Numerical Methods for Uncertainty Quanti cation · Perspectives of model order reduction for UQ...
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Workshop Numerical Methods for Uncertainty Quantification Hausdorff Center for Mathematics University of Bonn May 13–17, 2013 Organizers Alexey Chernov (University of Bonn) Vincent Heuveline (Heidelberg University / Heidelberg Institute for Theoretical Studies (HITS)) Fabio Nobile (EPF Lausanne)
Transcript of Numerical Methods for Uncertainty Quanti cation · Perspectives of model order reduction for UQ...
Workshop
Numerical Methods
for Uncertainty Quantification
Hausdorff Center for Mathematics
University of Bonn
May 13–17, 2013
Organizers
Alexey Chernov (University of Bonn)
Vincent Heuveline (Heidelberg University /Heidelberg Institute for Theoretical Studies (HITS))
Fabio Nobile (EPF Lausanne)
Program overview
Monday Tuesday Wednesday Thursday Friday8:00 registration8:30 opening8:40 8:40
Le Bris Stuart Cohen Matthies Sloan
9:30 9:30
Hackbusch Maday Gittelson Ghattas Marzouk
10:20 10:20coffee break Gwinner
10:50 10:50coffee break
Elman Schwab Silvester Zabaras11:20
11:40 Harbrecht
Benner Najm Le Maıtre Ernst12:10
12:30 Tempone
lunch break13:00closing
14:30
Scheichl Borzi SudretFree time
15:20 15:20Pulch Koutsourelakis and Bansmer
15:50 coffee 15:50coffee excursion. coffee
+16:20 16:20
poster session UllmannDoostan Boat
16:50Zaspel Cottone
17:10 departure17:20
PowellShapiro time 15:00. Koumoutsakos
18:00Schick 18:10
18:30
20:00dinner
Numerical Methods
for Uncertainty Quantification
Computational Science and Engineering has emerged in the last years as an extremelypowerful discipline able to simulate the behavior and evolution of complex systemsdescribed by sophisticated multiscale / multiphysics mathematical models. All this ismade possible by the exponential growth of computer power that we have witnessedin the last decades. With the increasing use of computational tools, comes also anincreasing need of a systematic assessment of the reliability of computer simulations andquantification of all sources of uncertainties associated to them, including discretizationerrors, models inadequacy, incomplete knowledge of model parameters, etc.
Uncertainty Quantification (UQ) in Computational Science and Engineering is a rel-atively new area aiming at developing theory and methods to quantitatively describethe origin, propagation, and interplay of different sources of error and uncertainty inthe analysis and prediction of the behavior of complex systems, arising for instancein engineering, biology, chemistry, geophysics, life sciences, finance, etc. UQ bridgesseveral disciplines including statistics, numerical analysis, computational sciences.
The workshop aims at showcasing different aspects related to Uncertainty Quantifica-tion in differential models and the most recent and important progresses in the fieldboth at the theoretical and computational level. The workshop will bring togetherthe leading scientists and active young researchers working on Numerical Methods forUncertainty Quantification and initiate an intensive idea exchange between variousresearch fields.
The topics of the workshop include but are not limited to
• Forward uncertainty propagation
• Uncertainty propagation in time dependent problems
• Reduced order models and low rank approximations
• Inverse uncertainty characterization
• Optimization and optimal control problems under uncertainty
Workshop Venue
All talks of the workshop will take place at the Mathematics Center, Endenicher Allee 60,in the Lipschitz Hall (Room 1.016), first floor (European counting). The poster sessionwill be organized in the Plucker Hall (Room 1.015) on Tuesday afternoon. The posterswill be available for viewing till the end of the week.
Please feel free to use the mathematics library in the ground floor. It has a largecollection of books and journals.
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Getting there
The most convenient way to get to the workshop venue from the hotels “Kurfurstenhof”,“Villa Esplanade”, “Mozart” and “Krug” is walking (about 10–15 min). Alternatively,you might take buses 604, 605, 606, 607 and 631 to the stop “Kaufmannstrasse” (infront of the Mathematics Center) or a taxi.
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Mozartstraße 1Hotel Mozart
Baumschulallee 20Hotel Kurfürstenhof
Colmantstraße 47Hotel Villa Esplanade
Sternenburgstr. 15Hotel Krug
Bonngasse 30Restaurant „Im Stiefel”
Endenicher Allee 60Workshop Venue
„Alter Zoll”(boat trip)
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Lunch and coffee breaks
Coffee will be served during the coffee breaks in the Plucker Hall (Room 1.015), nextto the Lipschitz Hall where all talks will take place. For the lunch you might go toa place of your choice. Many restaurants are located in the Clemens-August streetor in the city center. Another option is the student restaurant (exit the MathematicsCenter, cross the street and turn to the right).
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Internet access via WLAN
We offer a free Internet access via WLAN in the Workshop Venue. For this, youneed an access certificate. You should have received it via email. Alternatively, youmay connect to the open wireless network MIgast. Access data are available at theregistration desk upon request.
