methodsfor differential equations

OrganisersJohn Pearson and Jennifer Pestana

On least-squares commutator (LSC) precondi-tioning for incompressible two-phase flow

Niall Bootland & Andy Wathen & Chris Kees(University of Strathclyde)

Modelling two-phase incompressible flow gives rise tovariable coefficient Stokes or Navier–Stokes equationsthat can be challenging to solve computationally. Inparticular, large jumps in the coefficients between thetwo phases can cause solver performance to degradesignificantly. The least-square commutator (LSC)method is a largely automatic way to build a blockpreconditioner for incompressible flow problems. How-ever, for two-phase flow we will see that weighting withthe variable coefficients must be included in order togain a more robust solver. Further, it is known thata weighting is also required on Dirichlet boundariesif one is to obtain mesh-independence. We show em-pirically that, unlike in the smooth coefficient case,for two-phase flow there is an interplay between suchweightings which can be detrimental to convergence.In this talk we introduce and explore weighted LSCpreconditioners and discuss practical aspects relatedto two-phase flow.

On Preconditioners and higher-order time dis-cretizations for PDE-constrained optimizationproblems

Santolo Leveque & John Pearson (University of Ed-inburgh)

PDE-constrained optimization problems form a fer-tile research area, for which preconditioned iterativemethods may be usefully applied. In fact, Krylovmethods for symmetric indefinite matrices are oftensuitable for the saddle point systems arising from thediscretization of such problems. However, the perfor-mance of these methods is related to the spectral prop-erties of the matrices involved, thus advanced tech-niques for finding a robust preconditioner are requiredto accelerate the convergence of the iterative scheme.

In this talk, we consider preconditioned iterative meth-ods for time-dependent PDE-constrained optimizationproblems (commencing with the optimal control ofthe heat equation), coupled with suitable discretiza-tions in the time variable. The state-of-the-art inthis field, whether finite element or finite differencediscretizations are used for the spatial variable, is toapply a backward Euler method in time, as this hasbeen found to work well with existing preconditioners.However, this discretization strategy results in meth-ods that are of second order in space and only firstorder in time, thus in order to obtain an accurate so-lution it is reasonable to choose ∆t = O(h2), where∆t is the time step and h is the mesh-size in space.This results in a matrix system of huge dimension, andhence a very high CPU time is required for its solution.

In this talk we present a new method for solving cer-tain time-dependent PDE-constrained optimizationproblems, that are of second order both in time andspace. This involves a Crank–Nicolson discretizationof the time derivative within the PDE, and a newpreconditioner for the resulting saddle point system(which now has a more complex structure). Further,we show through numerical experiments that this ap-proach can obtain a more accurate solution (in termsof discretization error) than the original (backwardEuler) method, in less CPU time.

Preconditioners for boundary control of ellip-tic PDE

Daniel Loghin (University of Birmingham)

We are interested in developing efficient precondition-ers for the solution of large sparse linear systems withcoefficient matrix having the following block form

K =M O AT

O αQ −BTA −B O

,

where M,A ∈ Rn×n, Q ∈ Rm×m, B ∈ Rn×m. Thematrix K arises in the discrete form of the weak for-mulation.

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Find (y∗, u∗, p∗) ∈ Y × U × Y such that ∀(z, v, r) ∈Y × U × Y m(y∗, z) + a(z, p∗) = m(yd, z),

αq(u∗, v)− b(v, p∗) = 0,a(y∗, r)− b(u∗, r) = `f (r),

where (y∗, u∗, p∗) represents the global minimizer ofa linear-quadratic programming problem expressed inweak form as follows minimize

1

2m(y − yd, y − yd) +

α

2q(u, u)

subject to a(y, v)− b(u, v) = `f (v) ∀v ∈ Y.

Solving linear systems with coefficient matrix K is of-ten challenging: while sparse, the matrix can be largein practice and thus the approach often considered isiterative or hybrid (iterative-direct) accelerated by agood preconditioner. However, all the blocks, exceptfor Q can be rank-deficient, depending on the under-lying problem. This poses additional difficulties whendesigning block preconditioners based on the matrixstructure.

In this talk, we consider a class of boundary precondi-tioners for solving the block systems arising from thefinite element discretization of linear-quadratic ellip-tic control problems with Neumann boundary control.We show that an optimal approach can be obtained bysolving an interior problem and a coupled boundaryproblem involving certain Steklov-Poincare operators.The latter boundary problem can be preconditionedby a block matrix with rational entries involving acertain representation of a discrete fractional Sobolevnorm associated with the Neumann part of the bound-ary.

Multigrid preconditioners for anisotropic space-fractional diffusion equations

Mariarosa Mazza† & Marco Donatelli† & RolfKrause‡ & Ken Trotti† (†University of Insubria, ‡

University of Italian Switzerland)

We focus on a two-dimensional time-space diffusionequation with fractional derivatives in space. Theuse of Crank-Nicolson in time and finite differencesin space leads to dense Toeplitz-like linear systems.Multigrid strategies that exploit such structure areparticularly effective when the fractional orders areboth close to 2. We seek to investigate how structure-based multigrid approaches can be efficiently extendedto the case where only one of the two fractional ordersis close to 2, i.e., when the fractional equation showsan intrinsic anisotropy. Precisely, we build a multigrid

(block-banded-banded-block) preconditioner that hasrelaxed Jacobi as smoother and whose grid transferoperator is obtained with a semicoarsening technique.A further improvement in the robustness of the pro-posed multigrid method is attained using the V-cyclewith semicoarsening as smoother inside an outer full-coarsening. Several numerical results confirm that theresulting multigrid preconditioner is computationallyeffective and outperforms current state of the art tech-niques.

Fast interior point solvers and preconditioningfor PDE-constrained optimization

John Pearson (University of Edinburgh) & JacekGondzio & Margherita Porcelli & Martin Stoll

In this talk we consider the effective numerical solu-tion of PDE-constrained optimization problems withadditional box constraints on the state and controlvariables. Upon discretization, these may give riseto problems of quadratic or nonlinear programmingform: a sensible solution strategy is to apply an in-terior point method, provided one can solve the largeand structured matrix systems that arise at each New-ton step. We therefore consider fast and robust pre-conditioned iterative methods for these systems, ex-amining two cases: (i) where L2 norms measure themisfit between state and desired state, as well as thecontrol; (ii) with an additional L1 norm term pro-moting sparsity in the control. Having motivated andderived our recommended preconditioners, and shownsome theoretical results on saddle point systems, wepresent numerical results demonstrating the potencyof our solvers.

Preconditioning for the 4D-Var method in dataassimilation

Alison Ramage (University of Strathclyde)

The 4D-Var method is frequently used for variationaldata assimilation problems in applications like numer-ical weather prediction and oceanographic modelling.One key challenge is that the state vectors used inrealistic applications contain a very large number ofunknowns so it can be impossible to assemble, storeor manipulate the matrices involved explicitly. Wepresent a multilevel limited-memory approximation tothe inverse Hessian and illustrate its effectiveness as a

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preconditioner within a Gauss-Newton iteration.

Preconditioning for partial integro-differentialequations

Ekkehard Sachs & Lukas Zimmer (Trier University,Germany)

As a motivating example we consider a system of par-tial integro-differential equations that occurs in amodel for cell adhesion. We also take a look at theresulting optimization problem and give some theo-retical results on necessary optimality conditions us-ing semigroup theory for the analysis.

In the case of a numerical solution for the system equa-tions we need to implement iterative solvers. However,the question of preconditioners poses some interestingchallenges. Integral operators are compact operatorsin the appropriate function spaces and therefore theirfinite dimensional counterparts need no precondition-ing. On the other hand some of the integral kernelsused lead to the property that the action of the inte-gral operator can also be viewed as an approximationof the Laplacian, hence preconditioning for discretiza-tions would be necessary.

We use results from the analysis of Toeplitz matricesto give some insight into our findings.

Optimal operator preconditioning for pseudo-differential boundary problems

Jakub Stocek & Heiko Gimperlein & Carolina UrzuaTorres (Heriot–Watt University)

This work considers the Dirichlet problem for an el-liptic pseudodifferential operator A in a bounded Lip-schitz domain Ω, where Ω is either a subset of Rn, or,more generally, in a Riemannian manifold Γ:

Au = f in Ω,

u = 0 in Γ \ Ω.

Such pseudodifferential boundary problems are of in-terest in applications. For instance, the integral frac-tional Laplacian A = (−∆)s and its variants A =div(c(x)∇2s−1u) in a domain Ω ⊂ Rn arise in thepricing of stock options, image processing, continuummechanics, and in the movement of biological organ-isms or swarm robotic systems. Boundary integralformulations of the first kind for an elliptic bound-ary problem lead to equations for the weakly singular

(A = V) or hypersingular (A = W) operators on acurve segment or open surface.

On the one hand, the bilinear form associated to Ais nonlocal, and its Galerkin discretization results indense matrices. On the other hand, the conditionnumber of these Galerkin matrices is of orderO(h−2|s|),where h is the size of the smallest cell of the mesh and2s is the order of A. Therefore, the solution of theresulting linear system via iterative solvers becomesprohibitively slow on fine meshes.

We propose an operator preconditioner for general el-liptic pseudodifferential equations in a domain in ei-ther Rn or a Riemannian manifold. For linear sys-tems of equations arising from low-order Galerkin dis-cretizations, we prove that the condition number is in-dependent of the mesh size and of the choice of basesfor test and trial functions.

The basic ingredient is a classical formula by Boggiofor the fractional Laplacian, extended analytically. Inthe special case of the weakly and hypersingular op-erators on a line segment or a screen, our approachgives a unified, independent proof for a series of re-cent results by Hiptmair, Jerez-Hanckes, Nedelec andUrzua-Torres.

Numerical examples illustrate the performance of theproposed preconditioner on quasi-uniform, graded andadaptively generated meshes. They show the increas-ing relevance of the regularity assumptions on themesh with the order of the operator.

Parallel preconditioning for time-dependentPDE problems

Andy Wathen (Oxford University)

Monolithic (or all-at-once) discretizations of evolution-ary problems most often give rise to nonsymmetriclinear(ised) systems of equations which can be of verylarge dimension for PDE problems. In this talk wewill describe preconditioners for such systems withguaranteed fast convergence via use of MINRES (notLSQR or CGNE) or GMRES. These results apply withstandard time-stepping schemes and relate to Toeplitzmatrix technology and preconditioning via circulantsusing the FFT. Simple parallel computational resultswill be shown for the heat equation and the wave equa-tion.

This is joint work with Elle McDonald (CSIRO, Aus-tralia), Jennifer Pestana (Strathclyde University, UK)and Anthony Goddard (Durham University, UK).

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Minisymposium M5

Finite element methods forNavier-Stokes equations: theory and

algorithmsOrganisers

Gabriel Barrenechea and Julia Novo

A posteriori error analysis for stabilized meth-ods: a nonlinear Boussinesq problem, anda Stokes model under singular sources

Alejandro Allendes1 & Cesar Naranjo2 & EnriqueOtarola3 & Abner J. Salgado4

1,2 (Universidad Tecnica Federico Santa Marıa, Chile)3 (Universidad Tecnica Federico Santa Marıa, Chile)4 (University of Tennessee)

In this talk, we will present the design and analysis ofa posteriori error estimators for low-order stabilizedfinite element approximations of a generalized Boussi-nesq problem, and also for a Stokes model with sin-gular sources in two and three dimensional Lipschitz,but not necessarily convex, polytopal domains. Thedesigned error estimators, which are mainly based ona Ritz projection of the errors, are proven to be re-liable and locally efficient. The standard smallnessassumption on the solution for the nonlinear problemis required. On the basis of these estimators we designa simple adaptive strategy that yields optimal rates ofconvergence for the numerical examples that we per-form.

An a posteriori error estimator for the MHMmethod applied to Stokes and Brinkman prob-lems

Rodolfo Araya & Ramiro Rebolledo & FredericValentin (Universidad de Concepcion, Chile)

In this work, we propose and analyze a residual typea posteriori error estimator for the Multiscale Hybrid-Mixed (MHM) method applied to Stokes and Brink-man equations. This error estimator relies on themulti-level structure of the MHM method, giving valu-able information about the error of approximation. Asa result, the error estimator accounts for a first-levelglobal estimator defined on the skeleton of the parti-tion and second-level contributions from element-wiseapproximations. The analysis establishes local effi-ciency and reliability of the complete estimator. Also,it yields a new face-adaptive strategy on the mesh’s

skeleton which avoids changing the topology of theglobal mesh. Specially designed to work on multiscaleproblems, the present estimator can leverage parallelcomputers since local error estimators are independentof each other.

Low-order divergence-free finite elementmethods in fluid mechanics

Alejandro Allendes?, Gabriel R. Barrenechea†,Cesar Naranjo? & Julia Novo‡ ? (Universidad TecnicaFederico Santa Marıa, Chile) † (University of Strath-clyde) ‡ (Universidad Autonoma de Madrid)

It is a well-known fact that the finite element ap-proximation of equations in fluid mechanics (e.g., theNavier-Stokes equations) is simpler, and more accu-rate, if the finite element method delivers an exactlydivergence-free velocity. In fact, in such a case theconvective term remains antisymmetric in the discretesetting, without the need to rewrite it, and the stabil-ity analysis can be greatly simplified. Now, when thefinite element spaces used are the conforming P1 forvelocity and P0 for pressure, then, in addition to addappropriate terms to stabilise the pressure, some ex-tra work needs to be done in order to compensate forthe lack of incompressibility of the discrete velocity.

In this talk I will present two recent applications of atechnique developed previously in [2] to overcome thelimitations described in the previous paragraph. Thefirst example presented is the application of this ideato the steady-state Boussinesq equation [1]. Then,the transient Navier-Stokes equations will be anal-ysed, where some error estimates independent of theReynolds number will be shown.

References

[1] A. Allendes, G.R. Barrenechea, C. Naranjo : Adivergence-free low-order stabilized finite element me-thod for the steady state Boussinesq problem. Com-puter Methods in Applied Mechanics and Engineering,340, 90–120, (2018).

[2] G.R. Barrenechea and F. Valentin : Consistent lo-cal projection stabilized finite element methods. SIAMJournal on Numerical Analysis, 48(5), 1801-1825,(2010).

Local pressure correction for the Stokes equa-tions

17

Malte Braack & Utku Kaya (University of Kiel)

Pressure correction methods constitute the mostwidely used solvers for the time-dependent Navier-Stokes equations. There are several different pres-sure correction methods, where each time step usuallyconsists in a predictor step for a non-divergence-freevelocity, followed by a Poisson problem for the pres-sure (or pressure update), and a final velocity correc-tion to obtain a divergence-free vector field. In somesituations, the equations for the velocities are solvedexplicitly, so that the numerical most expensive stepis the elliptic pressure problem. We here propose tosolve this Poisson problem by a domain decompositionmethod which does not need any communication be-tween the sub-regions. Hence, this system is perfectlyadapted for parallel computation. We show under cer-tain assumptions that this new scheme has the sameorder of convergence as the original pressure correctionscheme (with global projection). Numerical examplesfor the Stokes system show the effectivity of this newpressure correction method. The convergence orderO(k2) for resulting velocity fields can be observed inthe norm l2(0, T ;L2(Ω)).

Conforming mixed finite element approxima-tion of the Navier-Stokes/Darcy-Forchheimerproblem

Marco Discacciati & Sergio Caucao & Gabriel N.Gatica & Ricardo Oyarzua (Loughborough University)

We present a mixed finite element approximation ofthe coupled system formed by the Navier-Stokes equa-tions and either Darcy or Darcy-Forchheimer equa-tions to model the filtration of an incompressible fluidthrough a porous medium. We consider the stan-dard mixed formulation in the Navier-Stokes domainand the dual-mixed one in the porous-medium region,which yield the introduction of the trace of the pres-sure in the porous medium as a suitable Lagrange mul-tiplier. We propose a stable Galerkin approximationthat employs Bernardi-Raugel and Raviart-Thomaselements for the velocities, piecewise constants for thepressures, and continuous piecewise linear elements forthe Lagrange multiplier. Numerical results illustratethe performance of the method and confirm the theo-retical convergence rates. Finally, we discuss possiblesolution techniques based on domain decompositionmethods that exploit the multi-physics nature of theproblem to characterize optimal preconditioners.

Mixed finite elements applied to continuousdata assimilation for the Navier-Stokes equa-

tions

Bosco Garcıa-Archilla & Julia Novo & Edriss S.Titi (Universidad de Sevilla)

We summarize the results presented in [2] and [3],where we introduce and analyze a finite elementmethod applied to a continuous downscaling data as-similation algorithm for the numerical approximationof the two and three dimensional Navier-Stokes equa-tions corresponding to given measurements on a coarsespatial scale, recently introduced in [1]. For represent-ing the coarse mesh measurements we consider differ-ent types of interpolation operators including a La-grange interpolant. We obtain uniform-in-time esti-mates for the error between a finite element approxi-mation and the reference solution corresponding to thecoarse mesh measurements. We consider both the caseof a standard Galerkin method and a Galerkin methodwith grad-div stabilization. For the stabilized methodwe prove error bounds in which the constants do notdepend on inverse powers of the viscosity. Some nu-merical experiments illustrate the theoretical results.