If you have difficulties with the Internet connection, please contact the registrationdesk or the organizers.
Excursion on Wednesday
Wednesday afternoon is free and we welcome you to join us for an excursion toKonigswinter. The excursion starts with a boat trip at 15:00. The boat departsfrom “Alter Zoll” (see the city map) and you will be expected to come there atleast 15 minutes in advance. In Konigswinter we will take a historic cog railway(www.drachenfelsbahn-koenigswinter.de) which will bring us to the top of the hillDrachenfels with a wonderful view over the Rhine valley, the Siebengebierge hill rangeand Bonn. On the way back to Konigswinter we will pass the Castle Drachenburg(www.schloss-drachenburg.de). As an alternative to the trip to Drachenfels, you mayexit the cog railway at the intermediate station at Schloss Drachenburg and visit thecastle instead. We will return to Bonn by boat departing at 18:20 from Konigswinter.
A fee depending on the number of participants applies for different parts of the excur-sion (between 6 and 10 Euro for the boot trip, cog railway or a round tour at SchlossDrachenburg respectively). We kindly ask you to inform us during the registration, orby Monday evening the latest, whether you want to join us for the excursion.
Dinner on Thursday
On Thursday evening we welcome you to join us at 20:00 for the Workshop Dinner at the“typical” restaurant Im Stiefel in the center of Bonn (Bonngasse 30). Unfortunately,due to our budget rules we are not able to cover the costs for the Dinner. Everybodywill have to pay himself / herself. The food prices (excluding drinks) range between 8and 15 Euro.
We kindly ask you to inform us during the registration, or by Monday evening thelatest, if you wish to join us for the Workshop Dinner.
Acknowledgement
The organizers would like to thank the Hausdorff Center for Mathematics for thegenerous financial support, which made this Workshop possible.
Special thanks go to Laura Siklossy, Anke Thiedemann and other members of the HIMand HCM Administration & Support teams for the strong an continuous support in alladministrative issues. Many thanks to Gunder-Lily Sievert and her team for organizingthe coffee breaks. We also thank Claudio Bierig, Anne Reinarz, Duong Pham, MichaelSchick and Lorenzo Tamellini for helping us with some practical issues before andduring the Workshop.
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Detailed program
Monday, May 13
8:40–9:30 Claude Le BrisVarious approaches for handling randomness in multiscale problems
9:30–10:20 Wolfgang HackbuschNumerical tensor calculus and elliptic PDE with stochastic coefficients
10:20–10:50 Coffee break
10:50–11:40 Howard ElmanReduced basis collocation methods for partial differential equations with randomcoefficients
11:40–12:30 Peter BennerPerspectives of model order reduction for UQ
12:30–14:30 Lunch break
14:30–15:20 Robert ScheichlMultilevel Markov chain Monte Carlo with applications in subsurface flow
15:20–15:50 Roland PulchModel order reduction for linear dynamical systems with random parameters
15:50–16:20 Coffee break
16:20–17:10 Alireza DoostanA compressive sampling approach to the solution of PDEs with high-dimensionalrandom inputs
17:10–18:00 Catherine PowellSolving saddle point formulations of elliptic PDEs on uncertain parameterised domains
18:00–18:30 Michael SchickA parallel multigrid spectral Galerkin solver for stochastic elliptic problems
Tuesday, May 14
8:40–9:30 Andrew StuartApproximate Guassian filters
9:30–10:20 Yvon MadayInterpolation through data assimilation: the generalized empirical interpolation method
10:20–10:50 Coffee break
10:50–11:40 Christoph SchwabInfinite dimensional quadrature approach to Bayesian inverse problems
11:40–12:30 Habib NajmParameter estimation with partial information
12:30–14:30 Lunch break
14:30–15:20 Alfio BorzıA Fokker-Planck-Kolmogorov control framework for stochastic processes
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Tuesday, May 14 (continued)
15:20–15:50 Phaedon-Stelios KoutsourelakisHigh-dimensional optimization in the presence of uncertainty: applications in randomheterogeneous media
15:50–16:50 Coffee break and poster session
Claudio BierigConvergence analysis of multilevel variance estimators in Multilevel Monte Carlo Meth-ods and application for random obstacle problems
Peng ChenAccurate and efficient evaluation of failure probability for system modeled by partial dif-ferential equations with random inputs
Abdul-Lateef Haji-AliAdaptive Multilevel Monte Carlo with simultaneous weak and strong error control
Ekaterina KostinaRobustification of optimal experiment designs against parameter uncertainties
Abhishek KunduIsoparametric finite element formulation for randomly parametrized dynamical systemswith random boundary topology
Duong PhamSparse spherical harmonic approximation for Dirichlet-to-Neumann equations on prolatespheroids with random loading
Francesco TeseiMultilevel Monte Carlo methods with control variate for elliptic SPDEs
16:50–17:20 Peter ZaspelKernel-based multi-GPU parallel uncertainty quantification with applications in com-putational fluid dynamics
17:20–18:10 Alexander ShapiroRisk neutral and risk averse approaches to multistage stochastic programming