References

[1] A. Azouani, E. Olson, and E. S. Titi. Contin-uous data assimilation using general interpolant ob-servables. J. Nonlinear Sci., 24(2):277–304, 2014.

[2] B. Garcıa-Archilla, J. Novo, and E. S. Titi. Uni-form in time error estimates for a finite element methodapplied to a downscaling data assimilation algorithmfor the Navier-Stokes equations. arXiv e-prints, pagearXiv:1807.08735, Mar. 2018.

[3] B. Garcıa-Archilla and J. Novo. Error analysisof fully discrete mixed finite element data assimila-tion schemes for the Navier-Stokes equations. arXive-prints, page arXiv:arXiv:1904.06113, Apr. 2019.

hp-Version discontinuous Galerkin methods onessentially arbitrarily-shaped elements

Emmanuil Georgoulis (University of Leicester)

We extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretiz-ing advection-diffusion-reaction problems to meshescomprising of extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis al-lows for curved element shapes, arising, without theuse of (iso-)parametric elemental maps. The feasibil-

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ity of the method relies on the definition of a suit-able choice of the discontinuity-penalization parame-ter, which turns out to be essentially independent onthe particular element shape. A priori error boundsfor the resulting method are given under very mildstructural assumptions restricting the magnitude ofthe local curvature of element boundaries. Numericalexperiments are also presented, indicating the prac-ticality of the proposed approach. This work gen-eralizes our earlier work detailed in the monograph[A.Cangiani, Z.Dong, E.H. Georgoulis and P.Houston,hp-version discontinuous Galerkin methods on polyg-onal and polyhedral meshes, SpringerBriefs in Math-ematics, Springer, Cham, 2017] from polygonal/poly-hedral meshes to essentially arbitrary element shapesinvolving curved faces without imposing any additionalmesh conditions.

The talk is based on joint work with A. Cangiani (Not-tingham) and Z. Dong (IACM-FORTH, Crete).

Adaptive augmented mixed methods for theOseen problem

M. Gonzalez & T.P. Barrios & J.M. Cascon (Uni-versidade da Coruna)

The problem of computing the flow of a viscous andincompressible fluid at small Reynolds numbers is de-scribed by the Oseen equations. In the recent paper[1], we introduced a new augmented variational for-mulation for this problem in the pseudostress-velocityvariables under homogeneous Dirichlet boundary con-ditions for the velocity, and we developed a simple aposteriori error analysis.

In this talk we will discuss a related method for theOseen problem when non-homogeneous mixed bound-ary conditions are imposed. We will describe a newaugmented dual-mixed variational formulation of theproblem. Then, we will analyze the correspondingGalerkin scheme, and provide the rate of convergencewhen each row of the pseudostress is approximated byRaviart-Thomas elements and the velocity is approx-imated by continuous piecewise polynomials. More-over, we will derive an a posteriori error indicatorwhich is reliable and locally efficient, and show theperformance of the corresponding adaptive algorithmthrough some numerical examples.

References

[1] T.P. Barrios, J.M. Cascon and M. Gonzalez, Aug-mented mixed finite element method for the Oseenproblem: A priori and a posteriori error analyses.

Comput. Methods Appl. Mech. Engrg. 313 (2017),216–238.

Higher order variational time discretisationsfor the incompressible flow problems

Gunar Matthies & Simon Becher (Technische Uni-versitat Dresden)

We present variational time discretisation schemes ofhigher order applied to incompressible flow problemsthat are described by Stokes, Oseen, or Navier–Stokesequations. As spatial discretisation, we will considerboth inf-sup stable and equal-order pairs of finite el-ement spaces for approximating velocity and pressure.

In cases of dominant convection, a spatial stabiliza-tion is needed. We will concentrate on local projec-tion stabilization methods which allow to stabilise thestreamline derivative, the divergence constraint and,if needed for equal-order pairs, the pressure gradientseparately.

Starting from the well-known discontinuous Galerkin(dG) and continuous Galerkin-Petrov (cGP) methodswe will present a two-parametric family of time dis-cretisation schemes which combine variational and col-location conditions. The first parameter correspondsto the ansatz order while the second parameter is re-lated to the global smoothness of the numerical solu-tion. Hence, higher order schemes with higher orderregularity in time can be obtained by adjusting thefamily parameters in the right way.

All members of the considered family show the samestability properties as either dG or cGP. Furthermore,the considered schemes provide a cheap post-processingto achieve better convergence orders in integral-basednorms. In addition, the post-processing could be usedfor adaptive time-step control.

Two-grid based postprocessing of mixed finiteelement approximations to the Navier-Stokesequations

Julia Novo & Francisco Durango (UniversidadAutonoma de Madrid)

In this talk we revise some postprocessing techniquesthat have been applied to mixed finite element ap-proximations to the Navier-Stokes equations and in-troduce a generalized postprocessed method that de-pends on several parameters. The idea of postpro-cessing is the following: once a standard Galerkin

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mixed finite element approximation is computed overa coarse mesh of size H at a fixed time T , the ap-proximation is postprocessed over a finer mesh of sizeh < H. In the literature one can find different waysto postprocess the Galerkin approximation dependingon the problem that is solved over the fine mesh atthe final time T . Basically, one can solve a Stokesproblem, an Oseen type problem or a Newton typeproblem. In the present talk we present a methodbased on several parameters that generalize the abovepostprocessing techniques. Depending on the valueschosen for the different parameters one can recoverone of the old (known) postprocessing procedures butalso produce a new (different) method. We get errorbounds for the generalized method valid for any of thevalues of the different parameters. In all the cases, thepostprocessed method has a rate of convergence oneunit bigger than the rate of convergence of the plainGalerkin method in terms of the coarse mesh H andoptimal in terms of the fine mesh h. The computa-tional added cost of postprocessing is however negligi-ble compared to the cost of solving the Navier-Stokesequations in the time interval [0, T ] by means of theGakerkin method. For the error analysis we do notassume nonlocal compatibility conditions for the truesolutions of the Navier-Stokes equations.

Numerical comparisons of finite element stabi-lized methods for a 2D vortex dynamics simu-lation at high Reynolds number

Samuele Rubino & Naveed Ahmed (Universityof Seville)

In this work, we consider up-to-date and classical Fi-nite Element (FE) stabilized methods for time-dependent incompressible flows. All studied methodsbelong to the Variational MultiScale (VMS) frame-work. So, different realizations of stabilized FE-VMSmethods are compared using a high Reynolds num-ber vortex dynamics simulation. In particular, a fullyResidual-Based (RB)-VMS method is compared withthe classical Streamline-Upwind Petrov–Galerkin(SUPG) method together with grad-div stabilization,a standard one-level Local Projection Stabilization(LPS) method, and a recently proposed LPS methodby interpolation. These procedures do not make useof the statistical theory of equilibrium turbulence, andno ad-hoc eddy viscosity modeling is required for allmethods. Applications to the simulation of a highReynolds number flow with vortical structures on rel-atively coarse grids are showcased, by focusing on atwo-dimensional plane mixing-layer flow. Both Inf-Sup Stable (ISS) and Equal Order (EO) H1-confor-ming FE pairs are explored using a second-order semi-

implicit Backward Differentiation Formula (BDF2) intime. Based on the numerical studies conducted, it isconcluded that the SUPG method using EO-FE pairsperforms best among all methods. Furthermore, thereseems to be no reason to extend the SUPG method bythe higher order terms of the RB-VMS method.

IFISS: A computational laboratory for investi-gating incompressible flow problems

David Silvester (The University of Manchester)

The IFISS package is a computational laboratory forthe interactive numerical study of incompressible flowproblems. It includes algorithms for discretization bymixed finite element methods and a posteriori errorestimation of the computed solutions, together withstate-of-the-art preconditioned iterative solvers for theresulting discrete linear equation systems. The utilityof the software is illustrated using two case studies.First, by computing a precise on-the-fly estimate ofthe discrete inf–sup constant when solving a Stokesflow problem. Second, by assessing the issues thatarise when conventional mixed methods are used toapproximate Beltrami flows or potential flows using astandard Navier–Stokes solver. The goal is to demon-strate that IFISS can be a valuable tool in both teach-ing and research.

References

[1] H. Elman, A. Ramage and D. Silvester. IFISS:A computational laboratory for investigating incom-pressible flow problems, SIAM Review, 56, 261273,2014. https://doi.org/10.1137/120891393

[2] V. John, A. Linke, C. Merdon, M. Neilan and L.Rebholz. On the divergence constraint in mixed finiteelement methods for incompressible flows, SIAM Re-view, 593, 492–544, 2017.https://doi.org/10.1137/15M1047696

Minisymposium M6

Computational methods formodel-driven optimization and control

under uncertaintyOrganisers

Dante Kalise and Mattia Zanella

Kinetic models for optimal control of wealthinequalities

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Bertram During (University of Sussex)

We discuss optimal control strategies for kinetic mod-els for wealth distribution in a simple market econ-omy, which are designed to reduce the variance of thewealth density among the population. Our analysisis based on a finite time horizon approximation, ormodel predictive control, of the corresponding con-trol problem for the microscopic agents’ dynamic andresults in an alternative theoretical approach to thetaxation and redistribution policy at a global level.

Hybrid differential games and their applicationto a match race problem

Adriano Festa (L’Aquila University) & Simone Ca-cace & Roberto Ferretti (Roma tre University)

We discuss the general setting of a two-player hybriddifferential game where the dynamics of the playersfollow some stochastic differential equations with thepossibility to switch, paying a cost, from one to an-other. We analyse the abstract problem and apply itto the modelling of a “match race” competition be-tween sailing boats. In a simplified version of thiscompetition, the goal of both players is to proceedon the windward direction while trying to slow downthe other player. We model the problem using thestructure of a stochastic, zero-sum differential game inpresence of hybrid dynamics, show that the game hasa value, provide a convergent approximation schemeand validate the approach on some typical racing sce-narios.

Controlling fluctuations in gas networks usingstochastic Galerkin formulations

Stephan Gerster & Michael Herty (RWTH AachenUniversity)

Since gas system operators assume nearly constantconsumption, there is a need to assess the risk ofstochastic fluctuations. Typically, simulating stochas-tic processes by a Monte-Carlo method is computa-tionally not feasible.The idea to represent stochastic processes by orthog-onal polynomials has been employed in uncertaintyquantification and inverse problems. This approach isknown as stochastic Galerkin formulation with a gen-eralized polynomial chaos (gPC) expansion. The gPCexpansions of the stochastic input are substituted intothe governing equations. Then, they are projected bya Galerkin method to obtain deterministic evolution

equations for the gPC coefficients.

So far, results for general hyperbolic systems are notavailable. A problem is posed by the fact that thedeterministic Jacobian of the projected system differsfrom the random Jacobian of the original system andhence hyperbolicity is not guaranteed. Applicationsto hyperbolic conservation laws are in general limitedto linear and scalar hyperbolic equations. We presenta stochastic Galerkin formulation for isothermal Eulerequations. Furthermore, we discuss numerically con-trol policies to damp fluctuations in the network.

References

S. Gerster, M. Herty, Entropies and Symmetrization ofHyperbolic Stochastic Galerkin Formulations, Preprint-No. 488 (2019), RWTH Aachen University.

S. Gerster, M. Herty, A. Sikstel, Hyperbolic Stochas-tic Galerkin Formulation for the p-System, Preprint-No. 477 (2018), RWTH Aachen University.

S. Gerster, M. Herty, Discretized Feedback Controlfor Systems of Linearized Hyperbolic Balance Laws,MCRF, Vol. 9, No 3, 2019.

Parameter estimation for macroscopic pedes-trian dynamics models

Susana N. Gomes (Warwick) & Andrew M. Stuart(Caltech) & Marie-Therese Wolfram (Warwick)

In this talk, we present a framework for estimatingparameters in macroscopic models for crowd dynam-ics using data from individual trajectories. We con-sider a model for the unidirectional flow of pedestriansin a corridor which consists of a coupling between adensity-dependent stochastic differential equation anda nonlinear partial differential equation for the den-sity. In the stochastic differential equation for the tra-jectories, the velocity of a pedestrian decreases withthe density according to the fundamental diagram.Although there is a general agreement on the basicshape of this dependence, its parametrization dependsstrongly on the measurement and averaging techniquesused as well as the experimental setup considered. Wewill discuss identifiability of the parameters appearingin the fundamental diagram, introduce optimisationand Bayesian methods to perform the identification,and analyse the performance of the proposed method-ology in various realistic situations. Finally, we dis-cuss possible generalisations, including the effect ofthe form of the fundamental diagram and the use of

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experimental data.

Polynomial approximation of Isaacs’ equationand robust control of parabolic PDEs

Dante Kalise & Sudeep Kundu & Karl Kunisch (Uni-versity of Nottingham)

We propose a numerical scheme for the approxima-tion of high-dimensional, nonlinear Isaacs PDEs aris-ing in robust optimal feedback control of nonlineardynamics. The numerical method consists of a globalpolynomial ansatz together with separability assump-tions for the calculation of high-dimensional integrals.The resulting Galerkin residual equation is solved bymeans of an alternating Newton-type/policy iterationmethod for differential games. We present numericalexperiments illustrating the applicability of our ap-proach in robust optimal control of nonlinear parabolicPDEs.

Kinetic models of traffic flow control via driver-assist vehicles

Andrea Tosin & Mattia Zanella (Politecnico diTorino)

In this talk, we present a hierarchical description ofcontrol problems for vehicular traffic, which aim tomitigate speed-dependent risk factors and to dampenstructural uncertainties responsible for scattered, thushardly predictable, aggregate trends. In particular, weimplement mathematically the idea that a few auto-mated cars can be controlled so as to align the speedsin the traffic stream either to each other or to somerecommended optimal speed.

First, we discuss the modelling of stochastic micro-scopic binary interactions among the vehicles, includ-ing a probabilistic description of the penetration rateof automated vehicles. When the interactions involveone of such vehicles, they are further subject to a bi-nary control problem, which aims to reduce the speedgap either with the leading vehicle or with a prescribedcongestion-dependent speed. Then, we upscale the in-teraction rules to the global flow by means of a kineticBoltzmann-type equation. Finally, we use the kineticequation to investigate the impact of the microscopiccontrol on the macroscopic flow. In particular, bymeans of suitable hydrodynamic limits, we show howto obtain consistent macroscopic traffic models, whichincorporate the action of the control from the micro-scopic scale.

Ours is a markedly multiscale approach, particularly

consistent with the idea that multi-agent systems, likevehicular traffic, need to be controlled by means ofbottom-up strategies rather than assuming that it ispossible to control directly their aggregate behaviour.

This talk is mainly based on the contents of the fol-lowing works:

References

[1.] A. Tosin, M. Zanella. Kinetic-controlled hydro-dynamics for traffic models with driver-assist vehi-cles, Multiscale Model. Simul., 2019. doi:10.1137/

18M1203766.

[2.] A. Tosin, M. Zanella. Control strategies for roadrisk mitigation in kinetic traffic modelling,IFAC-PapersOnLine, 51(9):67-72, 2018.

[3.] A. Tosin, M. Zanella. Uncertainty damping inkinetic traffic models by driver-assist controls, 2019(preprint). doi:10.13140/RG.2.2.35871.41124.

Uncertainty quantification for moving bound-ary problems with applications in compositesmanufacturing

M.V. Tretyakov (University of Nottingham)

The considered UQ problems are motivated by mod-elling of one of the main manufacturing processes forproducing advanced composites – resin transfer mould-ing (RTM). We consider models of the stochastic RTMprocess, which are formulated as random movingboundary problems. We study their properties, ana-lytically in the one-dimensional case and numericallyin the two-dimensional case. We will discuss Bayesianinversion algorithm for random moving boundary prob-lems and a stochastic control setting. The talk isbased on joint works with Marco Iglesias, MikhailMatveev, Andreas Endruweit, Andy Long, Minho Park.

Spectral methods for optimal control problemsfor equations of Fokker–Planck type

Urbain Vaes & Dante Kalise & Grigorios Pavliotis(Imperial College London)

In this talk we present a novel methodology for con-trolling the Fokker–Planck equation in periodic andunbounded domains. Our method is based on thederivation of first-order optimality conditions, leadinga forward-backward coupled system which is approxi-

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mated by an expansion in either Fourier series or ap-propriately rescaled Hermite functions, depending onthe domain. We propose a reduced gradient methodfor the synthesis of the optimal control signal, and westudy the properties of the proposed numerical schemeboth theoretically and by means of numerical experi-ments.

Structure preserving gPC methods for kineticequations with uncertainties

Mattia Zanella (Politecnico di Torino)

We introduce and discuss numerical schemes for theapproximation of kinetic equations that incorporateuncertain quantities. In contrast to a direct appli-cation of stochastic Galerkin generalized polynomialchaos (SG-gPC) methods which are widely consideredfor uncertainty quantification of differential equations,this class of schemes make use of a particle approachin the phase space coupled with a stochastic Galerkinexpansion in the random space. The proposed meth-ods naturally preserve the positivity of the statisticalmoments of the solution and are capable of achiev-ing spectral accuracy in the random space. Severaltests on kinetic models for collective phenomena vali-date the proposed methods both in the homogeneousand inhomogeneous setting, shading light on the in-fluence of uncertainties in phase transition phenomenadriven by noise such as their smoothing and confidencebands.

References

[1.] G. Dimarco, L. Pareschi, M. Zanella. Uncertaintyquantification for kinetic models in socio-economic andlife sciences. In Uncertainty Quantification for Hyper-bolic and Kinetic Equations, Editors S. Jin, and L.Pareschi, SEMA SIMAI Springer Series, vol. 14, pp151-191, 2017.

[2.] J. A. Carrillo, M. Zanella. Monte Carlo gPCmethods for diffusive kinetic flocking models with un-certainties. Preprint arXiv:1902.04518, 2019.

[3.] J. A. Carrillo, L. Pareschi, M. Zanella. Particlebased gPC methods for mean-field models of swarmingwith uncertainty. Communications in ComputationalPhysics, 25(2): 508-531, 2019.

Minisymposium M7

Probabilistic numerical computationand high-dimensional data analysis

OrganiserIvan Tyukin

A measure based construction of the finite el-ement method

Mark A. Girolami (University of Cambridge)

The finite element method provides the means to nu-merically approximate the solution to linear and non-linear systems of partial differential equations. It iswithout doubt one of the triumphs of applied mathe-matics in enabling the modelling and study of complexphysical and natural systems and processes. Howeverthe uncertainty inherent in such models is restrictedto that introduced by poorly defined values of phys-ical properties such that empirical measurements inthe form of data are used in the inverse-problem set-ting to infer plausible ranges of such values. How-ever the uncertainty induced by model misspecifica-tion is ignored. By defining probability measures overthe differential operators and the PDEs themselvesalong with a statistical representation of the model-reality mismatch a set of conditional probability mea-sures emerge which provide the means to update thefinite element solution in the light of empirical evi-dence and thus more informatively characterise un-certainty in the overall system. This talk will presentthe construction of such a statistical measure basedfinite element methodology and explore its propertiestheoretically and with applications.

Approximation with Gaussians of fixed andvarying scales

Jeremy Levesley & Stephen Walsh (University ofLeicester)

Approximation with Gaussians has advantages anddisadvantages. One of the main disadvantages is thatthe analytic nature of the function means that ap-proximation problems with Gaussians are often ill-conditioned. On the other hand, since Gaussians aresmooth, they carry the possibility of optimal conver-gence orders when the smoothness of the underlyingfunction is unknown (as it usually is in practice). Also,in high dimensions, most functions start to look likeGaussians.

23

In standard radial basis function applications Gaus-sians of a fixed width are used, and convergence isobtained by reducing the spacing between data pointswhile leaving the shape of the Gaussian fixed and con-stant. We would like to explore a different scenario,which is to use a multiscale method, where the shapedepends on the scale. We will explore this for a fixedshape at each scale and also a varying shape at eachscale. The shape is changed related to end-point sin-gularities in the function in order to mimic h−p adap-tivity in finite elements.

We will review results in the multilevel fixed shapeparadigm. Collaborators in this work are ManolisGeorgoulis, Peter Dong, Fuat Usta, Fazli Subhan (allat Leicester), and Simon Hubbert at Birkbeck. Wewill also show that the interpolation matrices in themultiscale, multishape scenario in one dimension areinvertible, a new result.

Kernel stochastic separation theorems andmathematics for making data-driven artificialintelligence systems better

I. Tyukin & A.N. Gorban (University of Leicester)

Large and growing streams of data are ubiquitous inmodern society and form the backbone of modernhealthcare, public safety, services, and science. Sus-tained functioning and progress in these essential areasdepend on the ability to extract and process informa-tion from large and growing data. Since processingoverwhelmingly large volumes of data can no longerbe accomplished by humans alone, we must rely onArtificial Intelligence (AI) systems built on state-of-the-art machine learning and data analytics technolo-gies.

With the explosive pace of progress in computing, cur-rent AI systems are now capable of spotting minutepatterns in large data sets and outperform humans inhighly complicated cognitive tasks like chess, Go, andmedical diagnosis. However, the super-human capa-bility of modern AIs to learn from massive volumes ofdata make their conclusions vulnerable to data incon-sistencies, poor data quality, and fundamental uncer-tainty inherent to any data. This uncertainty togetherwith engineering constraints on AI’s implementationlead to inevitable errors in data-driven AIs: incorrectface recognition matchings, incorrect cancer diagnosis,and Tesla and Uber crashes are just few examples.

In this talk we provide a mathematical formulation ofthe challenge of handling of AI errors and present a

mechanism to address the challenge: stochastic sep-aration theorems. We then further extend stochasticseparation theorems to infinite-dimensional settings.A general separability result for two random sets andrelaxed independence assumption is also established.We show that despite feature maps corresponding toa given kernel function may be infinite-dimensional,kernel separability characterizations can be expressedin terms of finite-dimensional volume integrals. Theseintegrals allow to determine and quantify separabilityproperties of an arbitrary kernel function. The theoryis illustrated with numerical examples.

Minisymposium M8

Matrix methods for NetworksOrganisers

Francesca Arrigo and Francesco Tudisco

Eigenvector-based centrality measures in mul-tilayer networks

Francesca Arrigo & Francesco Tudisco (Universityof Strathclyde)

Complex networks are used to model several differenttypes of interactions among various entities. Beingso versatile, they provide a useful tool to understandtoday’s world and answer questions about the under-lying phenomenon or structure. Often in practice thequestion that needs answered is: which is the mostimportant entity in the network? Several measures ofimportance, or centrality, have been introduced overthe years. Among these, one of the most popular onesbuilds on the idea that the importance of a node stemsfrom that of its neighbours. Mathematically this re-sults in considering the Perron eigenvector of somesuitable graph matrices. In this talk, we describe howto generalize this simple but effective idea to the morecomplicated setting of multi-layer networks, where en-tities interact on different levels and the higher-orderstructure is captured by tensors rather than matrices.

Spectral methods for certain inverse problemson graphs

Mihai Cucuringu (University of Oxford & The AlanTuring Institute)

We study problems that share an important commonfeature: they can all be solved by exploiting the spec-trum of their corresponding graph Laplacian matrix.

24

We first consider the classic problem of establishinga statistical ranking of a set of items given a set ofinconsistent and incomplete pairwise comparisons be-tween such items. Instantiations of this problem oc-cur in numerous applications in data analysis (e.g.,ranking teams in sports data), computer vision, andmachine learning. We formulate the above problem ofranking with incomplete noisy information as an in-stance of the group synchronization problem over thegroup SO(2) of planar rotations, whose least squaressolution can be approximated by either a spectral or asemidefinite programming relaxation. We also presenta simple spectral approach to the well-studied con-strained clustering problem. It captures constrainedclustering as a generalized eigenvalue problem withgraph Laplacians. The proposed algorithm works innearly-linear time, and consistently outperforms ex-isting spectral approaches both in speed and quality.Furthermore, we propose an algorithm for clusteringsigned networks, where the edge weights between thenodes of the graph may take either positive or neg-ative edges, and analyze it under a signed stochasticblock model.

Metaplex networks: influence of the exo-endostructure of complex systems on diffusion

Gissell Estrada-Rodriguez & Ernesto Estrada &Heiko Gimperlein (Maxwell Institute for Mathemati-cal Sciences and Heriot-Watt University)

In this talk I will introduce the concept of metaplexesand the dynamical systems on them. A metaplexcombines the internal structure of the entities of acomplex system with the discrete interconnectivity ofthese entities in a global topology. We focus here onthe study of diffusive processes on metaplexes, bothmodel and real-world examples. We provide theoreti-cal and computational evidence pointing out the roleof the endo- and exo-structure of the metaplexes intheir global dynamics, including the role played bythe size of the nodes, the location of sinks and sources,and the strength and range of the coupling betweennodes. We show that the internal structure of brainregions (corresponding to the nodes of the network)in the macaque visual cortex metaplex dominates al-most completely the global dynamics. On the otherhand, in the linear metaplex chain and the landscapemetaplex the diffusion dynamics display a combina-tion of the endo- and exo-dynamics, which we explainanalytically and in numerical results. Metaplexes areexpected to facilitate the understanding of complexsystems in an integrative way, which combine dynam-

ical processes inside the nodes and between them.

The random walk centrality, revised

Dario Fasino (University of Udine)

Let H be the matrix containing the hitting times ofthe random walk on a connected undirected graph onn nodes. It is well known that the matrix H+HT is amultiple of the resistance matrix. On the other hand,the matrix H − HT has rank 2. In fact, there existpositive numbers c1, . . . , cn such that

Hij −Hji =1

cj− 1

ci.

In a renowned paper by J. Noh and H. Rieger, ci isdefined in terms of an infinite series and is called therandom walk centrality of node i, since it quantifieshow central node i is regarding its potential to receiveinformation randomly diffusing over the network. Inpractice, the value of ci is quite good at recognizingnode i as being a “core” node or a ‘peripheral” nodein the network. In this talk I propose a novel inter-pretation of ci and new formulas to compute it fromthe pseudo-inverse of the Laplacian matrix of the net-work. Extensive numerical experiments show that cihas a striking correlation with the so-called randomwalk closeness centrality, C(i) = n/

∑nj=1(Hij +Hji).

The reason for that is, at the time of writing this ab-stract, not completely clear.

Bipartivity measures and methods

Philip Knight & Azhar Aleidan (University of Strath-clyde)

Bipartivity is a well-defined concept in graph theory,namely that a graph’s nodes can be divided into twosets in which there are no intra-set links. Often onefinds that real-world networks have a close-to-bipartitestructure, and there are several applications where itis helpful to both quantify and identify this structure.We compare measures based on links, walks and spec-tral information and show that these can give signif-icantly different quantifications. Each measure leadsto an algorithm for finding a nearby bipartite graph toany network. Finding the closest such graph is an NPhard problem, and we give some examples to show howclose some simple matrix-based algorithms can cometo finding an optimal solution in reasonable time.

Networks core–periphery detection with non-linear Perron eigenvectors

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Francesco Tudisco (University of Strathclyde) &Desmond J. Higham (University of Edinburgh)

Core–periphery detection is a highly relevant task inexploratory network analysis. Given a network ofnodes and edges, one is interested in revealing thepresence and measuring the consistency of a core–periphery structure using only the network topology.This mesoscale network structure consists of two sets:the core, a set of nodes that is highly connected acrossthe whole network, and the periphery, a set of nodesthat is well connected only to the nodes that are inthe core. Networks with such a core–periphery struc-ture have been observed in several applications, in-cluding economic, social, communication and citationnetworks.

In this work we propose a new core–periphery detec-tion model based on the optimization of the core–periphery quality function

f(x) =

n∑ij=1

Aij maxxi, xj (1)

on the unit sphere ‖x‖ = 1. While the quality measuref is highly nonconvex in general and thus hardly treat-able, we show that the global maximum of f coincideswith the nonlinear Perron eigenvector of a suitablydefined parameter dependent matrix M(x), i.e. thepositive solution to the nonlinear eigenvector problemM(x)x = λx.

Using recent advances in nonlinear Perron–Frobeniustheory, we show that (1) has a unique solution andwe propose a nonlinear power-method type schemethat (a) allows us to solve (1) with global convergenceguarantees and (b) effectively scales to very large andsparse networks. Finally, we present several numericalexperiments showing that the new method largely out-performs state-of-the-art techniques for core-peripherydetection.

Minisymposium M9

Recent Advances in ContinuousOptimisation

OrganisersNick Gould and Lindon Roberts

An analysis of biased stochastic gradient de-scent methods

Derek Driggs & Jingwei Liang (University of Cam-bridge) & Carola Schonlieb (University of Cambridge)

Although stochastic gradient methods have receivedmuch attention in the machine learning and optimisa-tion communities, almost all existing research consid-ers unbiased stochastic gradient estimators. We dis-cuss a framework for analysing several families of bi-ased stochastic gradient methods that include SAGA,SAG, SVRG, and SARAH as special cases. We usethis framework to develop a new biased stochastic gra-dient algorithm, Stochastic Average Recursive Gra-diEnt (SARGE), that combines SARAH’s recursivegradient estimate with the SAGA gradient estima-tor. We show that biased stochastic gradient estima-tors including SARGE often outperform their unbi-ased counterparts on a variety of tasks from machinelearning and imaging processing.

A block-coordinate Gauss-Newton method fornonlinear least squares

Jaroslav Fowkes & Coralia Cartis (University of Ox-ford)

We propose a block-coordinate Gauss-Newton methodfor nonlinear (nonconvex) least squares problems thatcomputes Jacobians only on a subset of the optimiza-tion variables at a time. We investigate globalisingthis approach using either a regularization term or atrust-region model and show global complexity resultsas well as extensive computational results on CUTEsttest problems. Furthermore, as our approach exhibitsvery slow rates of convergence on certain nonconvexproblems, we design adaptive block size variants of ourmethods that can overcome these difficulties.

Solution of quadratic programming problemsfor fast approximate solution of linear program-ming problems

Ivet Galabova & Julian Hall (University of Edin-burgh)

Linear programming problems (LP) are widely solvedin practice and reducing the solution time is essentialfor many applications. This talk will explore solvinga sequence of unconstrained quadratic programming(QP) problems to derive an approximate solution andbound on the optimal objective value of an LP. Tech-niques for solving these QP problems fast will be dis-

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cussed.

An exact line search algorithm for piecewisesmooth functions

Jonathan Grant-Peters (University of Oxford)

The problem of finding the local optimum of a 1Dsmooth function is one that has been solved for manyyears. Brent’s method solves this problem with super-linear convergence while the Golden Section algorithmsolves this problem robustly in linear time. However,when considering a piecewise smooth function, it isnot difficult to construct an example for which Brent’smethod fails to converge faster than the Golden Sec-tion. We present an algorithm designed to convergequickly for a class of piecewise smooth functions, whilebeing comparable to Brent’s method for smooth func-tions.

HiGHS: a high-performance linear optimizer

Julian Hall (University of Edinburgh)

This talk will present HiGHS, a growing open-sourcerepository of high-performance software for linear op-timization based on award-winning computational tech-niques for the dual simplex method. The talk will givean insight into the work which has led to the creationof HiGHS and then set out the features which allowsit to be used in a wide range of applications. Plansto extend the class of problems which can be solvedusing HiGHS will be set out.

On barrier and modified barrier multigrid meth-ods for 3d topology optimization

Michal Kocvara & Alexander Brune (University ofBirmingham)

Topology optimization of mechanical structures is achallenging problem due to its very large dimensioneasily reaching millions of variables. We propose tosolve it by the Penalty-Barrier Multiplier (PBM)method, introduced by R. Polyak and later studiedby Ben-Tal and Zibulevsky and others. The mostcomputationally expensive part of the algorithm isthe solution of linear systems arising from the New-ton method used to minimize a generalized augmentedLagrangian. We use a special structure of the Hes-sian of this Lagrangian to reduce the size of the lin-

ear system and to convert it to a form suitable fora standard multigrid method. This converted systemis solved approximately by a multigrid preconditionedMINRES method. The proposed PBM algorithm iscompared with the optimality criteria (OC) methodand an interior point (IP) method, both using a simi-lar iterative solver setup. We apply all three methodsto different loading scenarios. In our experiments, thePBM method clearly outperforms the other methodsin terms of computation time required to achieve acertain degree of accuracy.

A deterministic approach to avoid saddlepoints

Lisa Maria Kreusser (University of Cambridge) &Stanley J. Osher (UCLA) & Bao Wang (UCLA)

Loss functions with a large number of saddle points areone of the main obstacles to training many modernmachine learning models. For certain initial values,gradient descent (GD) converges to a saddle point.We call the region formed by these initial values the‘attraction region’. For quadratic functions, GD con-verges to a saddle point if the initial data is in a sub-space of up to n−1 dimensions. In this talk, we provethat a small modification of the recently proposedLaplacian smoothing gradient descent (LSGD) [Os-her, et al., arXiv:1806.06317] contributes to avoidingsaddle points without sacrificing the convergence rateof GD. In particular, we show for a class of quadraticfunctions that the dimension of the LSGD’s attractionregion is b(n−1)/2c which is significantly smaller thanGD’s (n-1)-dimensional attraction region.

Dimensionality reduction techniques for globaloptimization

Adilet Otemissov & Coralia Cartis (The Alan Tur-ing Institute and The University of Oxford)

We show that the scalability issues of Global Opti-mization algorithms can be alleviated for functionswith low effective dimensionality, which are constantalong certain linear subspaces. Such functions can befound in various applications, for example, in neuralnetworks and complex engineering simulations. Wepropose the use of random subspace embeddings withinany global minimization algorithm, extending Wanget al.’s (2013) approach. Using tools from randommatrix theory and conic integral geometry, we inves-tigate the efficacy and convergence of the method inits static and adaptive formulations, respectively. Us-

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ing popular global solvers, we illustrate our algorith-mic proposals/theoretical findings numerically. Jointwork with Coralia Cartis.

A geometric integration approach to nonsmooth,nonconvex optimisation

Erlend S. Riis & Matthias M. Ehrhardt & ReinoutQuispel & Carola-B. Schonlieb (University of Cam-bridge)

Discrete gradient methods are popular numerical meth-ods from geometric integration for solving systems ofODEs. They are well-known for preserving struc-tures of the continuous system such as energy dis-sipation/conservation. The preservation of dissipa-tion makes discrete gradient methods interesting foroptimisation problems. In this talk, we consider aderivative-free discrete gradient applied to dissipativeODEs such as gradient flow, thereby obtaining optimi-sation schemes that simultaneously are implementablein a black-box setting and retain favourable proper-ties of gradient flow. We give a theoretical analysis ofthese schemes in the nonsmooth, nonconvex setting,and conclude with numerical results for the bilevel op-timisation of regularisation parameters in image pro-cessing.

Improving the scalability of derivative-free op-timization for nonlinear least-squares problems

Lindon Roberts & Coralia Cartis & Jan Fiala &Benjamin Marteau (University of Oxford)

In existing techniques for model-based derivative-freeoptimization, the computational cost of constructinglocal models and Lagrange polynomials can be high.As a result, these algorithms are not as suitable forlarge-scale problems as derivative-based methods. Inthis talk, I will introduce a derivative-free methodbased on exploration of random subspaces, suitable fornonlinear least-squares problems. This method has asubstantially reduced computational cost (in terms oflinear algebra), while still making progress using fewobjective evaluations.

Sketching for sparse linear least squares

Zhen Shao1 & Coralia Cartis1 & Jan Fiala2

(1University of Oxford, 2NAG Ltd)

We discuss sketching techniques for sparse Linear LeastSquares (LLS) problems, that perform a randomiseddimensionality reduction for more efficient and scal-able solutions. We give theoretical bounds for the ac-curacy of the sketched solution/residual when hashingmatrices are used for sketching, quantifying carefullythe trade-off between the coherence of the original,un-sketched matrix and the sparsity of the hashingmatrix. We then use these bounds to quantify the suc-cess of our algorithm that employs a sparse factorisa-tion of the sketched matrix as a preconditioner for theoriginal LLS, before applying LSQR. We extensivelycompare our algorithm to state-of-the-art direct anditerative solvers for large-scale and sparse LLS, withencouraging results.

Model based optimisation applied to black-boxattacks in deep learning

Giuseppe Ughi & Prof. J. Tanner (University ofOxford)

Neural Network algorithms have achieved unprece-dented performance in image recognition over the pastdecade. However, their application in real world use-cases, such as self driving cars, raises the question ofwhether it is safe to rely on them.

We generally associate the robustness of these algo-rithms with how easy it is to generate an adversar-ial example: a tiny perturbation of an image whichleads it to be misclassified by the Neural Net (whichclassifies the original image correctly). Neural Netsare strongly susceptible to such adversarial examples,but when the architecture of the target neural net isunknown to the attacker it becomes more difficult togenerate these examples efficiently.

In this Black-Box setting, we frame the generationof an adversarial example as an optimisation prob-lem solvable via derivative free optimisation meth-ods. Thus, we introduce an algorithm based on theBOBYQA model-based method and compare this tothe current state of the art algorithm.

Minisymposium M10

Spectral MethodsOrganisers

Sheehan Olver, Alex Townsend andMarcus Webb

Spectral problems and new resolvent based

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methods

Matthew Colbrook (University of Cambridge)

I will present new results on spectral computationsfor linear operators on separable Hilbert spaces. Thisproblem has a rich history, leading to the Solvabil-ity Complexity Index hierarchy - a measure of thedifficulty of a computational problem. I will discussclassifications of spectral problems in this hierarchy.Some long-standing problems are solved (or shown tobe impossible) revealing potential surprises. For ex-ample, the problem of computing spectra of compactoperators, for which the method has been known fordecades, is strictly harder than the problem of com-puting spectra of Schrodinger operators with boundedpotentials, which has been open for more than half acentury. The latter type of problem can be solvedwith a precise form of error control and without spec-tral pollution by using computational estimates of re-solvent norms. Another resolvent based method I willintroduce is the first set of general algorithms for com-puting spectral measures of self-adjoint and unitaryoperators. Numerical examples are given throughout,demonstrating not only the theoretical convergenceguarantees of the algorithms, but also that they arecompetitive with “state-of-the-art” methods (whichdo not converge in general). These algorithms aremembers of a growing family of resolvent based meth-ods, moving away from the “discretise-then-solve”paradigm. They are parallelisable and have naturallinks with, as well as making use of, spectral methodsfor solutions of PDEs.

The ultraspherical spectral element method

Daniel Fortunato (Harvard University) & NicholasHale (Stellenbosch University) & Alex Townsend (Cor-nell University)

We introduce a novel spectral element method basedon the ultraspherical spectral method and the hier-archical Poincare–Steklov scheme for solving generalpartial differential equations on polygonal unstructuredmeshes. Whereas traditional finite element methodshave a computational complexity that scales as O(p6)with polynomial degree p, our method scales as O(p4),allowing for hp-adaptivity to be based on physical con-siderations instead of computational ones. Propertiesof the ultraspherical spectral method lead to almostbanded well-conditioned linear systems, allowing forthe element method to be competitive in the ultra-high-polynomial regime and to be robust to meshesthat contain elements with small aspect ratios. The

hierarchical Poincare–Steklov scheme enables precom-puted solution operators to be reused, allowing forfast elliptic solves in implicit and semi-implicit time-steppers. We develop an open-source software systemto demonstrate the flexibility and robustness of themethod.

A fast, robust solver for the Lippmann-Schwinger equation

Abinand Gopal & Per-Gunnar Martinsson (Univer-sity of Oxford)

An incredibly effective strategy for solving certain par-tial differential equations is to represent the solutionas an integral of a known kernel times an unknowndensity function and to then solve for the density bydiscretizing an integral equation. The resulting lin-ear systems are typically dense and, after quadraturecorrections necessary to achieve higher-order conver-gence, can be ill-conditioned and therefore challeng-ing for iterative solvers. We present a new high-ordersolver for the Lippmann-Schwinger equation, an inte-gral equation reformulation of the Helmholtz equationfor inhomogeneous scatterers. The key idea is to uti-lize rank structure in a fast, robust way.

A sparse spectral method for Volterra integralequations on the triangle

Timon Gutleb & Sheehan Olver (Imperial CollegeLondon)

We present a sparse spectral method to solve Volterraintegral equations

VKu = f or (λI + VK)u = f

with general kernels K(x, y) using bivariate orthogo-nal polynomials on a triangle domain. The Volterraintegral operator

(VKu)(x) :=

∫ x

0

K(x, y)u(y)dy

is shown to be sparse on a weighted Jacobi polyno-mial basis which allows us to achieve high efficiencyand exponential convergence without being restrictedto convolution kernel cases. We close by discussingconvergence proofs and showing solutions to sampleVolterra integral equations of first and second kind.

Resolvent techniques for computing the spec-trum of differential operators

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Andrew Horning & Alex Townsend (Cornell Uni-versity)

The spectral decomposition of a differential operator Lis fundamental to the study of differential equations:characterizing the spectrum and invariant subspacesof L allows one to construct simple solutions, analyzestability, and determine the asymptotic behavior ofsolutions. In classical and quantum physics, this de-composition often manifests in the familiar form ofeigenvalues and eigenfunctions. However, more so-phisticated radiative and dynamic models also admitresonant states, which are associated with the continu-ous spectrum and spectral measure of L. We present asuite of new methods for computing the spectrum andspectral measure directly, providing access to measur-able quantities of interest and an intuitive visualiza-tion of the operator L. Our technique is rooted inthe complex-analytic ideas of the resolvent-formalism,which describes the spectrum of L using the singular-ities of the resolvent R(z,L) = (zI − L)−1. We il-lustrate its advantages through a series of physicallymotivated examples.

Orthogonal polynomials on algebraic curvesand surfaces

Sheehan Olver (Imperial College)

Multivariate orthogonal polynomials on the circle andsphere are classical: they are precisely Fourier seriesand spherical harmonics. The circle and sphere arespecial cases of algebraic curves/surfaces, that is, thezero set of a polynomial. This raises the followingquestion: what are the properties of orthogonal poly-nomials on other algebraic curves? Can they be usedfor numerically solving surface PDEs? We will inves-tigate this question on simple quadratic curves andsurfaces (e.g. paraboloids and hyperboloids). We willalso discuss examples where the boundary of the do-main is specified by an algebraic curve, in particular,we will show how OPs can be used to solve PDEs ina half-disk.

Joint work with Ben Snowball (Imperial) and YuanXu (U. Oregon).

Continuous analogues of Krylov methods fordifferential operators

Marc Gilles & Alex Townsend (Cornell University)

Krylov subspace methods are a popular class of it-erative algorithms for solving Ax = b via matrix-vector products, with their convergence rates oftendepending on the condition number of A. In thistalk we develop analogues of the conjugate gradientmethod, MINRES, and GMRES for solving Lu =f via operator-function products, where L is an un-bounded elliptic differential operator. Our Krylovmethods employ operator preconditioning, have spec-tral convergence properties, and preserve the bilinearform associated to L. They are practical algorithmsthat are computationally competitive for spectral dis-cretizations.

A new Laplace solver for regions with corners

Nick Trefethen & Abi Gopal (University of Oxford)

Consider the Laplace equation with Dirichlet bound-ary data in a polygon or other domain with corners.We describe a new method for such problems of theflavour of the Method of Fundamental Solutions, butwith probably root-exponential convergence, i.e., er-ror = O(exp(−C

√N )) with C > 1. The idea is to

represent the function as the real part of a rationalfunction with poles exponentially clustered near eachcorner, leveraging a celebrated theorem in approxima-tion theory by Donald Newman in 1964. A short Mat-lab implementation solves problems amazingly fast, aswe will demonstrate.

Structured matrices and other desiderata forspectral methods

Marcus Webb (KU Leuven)

We discuss recent progress in designing the basis ofa spectral method so as to give rise to highly struc-tured linear systems, preserve self-adjointness of self-adjoint differential equations, and whose coefficientscan be computed by a fast algorithm (for example,utilising the Fast Fourier Transform). The advan-tages of these properties are more pronounced in time-dependent problems and eigenvalue problems than inboundary value problems. We conclude with openquestions about the feasibility of bases with these desi-derata. This is joint work with Arieh Iserles (Cam-bridge).

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Minisymposium M11

Numerical Methods for PartialDifferential Equations in Unbounded

DomainsOrganisers

Buyang Li and Chunxiong Zheng

Fast and memory efficient solution of boundaryintegral formulations of the Schrodinger equa-tion

Lehel Banjai & Maria Lopez Fernandez (Heriot-WattUniversity)

Fast and oblivious quadrature was introduced byLopez-Fernandez, Lubich and Schadle for convolutionswith a kernel whose Laplace transform is a sectorialoperator. The algorithm can compute N steps of aconvolution quadrature approximation of the convo-lution while using only O(logN) active memory andwith O(N logN) computational complexity.

In this talk we describe how oblivious quadrature canbe extended to some non-sectorial operators. In par-ticular we present an application to the integral formu-lation of a non-linear Schrodinger equation describingthe suppression of quantum beating.

A boundary integral equation method for lin-ear elastodynamics problems on unbounded do-mains

Silvia Falletta & Giovanni Monegato & Letizia Scud-eri (Politecnico di Torino)

We consider transient 3D elastic wave propagationproblems in unbounded isotropic homogeneous media,which can be reduced to corresponding 2D ones. Thisis the case, for example, of problems defined on the ex-terior of a bounded rigid domain, which are invariantin one of the Cartesian directions. Thus, the linearelastodynamics problem that characterizes small vari-ations of a displacement field u in a medium Ωe havingthe above properties, caused by a body force f , initialconditions u0v0 locally supported and a Dirichlet da-tum g, is defined by the following system:In [1] the two scalar equations are discretized by com-bining a Lubich time convolution quadrature formulawith a classical space collocation method (see [3] and[2]). Several numerical results, including comparisonswith the vector space-time boundary integral formu-

lation approach are presented and discussed.

References

[1] S. Falletta and G. Monegato L. Scuderi, Two bound-ary integral equation methods for linear elastodynam-ics problems on unbounded domains, submitted.

[2] S. Falletta and G. Monegato L. Scuderi, A space-time BIE methods for nonhomogeneous exterior waveequation problems. The Dirichlet case, IMA J. Nu-mer. Anal., 32(1) (2012), pp. 202–226.

[3] C. Lubich, On the multistep time discretizationof linear initial-boundary value problems and theirboundary integral equations, Numer. Math., 67(3)(1994), pp. 365–389.

Radiation boundary conditions for waves: ex-tensions and applications

Thomas Hagstrom (Southern Methodist University)

The radiation of energy to the far field is a centralfeature of wave physics. As such efficient, conver-gent domain truncation algorithms are a necessarycomponent of any wave simulation software. For thescalar wave equation and equivalent systems such asMaxwell’s equations in a uniform dielectric medium,Complete Radiation Boundary Conditions (CRBC),which are optimized local radiation boundary condi-tion sequences, provide a satisfactory solution: spec-tral convergence, rapid parameter selection based onsharp a priori error estimates, and effective corner/edge closures. Issues that arise in extending themethod to more general problems include the treat-ment of so-called reverse modes, waves whose groupand phase velocities are misaligned relative to the nor-mal direction at the radiation boundary, as well asproblems with inhomogeneities and nonlinearities.Here we will explore a number of ideas for construct-ing reliable and efficient domain truncation algorithmsin these more difficult settings:

• Modifications of the CRBC recursions to handlereverse modes - this approach is successful forproblems with single wave families;

• Rigorous a priori error analysis and optimiza-tion of ad hoc damping/stretching layers;

• Application of kernel compression/reduced or-der modeling to nonlocal formulations.

Specific applications to dispersive models of electro-magnetic waves, the elastic wave equation, as well aswaves in inhomogeneous media will be given.

Analyzing wave scattering problems in layered

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media by using perfectly matched layers

Wangtao Lu (Zhejiang University)

In this talk, I will report some recent progress on us-ing PMLs (Perfectly Matched Layers) for solving wavescattering problems in layered media.

The half-space matching method and its appli-cation to wave scattering in elastic plates

Yohanes Tjandrawidjaja & Vahan Baronian &Anne-Sophie Bonnet-Ben Dhia & Sonia FlissPOEMS (CNRS - INRIA - ENSTA Paristech)

We have developed a new method for scattering prob-lems in anisotropic media, where usual numerical meth-ods are either too expensive or even not applicable.This so-called Half-Space Matching method has beenfirst developed and validated in 2D. It consists in cou-pling several plane-wave representations of the solu-tion in half-spaces surrounding the scatterer with aFinite Element computation of the solution aroundthe scatterer. To ensure that all these representationsmatch in the infinite intersections of the half-spaces,the traces of the solution on the edges of the half-planes are linked by Fourier-integral equations. In thecase of a dissipative medium, the continuous problemis proved to be coercive plus compact, and the con-vergence of the discretization is ensured.

The method has been extended to the 3D case, for anapplication to non-destructive testing. The objectiveis to simulate the interaction of Lamb waves with adefect in an anisotropic elastic plate. The additionalcomplexity compared to the 2D case lies on the rep-resentations which are obtained semi-analytically bydecomposition on Lamb modes. In addition, the sys-tem of equations couples the FE representation in thebounded perturbed domain with not only the displace-ment, but also the normal stress of the solution on theinfinite bands limiting the half-plates. The first pre-liminary result has been obtained in the isotropic case.

Coupling methods for elliptic problems in theunbounded domain

Liwei Xu (University of Electronic Science and Tech-nology of China)

It is known that the coupling of finite element methods

and boundary integral equation methods is an efficientmethod for the numerical solution of elliptic problemsin the unbounded domain. In this talk, we first give aliterature review on the coupling method of finite ele-ment methods and boundary element methods. Thenwe present a new analysis result on one kind of cou-pling, the so-called DtN-FEM method, and presenta new coupling technique of finite element methodsand boundary integral equation methods, the so-calledDtD-FEM method. Numerical results will be pre-sented to demonstrate the accuracy of the methods.

Optimal control in a bounded domain for wavepropagating in the whole space

Wei Gong (Chinese Academy of Sciences) & BuyangLi (The Hong Kong Polytechnic University) & Huan-huan Yang (Shantou University)

We consider an optimal control problem in a bounded-domain Ω0 under the constraint of a wave equationin the whole space. The problem is regularized andthen reformulated as an initial-boundary value prob-lem of the wave equation in a bounded domain Ω ⊃ Ω0

coupled with a set of boundary integral equations on∂Ω taking account of wave propagation through theboundary. The well-posedness and stability of the re-formulated problem are proved. A fully discrete finiteelement method is proposed for solving the reformu-lated problem. In particular, the wave equation in thebounded domain is discretized by an averaged centraldifference method in time, and the boundary integralequations are discretized in time by using the convolu-tion quadrature generated by the second-order back-ward difference formula. The finite and boundary ele-ment methods are used for spatial discretization of thewave equation and the boundary integral equations,respectively. We prove the stability and convergenceof the numerical method, finally validate the spatialand temporal convergence rates numerically in 2D.

Rational approximation of the square root func-tion in the complex plane

Chunxiong Zheng (University College London)

The fast algorithm for the diffusion equation and theSchrodinger equation involves the efficient rational ap-proximation of the square root function in the complexplane. Many methods have been proposed to solve thisissue, and among them, Heron’s algorithm seems to bevery promising. In this talk, we make an error anal-ysis for Heron’s algorithm in suitable subdomains of

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the complex plane. As an application, we consider thefast algorithm for the artificial boundary condition ofthe one-dimensional convection diffusion equation inunbounded domains.

Minisymposium M12

Rational ApproximationOrganisers

Evan Gawlik and Yuji Nakatsukasa

Rational minimax iterations for computing frac-tional powers of matrices

Evan S. Gawlik (University of Hawaii at Manoa)

We construct a family of iterations for computing theprincipal square root of a square matrix A using Zolo-tarev’s rational minimax approximants of the squareroot function. We show that these rational functionsobey a recursion, allowing one to iteratively generateoptimal rational approximants of

√z of high degree

using compositions and products of low-degree ratio-nal functions. The corresponding iterations for thematrix square root converge to A1/2 for any input ma-trix A having no nonpositive real eigenvalues. In spe-cial limiting cases, these iterations reduce to knowniterations for the matrix square root: the lowest-orderversion is an optimally scaled Newton iteration, andfor certain parameter choices, the principal family ofPade iterations is recovered. Theoretical results andnumerical experiments indicate that the iterations per-form especially well on matrices having eigenvalueswith widely varying magnitudes.

After deriving iterations for the matrix square root,we generalize the above construction by deriving ratio-nal minimax iterations for the matrix pth root, wherep > 2 is an integer. The analysis of these iterations isconsiderably different from the case p = 2, owing tothe fact that when p > 2, rational minimax approx-imants of the function z1/p do not obey a recursion.Nevertheless, we show that several of the salient fea-tures of the Zolotarev iterations for the matrix squareroot, including equioscillatory error, order of conver-gence, and stability, carry over to case p > 2. A keyrole in the analysis is played by the asymptotic behav-ior of rational minimax approximants on short inter-vals.

Approximating the pth root by composite ra-tional functions

Evan S. Gawlik & Yuji Nakatsukasa (Oxford Uni-versity)

A landmark result from rational approximation the-ory states that x1/p on [0, 1] can be approximated bya type-(n, n) rational function with root-exponentialaccuracy. Motivated by the recursive optimality prop-erty of Zolotarev functions (for the square root andsign functions), we investigate approximating x1/p bycomposite rational functions of the form rk(x, rk−1

(x, rk−2(· · · (x, r1(x, 1))))). While this class of ratio-nal functions ceases to contain the minimax (best) ap-proximant for p ≥ 3, we show that it roughly achievespth-root exponential convergence with respect to thedegree. Moreover, crucially, the convergence is doubleexponential with respect to the number of degrees offreedom, suggesting that composite rational functionsare able to approximate x1/p and related functions(such as |x| and the sector function) with exceptionalefficiency.

Data-driven model order reduction forRayleigh-damped second-order systems

Igor Pontes Duff & Pawan Goyal & Peter Benner(Max Planck Institute for Dynamics of Complex Tech-nical Systems)

In this talk, we present a data-driven approach toidentify second-order systems of the form

Mx(t) + Dx(t) + Kx(t) = Bu(t),x(0) = 0, x(0) = 0,

y(t) = Cx(t),

with internal Rayleigh damping. This mean that thedamping matrix is given as a linear combination ofthe mass and stiffness matrices, i.e., D = αM + βK.These systems typically appear in order to performvarious engineering studies, e.g., vibrational analysis.The frequency response of the system can be given bythe following rational structured function:

H(s) = C(s2M + sD + K

)−1B,

which is also known as the transfer function. In anexperimental setup, the frequency response of a sys-tem can be measured via various approaches. As aconsequence, given frequency samples, the identifica-tion of the underlying system relies on rational struc-tured approximation. To that aim, we propose aninterpolation-based approach, extending the Loewnerframework for this class of systems. The efficiencyof the proposed method is demonstrated by means of

33

various numerical benchmarks.

Minisymposium M13

Hierarchical methods in stochasticnumerical approximations

OrganiserAbdul-Lateef Haji-Ali (Heriot-Watt)

Multilevel methods for fast Bayesian optimalexperimental design

Joakim Beck & Ben Mansour Dia & Luis Espath& Raul Tempone King Abdullah University of Scienceand Technology (KAUST)

We consider the problem of efficiently estimating theexpected Shannon information gain between prior andposterior probability distributions, which is a nestedintegral and appears as a design criterion in Bayesianoptimal experimental design for nonlinear models. Inthe context of partial differential equations with ran-dom input data, multilevel methods substantially re-duce the computational work complexity of their single-level counterparts. In this talk, we first present a mul-tilevel double loop Monte Carlo (MLDLMC) estima-tor and the associated optimal work complexity. Theoptimal values for the parameters of the MLDLMCestimator are determined by minimizing the averagecomputational work subject to some specified errortolerance that should be satisfied with some desiredprobability. Then, we present a multilevel double loopstochastic collocation (MLDLSC) method for high-dimensional integration by deterministic quadratureon sparse grids. The methods are demonstrated nu-merically for the design of electrical impedance tomog-raphy experiments with the experimental goal beingto infer fiber orientations in composite laminate ma-terials.

Multilevel Monte Carlo using approximate dis-tributions

Mike Giles & Oliver Sheridan-Methven (Universityof Oxford)

This talk will begin with a recap of the ideas andanalysis behind the multilevel Monte Carlo (MLMC)method [1, 2]. It will then discuss a new applicationof the method, building on prior ideas developed withHefter, Mayer and Ritter [3, 4, 5] to use different ap-

proximations of the inverse Normal CDF to generateincrements of a Brownian motion.

The advantage of this is that the new approximationscan be computed very efficiently, and the errors theyintroduce can be eliminated through a nested use ofMLMC. Numerical results will show the effectivenessof the approach, and also the additional benefits whichcome from using reduced precision computer arith-metic. The supporting numerical analysis will be out-lined briefly.

References

[1] M.B. Giles. Multilevel Monte Carlo path simula-tion. Operations Research, 56(3):607–617, 2008.

[2] M.B. Giles. Multilevel Monte Carlo methods. ActaNumerica, 24:259–328, 2015.

[3] M.B. Giles, M. Hefter, L. Mayer, and K. Ritter.Random bit quadrature and approximation of distri-butions on Hilbert spaces. Foundations of Computa-tional Mathematics, 19(1):205–238, 2019.

[4] M.B. Giles, M. Hefter, L. Mayer, and K. Ritter.Random bit multilevel algorithms for stochastic dif-ferential equations. Journal of Complexity, to appear,2019.

[5] M.B. Giles, M. Hefter, L. Mayer, and K. Ritter. Anadaptive random bit multilevel algorithm for SDEs. InMultivariate Algorithms and Information-Based Com-plexity. de Gruyter, 2019.

Inference with multilevel Monte Carlo

Kody J. H. Law & Ajay Jasra (University of Manch-ester)

Bayesian inference provides a principled and well-defined approach to the integration of data into ana priori known distribution, resulting in a posteriordistribution. This talk will concern the problem of in-ference when the posterior involves continuous modelswhich require approximation before inference can beperformed. Typically one cannot sample from the pos-terior distribution directly, but can at best only eval-uate it, up to a normalizing constant. Therefore onemust resort to computationally-intensive inference al-gorithms in order to construct estimators. These algo-rithms are typically of Monte Carlo type, and includefor example Markov chain Monte Carlo, importancesamplers, and sequential Monte Carlo samplers. Themultilevel Monte Carlo method provides a way of op-

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timally balancing discretization and sampling error ona hierarchy of approximation levels, such that cost isoptimized. Recently this method has been applied tocomputationally intensive inference. This non-trivialtask can be achieved in a variety of ways. This talkwill review 3 primary strategies which have been suc-cessfully employed to achieve optimal (or canonical)convergence rates ? in other words faster convergencethan i.i.d. sampling at the finest discretization level.Some of the specific resulting algorithms, and appli-cations, will also be presented.

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Abstracts of Contributed Talks

Numerical methods for PDE-constrained opti-mization problems in particle dynamics

Mildred Aduamoah & John Pearson & Ben God-dard (University of Edinburgh)

A vast number of important applications in mathe-matics and engineering are governed by mathemati-cal optimization problems. One crucial class of theseproblems that have gained significant recent attentionare PDE-constrained optimization problems, whichcan be used to model problems from fluid flow, in-dustrial processes, and biological mechanisms. How-ever, to date, relatively little has been done by wayof devising a systematic approach to tackle multiscalePDE-constrained optimization problems arising fromparticle dynamics. In this talk, we will introduce par-ticle dynamics optimization models, discuss the reallife applications they arise from, and explain the newnumerical methods that we have developed to solvethem.

Multilevel numerical algorithms for thin filmflow

Mashael Aljohani & Dr Mark Walkley & ProfessorPeter Jimack (University of Leeds)

The thin film flow equations that we consider in thisresearch are based upon a long-wave approximationto the Stokes equations. This results in a nonlinearparabolic system of two partial differential equations(PDEs) for two dependent variables (film thicknessand pressure). When discretised implicitly in time,and using a standard finite difference stencil in space,this results in a large system of nonlinear algebraicequations to be solved at each time step.

For suitable linear equation systems the use of multi-level algorithms is well-established as a means of ob-taining optimal (i.e.linear) time complexity. For somenonlinear equation systems multilevel algorithms mayalso be employed, and can also deliver optimal timecomplexity (or very close to optimal). However, thereappears to be little research that seeks to systemati-cally investigate which multilevel approach is the mosteffective for practical problems. In this work we con-sider three nonlinear multilevel algorithms to solve thediscretised thin film flow PDEs: the Full Approxima-tion Scheme (FAS), Newton-Multigrid (Newton-MG)and a new preconditioned Newton-Krylov variant.

Each of the nonlinear multilevel solvers will be de-scribed, and will be demonstrated to show linear (orclose to linear) time complexity. For the Newton-Krylov algorithm, we will introduce our novel block-preconditioner based on the use of a multilevel alge-braic multigrid (AMG) package. We will then presentsystematic numerical results to compare the three mul-tilevel algorithms to show that, while all three are op-timal, there are differences in performance that demon-strate the superiority of the new preconditionedNewton-Krylov approach.

Computational paradoxes in deep learning

Alexander Bastounis & Anders Hansen & VernerVlacic (University of Cambridge)

One of the biggest problems in modern AI is that neu-ral networks based on deep learning become highlyunstable, and currently there does not exist any curefor this problem. The instabilities are typically so se-vere that images that look identical to the human eyemay be classified with wildly different labels like a catand a fire truck. In this talk we discuss theoreticalparadoxical results in deep learning that shed light onthe mysterious behaviour of modern AI.

1. Any given algorithm for computing the optimalneural network given a training set will fail tofind the best neural network with high proba-bility on uncountably many training sets, evenif such an algorithm is given access to an oraclethat allows the algorithm to avoid local but notglobal minima.

2. For a given arbitrarily small distance ε > 0,there exists uncountably many classificationproblems (all of which are stable under ε per-turbations) for which one can construct a neuralnetwork which correctly identifies an (arbitrarilylarge) training set and an (arbitrarily large) testset but such a neural network must be unstableunder perturbations of size ε.

3. For any given ε > 0, one can construct problemsfor which solving a neural network optimisationproblem will yield an incorrect classification on amember of the training set, however, if one per-turbs the training set by at most ε, solving theoptimisation problem will yield a neural networkthat succeeds on the new training set.

The first of these paradoxes highlights an issue withthe current understanding of neural networks – namely,the universal approximation theorem guarantees the

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existence of a neural network for a given classificationproblem, but gives no method to find it and in factour result shows that in general this is not possible.

The second paradox provides a theoretical demonstra-tion of adversarial examples: that is, examples wherea small change in the data yields to a wildly differentoutput of the neural network. In this case, the train-ing process has forced the choice of an unstable neuralnetwork.

Finally, the third paradox shows that quantifying thesuccess of neural networks is a difficult task – whetheror not a neural network will be successful can changedrastically depending on arbitrarily small perturba-tions of the training set.

Convergence analysis of adaptive stochasticGalerkin FEM for elliptic PDEs with paramet-ric uncertainty

Alex Bespalov 1 & Dirk Praetorius 2 & LeonardoRocchi 1 & Michele Ruggeri 2 1 (University of Birm-ingham) 2 (Vienna University of Technology)

We study adaptive stochastic Galerkin finite elementapproximations for a class of elliptic partial differentialequations with parametric or uncertain coefficients.The proposed adaptive algorithm is steered by a novelreliable and efficient a posteriori error estimator thatcombines a two-level spatial estimator and a hierarchi-cal parametric estimator, see [1]. The structure of theestimator is exploited in the algorithm to perform abalanced adaptive refinement of the spatial and para-metric components of Galerkin approximations. Inthis talk, we present recent convergence results forthis adaptive refinement procedure [2]. In particu-lar, we show that the proposed algorithm drives theunderlying energy error estimate to zero. We also re-port the results of numerical experiments that focuson comparison of the computational cost associatedwith employing different marking criteria for adaptiverefinement.

References

[1] A. Bespalov, D. Praetorius, L. Rocchi, and M.Ruggeri, Goal-oriented error estimation and adaptiv-ity for elliptic PDEs with parametric or uncertain in-puts, Comput. Methods Appl. Mech. Engrg., 345,pp. 951–982, 2019.

[2] A. Bespalov, D. Praetorius, L. Rocchi, and M.Ruggeri, Convergence of adaptive stochastic Galerkin

FEM, Preprint, arXiv:1811.09462, 2018.

High order compact finite difference schemeson nonuniform grids for nonstationary PDEs

Raimondas Ciegis (Vilnius Gediminas Technical Uni-versity)

In this talk we consider high-order compact finite dif-ference schemes constructed on 1D non-uniform grids.We apply them to the parabolic equation:

∂u

∂t=∂2u

∂x2− q(x)u+ f(x, t),

and the Schrodinger equation:

∂u

∂t− i(∂2u

∂x2− q(x)u

)= 0,

where t and x are time and space variables, u(x, t) is areal valued function for the parabolic equation and acomplex valued function for the Schrodinger equation.

We consider a high-order compact semi-discrete finitedifference scheme on a non-uniform grid defined onthe three-point template for the parabolic problem

Bh(Unt + qUn−1/2 − fn−1/2

)= AhDU

n−1/2,

and for the Schrodinger problem

Bh(Unt + iqUn−1/2

)= iAhDU

n−1/2.

Here the operator Bh is defined as

BhUj := αjUj−1 + (1− αj − βj)Uj + βjUj+1.

Stability of these schemes is investigated by using thespectral method. Computer experiments are appliedin order to find critical grids for which the stabilitycondition is violated.

We have proved that for the Schrodinger problem theproposed scheme is not A-stable. Examples of criti-cal non-uniform grids are obtained. The analysis ofρ-stability is presented, it shows that ρ-stability canbe guaranteed only if the time step is not too smallτ ≥ k0(h). This nonclassical inverse stability condi-tion regularizes the discrete scheme, but reduces theconvergence accuracy to the second order.

A similar stability analysis is done also for the parabolicproblem. It is interesting to note that in this case thelarge scale computational experiments have not pro-duced any critical non-uniform grid when the stabilitycondition Reλj < 0 is violated for some eigenvalue.

37

In addition to FDS a compact finite element schemewith quadratic elements is constructed. This schemecan be seen as a high-order finite difference scheme, ifvalues of the solution are considered only on the gridpoints. Basic stability and energy conservation prop-erties of the obtained discrete scheme are obtained.

Numerical examples supporting our theoretical anal-ysis are provided and discussed.

The MRE inverse problem for the elastic shearmodulus

Penny J Davies & Eric Barnhill & Ingolf Sack(University of Strathclyde)

Magnetic resonance elastography (MRE) is a powerfultechnique for noninvasive determination of the biome-chanical properties of tissue, with important applica-tions in disease diagnosis. A typical experimental sce-nario is to induce waves in the tissue by time-harmonicexternal mechanical oscillation and then measure thetissue’s displacement at fixed spatial positions 8 timesduring a complete time-period, extracting the domi-nant frequency signal from the discrete Fourier trans-form in time. Accurate reconstruction of the tissue’selastic moduli from MRE data is a challenging inverseproblem, and I will describe a new approach basedon combining approximations at different frequenciesinto a single overdetermined system.

Some highlights from NLAFET WP3 on par-allel sparse direct solution

Iain Duff & Sebastien Cayrols & Florent Lopez &Stojce Nakov (STFC Rutherford Appleton Laboratoryand Cerfacs)

The FET-HPC EU Horizon 2020 Project NLAFETjust ended at the end of April. STFC-RAL was thelead partner for Workpackage 3 on sparse direct meth-ods. We discuss some highlights from this work.

While it is recognized that direct methods are verypowerful over a wide range of applications, they some-times have issues on extremely large problems, partic-ularly in 3 dimensions. Our approaches overcome thislimitation by subdividing the problem through parti-tioning the matrix. The direct solver is then used ona submatrix of the original problem.

We illustrate this in two quite different ways. One isby preordering to singly bordered block diagonal form

and the other is using an approach based on the blockCimmino algorithm.

The added bonus is that such a subdivision also in-troduces another level of parallelism and we are thenable to combine distributed memory parallelism usingMPI with shared memory parallelism using OpenMPor other runtime systems. We illustrate this with runsof our codes on industrial strength large test problemson a heterogeneous platform where we show that ourcodes, which are available on github, outperform otherstate-of-the-art codes.

Approximation of time domain boundary inte-gral equations

Dugald B Duncan (Heriot-Watt University)

Time domain boundary integral equations (TDBIEs)are used to model time dependent wave scatteringfrom surfaces in an infinite, homogeneous medium.We concentrate on the acoustic scattering case. Thesolution can be reconstructed anywhere in space andtime in terms of a potential which is computed only onthe surface. The approximation of the time dependentsurface potential by numerical methods is expensiveand sensitive to stability issues, but it can be cheaperthan solving the wave equation in a very large spacedomain. We describe a practical implementation of afull space-time Galerkin TDBIE method derived andanalysed by Ha Duong, and its connection with a sim-pler method, derived by Davies and Duncan, using“backward-in-time” collocation coupled with Galerkinin space.

Preconditioning techniques for the Darcy-Forchheimer model using Block-Centered finitedifference method

Faisal A. Fairag & Mohsen G. Alshahrani (KingFahd University of Petroleum and Minerals)

We consider the Darcy-Forchheimer problem in twodimensions describing the fluid flow such as oil andgas in porous media

µk−1u+ βρ|u|u+∇p = 0, in Ω,

∇ · u = f, in Ω, u · n = 0, on ∂Ω, (1)

with the compatibility condition∫

Ωf dxdy = 0. The

unknowns are the pressure p and the velocity u =(u1, u2). Here µ, k and β denote the viscosity coef-ficient, the permeability tensor and the Forchheimer

38

number, respectively. n denotes the outward normalunit vector to ∂Ω and |.| denotes the Euclidean norm.The function f ∈ L2(Ω) and Ω ∈ R2. If there exist

positive constants C1 and C2 such that: C1 ≤µ

k≤

C2, C1 ≤ βρ ≤ C2, then (1) is well-posed. If β = 0,then (1) becomes the Darcy’s equation which describesa linear relation between the velocity and the pressuregradient.

Block-Centered Finite Difference discretization(BCFDM) and Newton-Raphson method yield a sparseand indefinite linear system of the form D1 MT

3 I ⊗ ETM3 D2 ET ⊗ II ⊗ E E ⊗ I 0

U1

U2

P

=

b1b2b3

,

E =

−1

1. . .

. . . −11

(2)

where D1 and D2 are diagonal matrices and ⊗ de-notes the Kronecker product. The focus of this studyis to study the efficient solution of the linear system(2). For large problems with a sparse coefficient ma-trix, iterative methods are preferable. In this talk, wewill investigate the performance of a preconditionedMINRES method which utilises the structure of thecoefficient matrix (2).

Acknowledgement:

The authors acknowledge the support provided by KingAbdulaziz City for Science and Technology (KACST)(project no. 1-17-003-0002) and KFUPM.

Two-step implicit higher order numerical inte-grator for stiff systems of initial – boundaryvalue problems of ordinary differential equa-tions

Johnson O. Fatokun* & Samuel I. Okoro (AnchorUniversity)

In this paper, we consider the derivation of a Two-Step Implicit Higher Order Numerical Integrator forStiff Systems of Ordinary Differential Equations. Theexponentially fitted numerical method is developedusing the method of trapezoidal interpolant for thenumerical integration for stiff system. This methodpreserves the A-stability property of the numericalscheme and is also L-stable. Two theorems and oneLemma were proposed and proved which establish theA-stability and L-stability properties of the derived

method. The local truncation error of the methodis estimated from the continuous form of the methodderived and presented. The analysis and numericalexperiments show clearly that the method competesfavourably with other known methods when appliedto stiff systems of initial value problems of ordinarydifferential equations. The method is of at least or-der four (Oh4) in accuracy. Key words: Stiff systems,trapezoidal integrators, A-Stability, L-Stability, Im-plicit two-step method.

AMS Classification: 65L05

An adaptive primal-dual framework for nons-mooth convex minimization

Olivier Fercoq & Quoc Tran-Dinh & Ahmet Ala-caoglu & Volkan Cevher (LTCI, Telecom ParisTech,Universite Paris-Saclay)

We propose a new self-adaptive, double-loop smooth-ing algorithm to solve composite, nonsmooth, and con-strained convex optimization problems.

Our algorithm is based on Nesterov’s smoothing tech-nique via general Bregman distance functions. It self-adaptively selects the number of iterations in the innerloop to achieve a desired complexity bound withoutrequiring the accuracy a priori as in variants of Aug-mented Lagrangian methods (ALM). We prove O( 1

k )-convergence rate on the last iterate of the outer se-quence for both unconstrained and constrained set-tings in contrast to ergodic rates which are commonin ALM as well as alternating direction method-of-multipliers literature.

Compared to existing inexact ALM or quadraticpenalty methods, our analysis does not rely on theworst-case bounds of the subproblem solved by the in-ner loop. Therefore, our algorithm can be viewed as arestarting technique applied to the ASGARD method(Tran-Dinh et al, 2018) but with rigorous theoreticalguarantees or as an inexact ALM with explicit innerloop termination rules and adaptive parameters. Ouralgorithm only requires to initialize the parametersonce, and automatically updates them during the it-eration process without tuning.

We illustrate the superiority of our methods via sev-eral examples as compared to the state-of-the-art.

Wilkinson test matrices

Carla Ferreira & Beresford Parlett (Universityof Minho)

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In the 1950’, J. H. Wilkinson introduced two fami-lies of symmetric tridiagonal integer matrices W±2m+1,for each positive integer m. All the next-to-diagonalentries are one. The diagonal entries are given by

diag(W±2m+1)

= diag(m,m− 1, . . . , 1, 0,±1,±2, . . . ,±m).

Each matrix illustrated subtle difficulties in the auto-matic computation of eigenvectors and eigenvalues.

Most of the eigenvalues of W±2m+1 are close to diagonalentries. We developed the structure of their eigenvec-tors in a natural way which reveals that the envelopesof these eigenvectors all look the same to the nakedeye. The shape is a badly dented bell curve. We alsoanalyzed the eigenvectors of the remaining non-integereigenvalues.

Variational time integration for evolutionarysystems

S. Franz (TU Dresden)

Evolutionary systems are time dependent systems offirst order partial differential equations. For a HilbertspaceH and a weighted space-time Hilbert space givenby

Hρ(R;H) :

f : R→ H : f meas.,

∫R

, 〈f(t), f(t)〉He−2ρt dt <∞

with an associated inner product

〈f, g〉ρ :

∫R〈f(t), g(t)〉He−2ρt dt

their simplest form is

(∂tM0 +M1 +A)U = F,

where

• M0, M1 are bounded, linear, self-adjoint opera-tors on H,

• A is an unbounded, skew-selfadjoint operator onH,

• ∃ρ0 > 0, γ > 0 such that for all ρ ≥ ρ0, x ∈ Hit holds 〈(ρM0 +M1)x, x〉ρ ≥ γ〈x, x〉ρ.

Then the system has for each ρ ≥ ρ0 and F ∈ Hρ(R, H)a unique solution U ∈ Hρ(R, H), see [1], and it holds

‖U‖ρ ≤1

γ‖F‖ρ.

Examples of such systems are wave-equations, heat-equations, Maxwell-equations, linear elasticity, acous-tic equations, combinations thereof and many manymore.

In this talk we investigate the numerical approxima-tion of these systems by using variational timeintegration methods like discontinuous Galerkin andcontinuous-Petrov-Galerkin. As examples we considerproblems, where the type of the partial differentialequation changes in the domain, see [2], or the ho-mogenisation of rapidly changing coefficients, see [3,4].

References

[1] R. Picard, A structural observation for linear ma-terial laws in classical mathematical physics, Math.Methods Appl. Sci., 32, 1768–1803, 2009.

[2] S. Franz, S. Trostorff, and M. Waurick, Numericalmethods for changing type systems, IMAJNA, 2018,doi.org/10.1093/imanum/dry007.

[3] S. Franz and M. Waurick, Resolvent estimatesand numerical implementation for the homogenisa-tion of one-dimensional periodic mixed type problems,ZAMM, 98(7):1284–1294, 2018.

[4] S. Franz and M. Waurick, Homogenisation of para-bolic/hyperbolic media, accepted for publication inBAIL2018 Proceedings, 2018.

A new two-step stable high accuracy implicitmethod for general second order nonlinear initial-value problems on a graded mesh

Bishnu Pada Ghosh (South Asian University, In-dia)

A new two-step method of order three on a gradedmesh for the numerical solution of general second or-der nonlinear initial-value problems u′′ = f(t, u, u′),u(t0) = γ0, u

′(t0) = γ1 is discussed. In practice, onlya monotonically decreasing mesh will be employed.When applied to a test equation u′′ + 2αu′ + β2u =g(t), α > β ≥ 0 ≥, the derived method is absolutelystable and superstable for a uniform mesh. The pro-posed method is applicable to both singular problemsand problems containing boundary layers that arise

40

close to the point where one wishes to compute. Com-putational results are presented to demonstrate thevalidity of the proposed method. The numerical re-sults clearly indicate that the proposed method pro-duces better results in comparison with the existingmethods.

Domain decomposition preconditioners for het-erogeneous Helmholtz problems

Shihua Gong & Ivan G. Graham & Euan A. Spence(University of Bath)

We consider one-level additive Schwarz precondition-ers for a family of Helmholtz problems of increasingdifficulty characterized by wave number k. As k in-creases, the solution becomes more oscillatory andthus meshes need to be increasingly refined, leadingto huge linear systems. Moreover, these linear sys-tems become more indefinite and the minors of the lin-ear systems may be even singular, which prevents theusage of many standard preconditioning techniques.We discuss additive Schwarz preconditioners, of whichthe action requires solution of independent subprob-lems with absorbing boundary conditions and (possi-bly) absorption added in the domain. Then the localsolutions are glued together using prolongation oper-ators defined by a partition of unity and nodal inter-polation. Supporting theory for this preconditionerin the case of homogeneous Helmholtz problems isgiven by I. G. Graham, E. A. Spence and J. Zouin [Preprint arXiv:1806.03731.]. In this talk, we willpresent preliminary theoretical results and numericalexperiments for this method applied to heterogeneousHelmholtz problems. We also show that the methodis independent of the underlying finite element dis-cretizations.

Analysis of circulant embedding methods forsampling stationary random fields

Ivan Graham (University of Bath)

This talk contains joint work with Markus Bachmayr,Frances Kuo, Van Kien Nguyen, Dirk Nuyens, RobScheichl and Ian Sloan ([1],[2]).

A standard problem in uncertainty quantification andin computational statistics is the rapid sam- pling ofstationary Gaussian random fields with given covari-ance in a d-dimensional (physical) domain. In manyapplications it is sufficient to perform the sampling ona regular grid on a cube enclosing the physical domain,

in which case the corresponding covariance matrix isnested block Toeplitz. After extension to a nestedblock circulant matrix, this can be diagonalised byFFT and, provided the circulant is positive definite,this provides a finite expansion of the field in terms ofa deterministic basis, with coefficients given by i.i.d.standard normals. This is a discrete counter- partof the well-used Karhunen-Loeve expansion, but (be-cause of the FFT) it can be computed in log-optimaltime.

Questions at the interface of analysis and numericallinear algebra which arise from this process are: (i)What is the most efficient way to extend the blockToeplitz matrix to a block circulant, so that positivedefiniteness is maintained and (ii) What is the decayrate of the eigenvalues of the block circulant (sincethis determines the efficiency of several UQ algorithmswhich use this type of expansion).

We give an overview of the answers to these questions,both for classical (non-smooth) circulant embedding[1] and for circulant embedding using smooth cut-offfunctions [2].

References

[1] I.G. Graham, F.Y. Kuo, D. Nuyens, R. Scheichl,and I.H. Sloan, Analysis of circulant embed- ding meth-ods for sampling stationary random fields, SIAM J.Numer. Anal. 56(3), 1871 1895, 2018.

[2] M. Bachmayr, I.G. Graham, V.K. Nguyen and R.Scheichl, Unified analysis of periodisation- based sam-pling methods for Matern covariances, in preparation.

Bayesian network PDEs for multiscale repre-sentations of porous materials

Eric Hall (RWTH Aachen University) & Kimoon Um(Stanford University) & Markos Katsoulakis (Univer-sity of Massachusetts Amherst) & Daniel Tartakovsky(Stanford University)

Microscopic (pore-scale) properties of porous mediaaffect and often determine their macroscopic (con-tinuum- or Darcy-scale) counterparts. Understandingthe relationship between processes on these two scalesis essential to both the derivation of macroscopic mod-els of, e.g., transport phenomena in natural porousmedia, and the design of novel materials, e.g., for en-ergy storage. Microscopic properties exhibit complexstatistical correlations and geometric constraints thatpresent challenges for the estimation of macroscopicquantities of interest (QoIs), e.g., in the context of

41

global sensitivity analysis (GSA) of macroscopic QoIswith respect to microscopic material properties. Wepresent a systematic way of building correlations intostochastic multiscale models through Bayesian Net-works. The proposed framework allows us to constructthe joint probability density function (PDF) of modelparameters through causal relationships that are in-formed by domain knowledge and emulate engineeringprocesses, e.g., the design of hierarchical nanoporousmaterials. These PDFs also serve as input for the for-ward propagation of parametric uncertainty therebyyielding Bayesian Network PDE. To assess the im-pact of causal relationships and micro-scale correla-tions on macroscopic material properties, we proposea moment-independent GSA and corresponding effectrankings for Bayesian Network PDE, based on the dif-ferential Mutual Information, that leverage the struc-ture of Bayesian Networks and account for both cor-related inputs and complex non-Gaussian QoIs. Ourfindings from numerical experiments, which feature anon-intrusive uncertainty quantification workflow, in-dicate two practical outcomes. Firstly, the inclusion ofcorrelations through structured priors based on causalrelationships informed by domain knowledge impactspredictions of QoIs which has important implicationsfor engineering design. Secondly, we observe the inclu-sion of structured priors with non-trivial correlationsyields different effect rankings than independent pri-ors and moreover these rankings are more consistentwith the anticipated physics of a model problem.

Numerical solution of the Hemker problem

Alan F. Hegarty (University of Limerick) & EugeneO’Riordan (Dublin City University)

In [1] Hemker proposed a model test problem in twospace dimensions, comprising the simple constant co-efficient linear singularly perturbed convection-diffusionequation

−ε4u+ ux = 0, for x2 + y2 > 1;

on the unbounded domain R2 \ x2 + y2 ≤ 1, ex-terior to the unit circle, with the boundary condi-tions u(x, y) = 1, if x2 + y2 = 1 and u(x, y) → 0as x2 + y2 →∞.

[1] notes that the domain is sufficiently complex to al-low an exponential boundary layer as well as parabolicinterior layers, and that to find a proper mesh is “nottrivial at all”.

In neighbourhoods of the points on the circle wherethe characteristics of the reduced problem vx = 0 aretangential to the circle, there are transition regions,

where the steep gradients in the solution migrate fromthe exponential boundary layers (located on the leftside of the unit disk) to the parabolic internal layersemerging from the characteristic points (0,±1).

The challenge proposed in [1] is to design a numericalmethod, which produces stable and accurate approx-imations over the entire domain, for arbitrary smallvalues of the perturbation parameter ε. We proposehere such a method, based on appropriately chosenShishkin meshes in different coordinate systems.

References

[1] P. W. Hemker, A singularly perturbed model prob-lem for numerical computation, J. Comp. Appl. Math,76, 1996, 277–285.

On numerical simulations and a posteriori anal-ysis for algebraic flux correction schemes

Abhinav Jha & Volker John (Freie Universitat Berlin,Weierstrass Institute, Berlin)

Non-linear discretizations are necessary for convection-diffusion-reaction equations for obtaining accurate so-lutions that satisfy the discrete maximum principle(DMP). Algebraic stabilizations, also known as Alge-braic Flux Correction (AFC) schemes, belong to thevery few finite element discretizations that satisfy thisproperty. The non-linearity of the system arises be-cause of the presence of solution dependent limiters.Because of the non-linear nature of the problem, anew issue arises, which is the efficient solution of thesystem of equations.

The first part of the talk will address this issue andseveral methods will be discussed for solving the non-linear problem with a major focus on fixed point it-erations and Newton’s method. Different algorithmiccomponents such as Anderson acceleration and dy-namic damping will be discussed as well. The meth-ods are compared on different parameters such as thenumber of iterations and computing time needed. Nu-merical examples are presented in 2d as well as 3dwhich will assess the different solvers.

The second part of the talk is devoted to the pro-posal of a new a posteriori error estimator for theAFC schemes. With a mild assumption on the in-terpolation error, we find a global upper bound in theenergy norm of the system which is independent of thechoice of limiters. Numerical results are presented in

42

2d with two different types of limiters.

Iterative methods for elliptic PDEs based onoperator preconditioning

Janos Karatson (ELTE University)

A desirable property of an iterative method for a dis-cretized partial differential equation is mesh indepen-dence. Then, if the auxiliary problems can be solvedwith optimal order of arithmetic operations (i.e. thecost O(N) is proportional to the degrees of freedom),then the overall iteration also has this advantage. Suchefficient preconditioners for discretized elliptic prob-lems can be obtained via equivalent operator precon-ditioning, that is, the preconditioner shall be chosenas the discretization of a suitable auxiliary operatorthat is equivalent to the original one. This conceptwas elaborated by T. Manteuffel et al. [4] and thenapplied in several situations, see, e.g., [6, 7, 8]. Theauthor’s work is mainly based on joint research withO. Axelsson, e.g., [1, 2, 3], see also [5] for nonlinearproblems.

This talk first gives a brief theoretical summary, in-cluding both linear and superlinear mesh independentconvergence. Then various applications are shown,such as parallel preconditioning of transport type sys-tems, shifted Laplace preconditioners for Helmholtzequations, and streamline diffusion preconditioningboth for linear convection-diffusion equations and forsemilinear parabolic systems.

References

[1] Axelsson, O., Karatson, J.,. Equivalent operatorpreconditioning for elliptic problems, Numer. Algor.,50:297–380 (2009).

[2] Axelsson, O., Karatson, J., Robust preconditioningestimates for convection-dominated elliptic problemsvia a streamline Poincare-Friedrichs inequality, SIAMJ. Numer. Anal., 52 (2014), No. 6, pp. 2957-2976.

[3] Axelsson, O., Karatson, J., Superlinear conver-gence of the GMRES for PDE-constrained optimiza-tion problems, Numer. Funct. Anal. Optim. 39(2018), no. 9, 921–936.

[4] Faber, V., Manteuffel, T., Parter, S.V., On thetheory of equivalent operators and applications to thenumerical solution of uniformly elliptic partial differ-ential equations, Adv. in Appl. Math., 11 (1990),109-163.

[5] Farago I., Karatson J., Numerical Solution of Non-linear Elliptic Problems via Preconditioning Opera-tors: Theory and Application. Advances in Compu-tation, Volume 11, NOVA Science Publishers, NewYork, 2002.

[6] Kirby R. C., From Functional Analysis to IterativeMethods, SIAM Review, 52(2) 269-293 (2010).

[7] Loghin, D., Wathen, A. J., Analysis of precondi-tioners for saddle-point problems, SIAM J. Sci. Com-put. 25 (2004), no. 6, 2029–2049.

[8] Mardal, K-A.,. Winther, R., Preconditioning dis-cretizations of systems of partial differential equations,Numer. Linear Algebra Appl. 18 (2011), no. 1, 1–40.

Numerical methods for Cordial Volterra inte-gral equation with vanishing delays

Melusi Khumalo (University of South Africa)

In this talk, we consider some numerical approachesto solving Cordial Volterra integral equations (CVIEs)with vanishing delays. A number of authors have es-tablished an inextricable link between the compact-ness of a Cordial Volterra integral operator and theexistence and uniqueness of the associated CVIE. Thevalidity of such results are investigated in the case ofCVIEs incorporating vanishing delays.

Collocation methods with uniform mesh as well asgraded mesh are then implemented on some strate-gic examples. Convergence rates and existence anduniqueness of the collocation solution are considered.

Structured Sylvester and T -Sylvester equations

Ivana Kuzmanovic Ivicic & Ninoslav Truhar (Uni-versity of Osijek)

Sylvester and T -Sylvester equations are matrix equa-tions of the form

AX +XB = E

and

AX +XTB = E,

where A, B and E are given and X is an unknown ma-trix. Sylvester equations appear frequently in manyareas of applied mathematics. For example, Sylvesterequations play vital roles in matrix eigen-decom-

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positions, control theory, model reduction, numeri-cal solution of matrix differential Riccati equationsand algebraic Riccati equations, image processing, andmany more. On the other hand, T -Sylvester matrixequations have recently attracted the attention of re-searchers because of their relationship with palindromiceigenvalue problems.

We will present results about structured Sylvester andT -Sylvester equations, especially for structured prob-lems with the system matrices of the form A = A0 +U1V1 and B = B0 + U2V2 where U1, U2, V1, V2 aresmall rank update matrices. We will give the Sherman-Morrison-Woodbury-type formula for solutions of thistype of equation. The obtained formulas are used forconstruction of algorithms which solve the equationsof the above form much more efficiently than the stan-dard algorithms. Application of obtained algorithmswill be illustrated in several examples.

This is joint work with Ninoslav Truhar.

Robust solvers for highly heterogeneous Max-well’s equations

Alexandros Kyriakis (University of Strathclyde)

Maxwell’s equations are the fundamental equationswhich govern and explain the behaviour of electromag-netic waves. A key topic of interest is the investigationof highly heterogeneous problems with differing con-ductivities, which vary highly in magnitude along thedomain. An analogue to this is Darcy’s problem witha highly heterogeneous diffusion coefficient. Our inter-est lies in the stability and convergence to the solutionwhen building an appropriate coarse space. Empha-sis will be given to exposing some of the investigatedcases. A few practical test cases will be presented,starting with Darcy’s problem, using a two-level do-main decomposition method. For each case we usethe GenEO coarse space and investigate numericallyusing the specialised software FreeFem++.

Improving the condition number of sample co-variance matrices for use in variational dataassimilation

Amos S. Lawless, Jemima M. Tabeart, Sarah L.Dance, Nancy K. Nichols & Joanne A. Waller (Uni-versity of Reading & National Centre for Earth Ob-servation)

Variational data assimilation is used on a daily basiswithin numerical weather and ocean prediction, com-

bining the latest observational data with numericalmodels to estimate the current state of the system.It formulates the problem as the minimisation of aweighted nonlinear least-squares objective function, inwhich the fit to the observations is balanced againstthe fit to a previous model forecast, with covariancematrices weighting the different sources of informa-tion. Until recently most operational forecasting cen-tres have assumed that the errors in the observationsare uncorrelated. However, this is not always true, es-pecially when considering multichannel satellite data,and it is becoming more important to specify obser-vation error correlations.

As observation error covariance matrices must usu-ally be obtained by sampling methods, estimates areoften degenerate or ill-conditioned, making it impos-sible to invert an observation error covariance matrixwithout the use of techniques to reduce its conditionnumber. In this work we present new theory for twoexisting methods that can be used to ‘recondition’ anycovariance matrix: ridge regression, and the minimumeigenvalue method. We compare these methods withmultiplicative variance inflation, which cannot alterthe condition number of a matrix, but is often usedto account for neglected correlation information. Weinvestigate the impact of reconditioning on variancesand correlations of a general covariance matrix in botha theoretical and practical setting. The new theoryshows that, for the same target condition number,both methods increase variances compared to the orig-inal matrix, with larger increases for ridge regressionthan the minimum eigenvalue method. We prove thatthe ridge regression method strictly decreases the ab-solute value of off-diagonal correlations. Theoreticalcomparison of the impact of reconditioning and mul-tiplicative variance inflation on the data assimilationobjective function shows that variance inflation altersinformation across all scales uniformly, whereas recon-ditioning has a larger effect on scales corresponding tosmaller eigenvalues.

We then consider two numerical examples: a spatialcorrelation function, and an observation error covari-ance matrix arising from satellite interchannel corre-lations. The minimum eigenvalue method results insmaller overall changes to the correlation matrix thanridge regression, but can increase off-diagonal correla-tions. We find cases where standard deviations doublein size, which will have implications for the solutionof a data assimilation problem.

Doubly-stochastic scaling of adjacency matri-ces for community detection

Luce le Gorrec & Sandrine Mouysset & Daniel Ruiz

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(Universite de Toulouse)

Community detection in networks consists of findinggroups of individuals such that a same group containselements that are similar (behave similarly, share com-mon features, ...), whereas elements of two differentgroups are different (have different features, ...). Var-ious modularity measures have been designed to eval-uate the correctness of a community structure for agiven network. Most of these measures have been de-signed on simple unweighted networks, and some ofthem have been extended to be applied on weightedgraphs. But it may happen that this generalisationpresents a lack of connection with the native philos-ophy of the measure, making hard to know in whatextend it still evaluates the correctness of communitystructures. We propose to revisit some similarity mea-sures generalisation, or validate an existing one.

There exists a matrix transformation that has prop-erties that may highlight the community structure ofa network if applied on the network adjacency ma-trix: the doubly-stochastic scaling [3]. This scaling isinteresting for community detection because:

• It leverages small and big communities by givingto the first higher edge weights.

• If a network has a flow—i.e. an imbalance be-tween edges that enter and leave a community—the doubly-stochastic scaling erases the imbal-ance and highlights the community.

• The scaled graph is 1-regular—all its node de-grees are 1.

In our study, we hence use the doubly-stochastic scal-ing of the adjacency matrix as a preprocessing forthe Louvain’s algorithm [1]. This algorithm has beenproved to be one of the best community detection al-gorithms in the recent study [6]. A generic version ofthis algorithm proposed in [2] allows the user to max-imise some other modularities. We use this versionon some well-known benchmark test cases [5, 4] inorder to first validate our modularity measures gen-eralisations, then to compare the efficiency of thesemeasures, and finally to show the added value of thedoubly-stochastic scaling

References

[1] V. D. Blondel, J.-L. Guillaume, R. Lambiotte, andE. Lefebvre, Fast unfolding of communities in largenetworks, Journal of Statistical Mechanics: Theoryand Experiment, 2008 (2008), p. P10008.

[2] R. Campigotto, P. Conde-Cespedes, and J. Guil-laume, A generalized and adaptive method for com-

munity detection, CoRR, abs/1406.2518 (2014).

[3] P. A. Knight, D. Ruiz, and B. Ucar, A symmetrypreserving algorithm for matrix scaling, SIAM Jour-nal on Matrix Analysis and Applications, 35 (2014),pp. 931955.

[4] A. Lancichinetti and S. Fortunato, Benchmarks fortesting community detection algorithms on directedand weighted graphs with overlapping communities,Phys. Rev. E, 80 (2009), p. 016118.

[5] A. Lancichinetti, S. Fortunato, and F. Radicchi,Benchmark graphs for testing community detectionalgorithms, Physical Review E, 78 (2008), p. 046110.

[6] Z. Yang, R. Algesheimer, and C. Tessone, A com-parative analysis of community detection algorithmson artificial networks, Scientific Reports, 6 (2016).

Multigrid-based augmented block-Cimminomethod

Philippe Leleux (CERFACS) & Dr. Daniel Ruiz(IRIT) & Pr.Ulrich Rude (FAU, CERFACS)

The Augmented Block Cimmino Distributed Solver(ABCD Solver1) is a hybrid method designed to solvelarge sparse unsymmetric linear systems of the form:Ax = b, where A is a full row rank m× n sparse ma-trix, x is a vector of size n and b is a vector of size m.The approach is based on the block Cimmino row pro-jection method (BC) [2]. BC is applied on the systemwhich is partitioned in row blocks. Convergence rateof BC is known to be slow and we rather solve thesymmetric semi-positive definite and consistent sys-tem Hx = k obtained when considering the fix pointof the iterations. To accelerate the convergence of theblock Cimmino method, we solve instead this systemusing a stabilized block Conjugate Gradient (BCG)[3].

The solver also offers the possibility to construct a

larger system with an enlarged matrix[A CB S

]where

the numerical orthogonality between partitions is en-forced. As a result, the augmented block Cimminomethod converges in one single iteration. This resultsin a pseudo-direct method (ABCD) [1] with the so-lution depending on projections, as in BC, and on thedirect solution of a condensed system involving ma-trix S. Implementation of both ABCD and BC areavailable in the ABCD Solver package.

1http://abcd.enseeiht.fr

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For large PDE problems, we investigate extensions ofthis augmentation approach, in which we relax thestrict orthogonality between blocks so as to reducethe size of the augmentation matrices C, B, and S.The purpose is to control better the memory needsand computations. To do so, we exploit ideas fromthe multigrid framework. Assuming that we have athand several levels of grids for the system, the originalsystem is then augmented very similarly to the ABCDmethod, by enforcing the orthogonality between par-titions on a coarse level. The result is an augmentedsystem with approximated orthogonality on the finegrid level discretized operator. While in the exactABCD approach the size of the augmentation can belarge for highly connected subdomains, the new ap-proach gives a way to control explicitly this size bychoosing coarser levels of grids. This system can besolved with the classical block-Cimmino with fast lin-ear convergence for a wide range of systems comingfrom PDE problems. We demonstrate the efficiencyof the method on Helmholtz and Convection-Diffusion2D problems.

References

[1] Iain S Duff, Ronan Guivarch, Daniel Ruiz, andMohamed Zenadi. The augmented block Cimminodistributed method. SIAM Journal on Scientific Com-puting, 37(3):A1248A1269, 2015.

[2] Tommy Elfving. Block-iterative methods for con-sistent and inconsistent linear equations. NumerischeMathematik, 35(1):112, 1980.

[3] Daniel Ruiz. Solution of large sparse unsymmet-ric linear systems with a block iterative method in amultiprocessor environment. CERFACS TH/PA/9, 6,1992.

Modeling round-off error in the fast gradientmethod for predictive control

Ian McInerney & Eric C. Kerrigan & George A.Constantinides (Imperial College London)

We present a method to determine the smallest datatype for the Fast Gradient Method (FGM) to con-verge when solving a linear Model Predictive Control(MPC) problem in fixed-point arithmetic. We derivetwo models for the round-off error present in fixed-point arithmetic: a generic model and a structure-exploiting parametric model. We also propose a met-ric for measuring the amount of round-off error theFGM can tolerate before diverging. Using this met-ric and the round-off error models, we compute the

minimum number of fractional bits needed for thefixed-point data type. We show that the structure-exploiting parametric model nearly halves the num-ber of fractional bits needed to implement an exampleproblem, significantly decreasing the resource usagefor an implementation on a Field Programmable GateArray (FPGA).

Recently, MPC has grown in popularity due to its abil-ity to incorporate operating constraints in the com-putation of an optimal control action. At its core,MPC solves an optimization problem to determine thenext control action, which for the Constrained LinearQuadratic Regulator (CLQR) is a Quadratic Program(QP). This popularity has led to MPC being imple-mented on smaller processors and FPGAs that do notcontain hardware to accelerate floating-point compu-tations.

The lack of floating-point acceleration necessitates theuse of fixed-point representation, where the number isstored using a fixed number of bits separated into twosegments: integer bits and fraction bits. When imple-menting the QP solver in fixed-point, the number offractional bits must be large enough to ensure that theround-off error does not cause the FGM iterations todiverge. Prior work has shown that the convergence ofthe FGM in fixed-point is dependent upon the eigen-values of the fixed-point representation of the QP’sHessian lying in the range (0, 1).

In this work we present a new metric, called the round-ing stability margin, that uses the pseudospectrumof the Hessian to compute how much round-off er-ror can be introduced before the FGM iterations di-verge. Since the Hessian H of the QP is normal,the pseudospectrum can provide direct informationabout the eigenvalue spectrum. We compute the ε-pseudospectrum of H at the points λ ∈ 0, 1 to de-termine how large a perturbation H must undergo toshift its spectrum to include those points. This is thenused to bound the spectrum of an additive perturba-tion matrix on H that represents the round-off errors.

We present two models for the round-off error intro-duced by moving the Hessian into a fixed-point repre-sentation. The first is a generic model that finds theworst-case round-off error for any CLQR problem byusing the worst-case perturbation for every element ofH. The second is a parametric model that exploitsthe Toeplitz structure of the Hessian for Schur-stablesystems to find the cutoff diagonal beyond which allelements of H round to zero. The round-off error isthen modeled as the worst-case for all diagonals be-fore the cutoff, and as the actual element value for alldiagonals after the cutoff.

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These round-off error models are then used to com-pute the spectrum of the additive perturbation matrixrepresenting the transformation to fixed-point. Com-bining these models with the rounding stability mar-gin allows for the computation of the number of frac-tional bits required for the FGM to be convergent.We demonstrate that exploiting the structure of theMPC problem can reduce the number of fractionalbits needed by 30–45%, and allow for a reduction inhardware usage and solution time by up to 77% and25%, respectively, for an implementation of FGM ona FPGA.

A first-order system method for boundary layerproblems

Niall Madden & J.H. Adler & S. MacLachlan & L.Zikatanov (National University of Ireland Galway)

We are interested in finite-element solutions of sin-gularly perturbed problems, whose solutions featureboundary layers. There has long been concern thatfinite-element analyses are not appropriate for suchproblems, since it is often the case that the energynorms naturally associated with standard Galerkinmethods are not strong enough to represent the lay-ers present in the solution. Lin and Stynes (SINUM,2012) were among the first to fully diagnose the keyissue that, for a reaction-diffusion problem, the en-ergy norm is not correctly “balanced”. They also pro-posed a remedy, in the form of a mixed finite elementmethod, for which the associated norm is balanced.

There have been various other attempts to either provethe convergence of solutions obtained with standardmethods, but in non-induced norms, or to devise newschemes; see, e.g., Roos (Proc. BAIL 2016) for anoverview. Our interest is in the latter approach and,in particular, the method proposed by Adler et al.(IMA J. Numer. Anal., 2016). This has advantagesover the Lin and Stynes approach, including that theassociated finite-element space is simplified, and thatone can apply fast solvers.

In this talk we will examine how that framework canbe enhanced so that it may provide further advan-tages, including being extendable to other singularlyperturbed problems, and allowing natural adaptive re-finement.

References

[1] Runchang Lin and Martin Stynes. A balanced fi-nite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal., 50(5):

2729–2743, 2012.

[2] Roos, Hans-G. Error estimates in balanced normsof finite element methods on layer-adapted meshes forsecond order reaction-diffusion problems. Proc. BAIL2016, Lect. Notes Comput. Sci. Eng., Springer, 2017.

[3] James Adler, Scott MacLachlan, and Niall Mad-den. A first-order system Petrov-Galerkin discretisa-tion for a reaction-diffusion problem on a fitted mesh.IMA J. Numer. Anal., 36 (3):1281-1309, 2016.

Computing the Jordan structure of totally non-negative matrices

Nicola Mastronardi & Teresa Laudadio & Paul VanDooren (Istituto per le Applicazioni del Calcolo “M.Picone”)

Given the factorization of a totally nonnegative ma-trix A of order n, into the product

A = B1B2 · · ·Bn−2Bn−1DCn−1Cn−2 · · ·C2C1,

with Bi, CTi lower bidiagonal totally nonnegative ma-

trices and D diagonal one [1, 2], an algorithm for com-puting the size of the Jordan block associated to thezero eigenvalue was proposed in [3] with high relativeaccuracy in floating point arithmetic and O(n4) com-putational complexity.

In this talk we propose a modification of the latteralgorithm that computes the Jordan structure [4] ofA with high relative accuracy in O(n3) computationalcomplexity.

References

[1] S. M. Fallat and C. R. Johnson, Totally nonnega-tive matrices, Princeton University Press, Princeton,NJ, (2011).

[2] P. Koev, Accurate computations with totally non-negative matrices, SIAM J. Matrix Anal. Appl., 29(2007), pp. 731–751.

[3] P. Koev, Accurate Eigenvalues and Exact Zero Jor-dan Blocks of Totally Nonnegative Matrices, Numer.Math., 141, (2019), pp. 693–713.

[4] N. Mastronardi, P. Van Dooren, Computing theJordan Structure of an Eigenvalue, SIAM J. MatrixAnal. Appl., 38 (2017), pp. 949–966.

Path-following methods for the matrix numer-ical range, waves on a shear flow, and other

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parametrised eigenvalue problems

Peter Maxwell (Norwegian University of Science andTechnology)

An initial review of the path-following method in [1]for calculating the field of values boundary of a matrixis given. Thereafter, an adaptation of that algorithmto instead calculate the dispersion relation for linearsurface waves on a shear flow is presented. Possiblegeneralisations to other problem types are described.

The Johnson algorithm for calculating the field of val-ues boundary of a matrix, A, is a parametrisationof the eigenproblem formed by projection of A ontothe real axis after rotation by angle θ. By differ-entiating the (Hermitian) eigenproblem with respectto the real scalar θ and including an additional con-straint, a system of ordinary differential equations isobtained. After an initial eigenproblem solve for thedominant eigenpair, Runge–Kutta numerical integra-tion can then integrate along the boundary curve tocalculate control points. Piecewise Hermite interpola-tion yields dense output. The numerical integrationrepresents a nominal setup cost after which very manypoints on the curve can be calculated accurately atnegligible cost.

For a given background velocity flow U(z) in the hor-izontal direction at depth z and wave number k ∈[ka, kb], the dispersion relation is the curve describedby the solution phase velocity c(k) to the eigenproblemformed by the Rayleigh equation with linearised free-surface boundary condition, parametrised by k. Usu-ally, the problem amounts to finding sufficiently manypoints on the curve. Using a collocation method withtwo point boundary row-replacement strategy to dis-cretise the problem, a parametrised quadratic eigen-value problem is obtained. A similar approach as be-fore –differentiating with respect to k and forming asystem of differential equations– permits integrationalong the dispersion relation curve. This has the ben-efit of swapping many QZ decompositions on a size2N companion matrix for one QZ decomposition andsome linear solves on a size N matrix. Again, denseoutput is obtained by piecewise Hermite interpolation.The resulting path-following algorithm preserves theaccuracy in the initial collocation calculation and isasymptotically two orders of magnitude faster thanother known numerical algorithms for this problem.

Finally, possible extensions of the method and uses insolving other problems are briefly mentioned.

Reference

[1] Loisel, S. and Maxwell, P. (2018), Path-FollowingMethod to Determine the Field of Values of a Matrixwith High Accuracy, SIAM J. Matrix Anal. Appl. 39-4 (2018), pp. 1726-1749, doi: 10.1137/17M1148608.

On divergence-free methods for double-diffusionequations in porous media

Paul E. Mendez & Raimund Burger (Universidadde Concepcion), & Ricardo Ruiz-Baier (Oxford Uni-versity)

A stationary Navier-Stokes-Brinkman model coupledto a system of advection-diffusion equations serves asa model for so-called double-diffusive viscous flow inporous media in which both heat and a solute withinthe fluid phase are subject to transport and diffu-sion. The solvability analysis of these governing equa-tions results as a combination of compactness argu-ments and fixed-point theory. In addition an H(div)-conforming discretisation is formulated by a modifi-cation of existing methods for Brinkman flows. Thewell-posedness of the discrete Galerkin formulation isalso discussed, and convergence properties are derivedrigorously. Computational tests confirm the predictedrates of error decay and illustrate the applicability ofthe methods for the simulation of bacterial bioconvec-tion and thermohaline circulation problems.

Keywords: Viscous flow in porous media; doubly-diffusive problems; cross-diffusion; fixed-point theory;mixed finite element methods; a priori error estima-tion.

Mathematics Subject Classifications (2010):65N30; 76S05; 76R50.

Relative perturbation bounds for regular quad-ratic eigenvalue problem

Suzana Miodragovic & Ninoslav Truhar & Xin Liang& Peter Benner (University of Osijek)

We present new relative perturbation bounds for theeigensubspaces for the quadratic eigenvalue problemλ2Mx+λCx+Kx = 0, where M and K are nonsingu-lar Hermitian and C is any Hermitian matrix. First,we derive the sin Θ type theorems for the eigensub-spaces of the regular matrix pairs (A,B), where bothA and B are Hermitian matrices. Using a proper lin-earization and new relative perturbation bounds forregular matrix pairs (A,B), we develop correspond-

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ing sin Θ type theorems for the eigensubspaces for theconsidered regular quadratic eigenvalue problem. Ourbound can be applied to the gyroscopic systems whichwill also be shown. The obtained bounds will be illus-trated by numerical examples.

An image restoration by a stochastic model

F.Z. Nouri & M. Benseghir (Badji Mokhtar Univer-sity)

In this work, we tackle an image restoration prob-lem by an approach based on a stochastic differentialequation (SDE) with reflection. The drift and diffu-sion terms are rigorously formulated to express theconcept of the stochastic anisotropic diffusion. Forthe numerical approximation, we consider the mod-ified Euler scheme with a random stop time and adiffusion parameter depending on the image geometrystructure. The numerical results obtained with theproposed algorithm, which clearly show the improve-ment of the edges, the smoothing of the noise and theconnection of the broken lines, are very encouragingand they demonstrate its competitiveness comparedto other methods.

Matrix equations techniques for time-dependentpartial differential equations

Davide Palitta (Max Planck Institute for Dynamicsof Complex Technical Systems)

In this talk we show how the linear system stemmingfrom the all-at-once approach for the heat equationcan be recast in terms of a Sylvester matrix equationwhich naturally encodes the separability of the timeand space derivatives.

Combining timely projection techniques for the spaceoperator together with a full exploitation of the struc-ture of the discrete time derivative, we are able toefficiently solve problems with a tremendous numberof degrees of freedom while maintaining a low storagedemand in the solution process.

Such a scheme can be easily adapted to solve differenttime-dependent PDEs and several numerical resultsare reported to illustrate the potential of our novelapproach.

Parallel cross interpolation for high–precisioncomputation of high–dimensional integrals

Dmitry Savostyanov & Sergey Dolgov (Universityof Brighton)

High–dimensional integrals appear in stochastic cal-culus, mathematical finance, quantum physics, etc.Textbook quadratures applied to d–dimensional in-tegrals require nd function evaluations, which is un-feasible for d > 10 (a notoriously known curse of di-mensionality). Typically such integrals are treated us-ing Monte Carlo (random samples) or quasi-MC algo-rithms (optimal lattice), but due to their relativelyslow convergence they struggle to deliver very accu-rate results.

An alternative approach is offered by tensor prod-uct algorithms, which separate variables and hencelift the curse of dimensionality. The multivariate isreconstructed from a small adaptively chosen set ofits univariate fibers, representing its behaviour alongall dimensions. The resulting cross interpolation algo-rithm converges faster than MC and qMC and enableshigh–precision integration. A parallel version of ten-sor interpolation algorithm is based on independentexploration of all dimensions of the given problem.

The efficiency of the proposed algorithm is demon-strated for Ising integrals related to magnetic suscep-tibility of two–dimensional spin lattices. This applica-tion encourages evaluation of integrals in dimensionsup to 1000 with very high precision, eventually allow-ing physicists to conjecture analytic formulae. MonteCarlo methods are not up to the challenge, while thetensor interpolation algorithm computes integrals to100 decimal digits.

The success of tensor cross interpolation for calcula-tion of Ising integrals encourages us to try it for awider class of applications, where high precision is re-quired, e.g. multivariate probability, stochastics anddata science. We are hopeful that the proposed ten-sor cross interpolation algorithm will demonstrate fastconvergence in these applications and eventually be-come a method of choice for high–dimensional inte-gration.

What kind of tensors are compressible?

Tianyi Shi & Alex Townsend (Cornell University)

Tensors often have too many entries to be stored ex-plicitly so it is essential to compress them into datasparse formats. I will identify three methodologies

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that can be used to explain when a tensor is com-pressible. Each methodology leads to bounds on thecompressibility of certain tensors, partially explainingthe abundance of low-rank tensors in applied mathe-matics. In particular, I will focus on tensors with a so-called displacement structure, showing that solutionsto Poisson equations on tensor-product geometries arehighly compressible. As the rank bounds are construc-tive, I will develop an optimal-complexity spectrally-accurate 3D Poisson solver withO(n(log n)2(log 1/ε)2)complexity for a smooth righthand side, where n×n×n is the tensor discretization of the solution.

Minimax approximation by ridge functions, andneural networks

David E. Stewart & Palle Jorgensen (University ofIowa)

Neural networks with a single hidden layer approxi-mate unknown functions given implicitly by a largeset of labelled data. The problem that is consideredhere is the question of approximating continuous func-tions by sums of ridge functions with weight vectors ina given fixed set W . More specifically, we aim to de-termine how large is the largest unapproximable func-tion by ridge functions of this type with Lipschitz con-stant one. This should give us a better understandingof how well sums of ridge functions with fixed weightvectors can approximate general functions. This infor-mation is particularly relevant to Extreme LearningMachines which do not train all weights.

Invertible piecewise rational interpolation

Jan Van lent (UWE Bristol)

This talk presents a simple method to interpolatestrictly monotone data using a monotone rationalspline that is continuous and has continuous deriva-tives. Each piece of the spline is a rational functionof degree one. The inverse of the approximation is aspline of the same type and is obtained at the sametime. This class of monotone and easily invertiblefunctions can be useful for approximating cumulativedensity functions, for example. Previous methods forconstructing splines of this type were based on solvingnonlinear systems of equations. In this talk, a methodwill be presented that only requires solving a bidiago-nal or tridiagonal linear system. The cost of construct-ing the spline and the cost of evaluating the spline areessentially the same as for a standard quadratic or cu-bic spline. However, such standard splines are not

necessarily monotone functions for monotone data.Insuring monotonicity requires extra work and evenif the spline function is monotone, evaluating the in-verse function is not straightforward and requires ex-tra computation. In summary, the method presentedrequires similar calculations to standard splines, butin addition insures monotonicity for monotone dataand obtains the inverse spline function at no extracost.

Compression properties in rank-structuredsolvers for Toeplitz linear systems

Heather Wilber & Bernhard Beckermann (Univer-site de Lille) & Daniel Kressner (EPFL)

Toeplitz matrices are abundant in computational math-ematics, and since they are also data sparse, it isnatural to consider fast and superfast methods forsolving linear systems involving them. If a matrixT is circulant in addition to being Toeplitz, then Tis diagonalized by a unitary discrete Fourier trans-form. If T is not circulant, then applying a discreteFourier transform results in a matrix with off-diagonalblocks that are well-approximated by low rank matri-ces. Surprisingly little is known about the compress-ibility of these highly structured matrices in a theoret-ically rigorous sense, even though this compressibilityis relied upon in practice in a number of solvers. Inthis talk, we show that the compression properties ofthese matrices can be thoroughly explained as a con-sequence of their displacement structure. Our resultslead to extremely efficient displacement-based com-pression strategies that can be used to formulate fullyadaptive superfast rank-structured Toeplitz solvers.Our ideas also lead to superfast solvers for other linearsystems involving matrices with special displacementstructures.

Backward errors incurred by the CORK lin-earizations

Kuan Xu (University of Science and Technology ofChina) & Hongjia Chen (Nanchang University)

In this presentation, we show a unified treatment forthe compact rational Krylov (CORK) linearization ex-pressed in various commonly used bases, includingTaylor, Newton, Lagrange, orthogonal, and rationalbasis functions. We construct one-sided factorizationsthat relate the eigenpairs of the CORK linearizationand those of the original polynomial or rational eigen-value problem. With these factorizations, we estab-lish upper bounds in terms of norms of the coefficient

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matrices and those of the basis functions for the back-ward error of an approximate eigenpair of the originaleigenvalue problem relative to the backward error ofan approximate eigenpair of the CORK linearization.These bounds suggest a scaling strategy to reduce thebackward error of the computed eigenpairs. Numeri-cal experiments show that the actual backward errorsare successfully reduced by scaling and the errors, be-fore and after scaling, are both well bounded by thebounds.

Optimal transport based reflection inversion:uncover model below reflectors

Yunan Yang & Bjorn Engquist (New York Univer-sity)

The least-squares norm (L2) as the objective functionin reflection-dominated full waveform inversion (FWI)typically results in migration like features in the ini-tial model updates. Further iterations only marginallyimprove the velocity model.

We will show that the failure is most likely that theL2 norm is relatively insensitive to the lower frequencycomponent of the data.

Inversions using the L2 norm update the high-wavenumber components in the model first, which isgood for migration but problematic for inversion.

On the other hand, with an intrinsic “transport” prop-erty, the optimal transport based misfit functions nat-urally exploit the lower frequency component in thedata and generates low-wavenumber model updatewhich is ideal for recovering the model kinematics.

Our focus is on recovering velocity even below the re-flecting interfaces by reflections only. Successful re-covery of several layered models, as well as a realis-tic benchmark with salt inclusion, demonstrate thatthe optimal-transport FWI is able first to reconstructthe low-wavenumber structure of the model and con-sequently accelerate the model convergence.

Fast iterative solver for stochastic Galerkin fi-nite element approximations

Rawin Youngnoi & Daniel Loghin & Alex Bespalov(University of Birmingham)

Stochastic Galerkin finite element method (SGFEM)provides an efficient alternative to sampling methodsfor the numerical solution of linear elliptic PDE prob-

lems with parametric or random inputs. In order tocompute the stochastic Galerkin solution for a givenproblem, one needs to solve a large coupled system oflinear equations. Thus, an efficient iterative solver isa key ingredient of any SGFEM implementation. Inthis talk, we discuss a new preconditioning techniquefor SGFEM. We will present the available analysis andthe results of numerical experiments for the model dif-fusion problems with affine and non-affine parametricrepresentations of the coefficient. Specifically, we lookat the efficiency of the solver (in terms of iterationcounts for solving the underlying linear system) andcompare the new preconditioner with other existingpreconditioners for stochastic Galerkin matrices, suchas the mean-based preconditioner from [C. E. Powelland H. Elman, IMA J. Numer. Anal., 29: 350–375,2009] and the Kronecker product preconditioner from[E. Ullmann, SIAM J. Sci. Comput., 32(2): 923-946,2010].

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