Wednesday, May 15
8:40–9:30 Albert CohenBreaking the curse of dimensionality in sparse polynomial approximation of parametricPDEs
9:30–10:20 Claude Jeffrey GittelsonAdaptive stochastic Galerkin finite element methods
10:20–10:50 Coffee break
10:50–11:40 David SilvesterA posteriori error estimation for elliptic PDEs with random coefficients
11:40–12:30 Olivier Le MaıtrePolynomial chaos expansions for the approximation of uncertain stochastic model so-lutions
12:30–14:30 Lunch break
14:30–18:30 Free time and excursion (boat departure time 15:00)
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Thursday, May 16
8:40–9:30 Hermann MatthiesInverse uncertainty quantification
9:30–10:20 Omar GhattasQuantification of uncertainty for large-scale inverse wave propagation problems
10:20–10:50 Coffee break
10:50–11:40 Nicholas ZabarasA probabilistic graphical model approach to uncertainty quantification
11:40–12:30 Oliver ErnstUQ for groundwater flow
12:30–14:30 Lunch break
14:30–15:20 Bruno SudretSparse polynomial chaos expansions in engineering applications
15:20–15:50 Stephan BansmerMeasurement uncertainty of ensemble-averaged flow data
15:50–16:20 Coffee break
16:20–16:50 Elisabeth UllmannMultilevel estimation of rare events
16:50–17:20 Giulio CottoneRepresentation and treatment of random functions with divergent integer moments
17:20–18:10 Petros KoumoutsakosBayesian uncertainty quantification and propagation in molecular dynamics simulationsof nanoscale flows
Friday, May 17
8:40–9:30 Ian H. SloanQMC lattice methods for PDE with random coefficients
9:30–10:20 Youssef MarzoukBayesian data assimilation and dimension reduction with optimal maps
10:20–10:50 Joachim GwinnerVariational inequalities and equilibrium problems with uncertain data
10:50–11:20 Coffee break
11:20–12:10 Helmut HarbrechtOn multilevel quadrature for elliptic stochastic partial differential equations
12:10–13:00 Raul TemponeAnalysis and computations for linear hyperbolic PDEs with stochastic coefficients
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Monday, 8:40–9:30
Various approaches for handling randomness
in multiscale problems
Claude Le Bris
Ecole des Ponts and INRIA, Paris
The talk will overview several recent works addressing multiscale problems with a cer-tain amount of randomness. Monte-Carlo type approaches and variance issues will beexamined. Methods that as much as possible avoid to explicitly treat randomness willalso be presented. The talk is based on a series of joint works with many collaborators:X. Blanc (Paris 6) and PL. Lions (College de France), F. Legoll and other coworkersat Ecole des Ponts and INRIA.
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Monday, 9:30–10:20
Numerical tensor calculus and elliptic PDE
with stochastic coefficients
Wolfgang Hackbusch
Max-Planck-Institut fur Mathematik in den Naturwissenschaften, Leipzig
An introduction into tensor formats and tensor operations is given ([1]). We discuss thetreatment of an elliptic boundary value problem with log-normal stochastic coefficients([2]).
References
[1] W. Hackbusch, Tensor spaces and numerical tensor calculus, Springer, Berlin,2012.
[2] M. Espig, W. Hackbusch, A. Litvinenko, H. G. Matthies, and P. Wahnert, Effi-cient low-rank approximation of the stochastic Galerkin matrix in tensor formats,Computers and Mathematics with Applications, published on-line.
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Monday, 10:50–11:40
Reduced basis collocation methods for partial
differential equations with random coefficients
Howard Elman
University of Maryland
Joint work with:Qifeng Liao
The sparse grid stochastic collocation method is a new method for solving partialdifferential equations with random coefficients. However, when the probability spacehas high dimensionality, the number of points required for accurate collocation solutionscan be large, and it may be costly to construct the solution. We show that this processcan be made more efficient by combining collocation with reduced-basis methods, inwhich a greedy algorithm is used to identify a reduced problem to which the collocationmethod can be applied. Because the reduced model is much smaller, costs are reducedsignificantly. We demonstrate with numerical experiments that this is achieved withessentially no loss of accuracy.
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Monday, 11:40–12:30
Perspectives of model order reduction for UQ
Peter Benner
Max Planck Institute for Dynamics of Complex Technical SystemsSandtorstr. 1
39016 MagdeburgGermany
We will discuss the usage of model order reduction (MOR) methods for uncertaintyquantification of dynamical systems. In particular, parametric MOR is a techniquethat is applicable directly to non-intrusive UQ methods such as Monte Carlo (MC)and (spectral) collocation. In such a setting, a reduced-order model is computed thatpreserves (deterministic or stochastic) parameters as symbolic quantities, so that, e.g.,MC simulations can be run with the reduced-order model instead of the full model,thereby accelerating the simulation drastically. We will discuss different approaches forparametric MOR based either on rational interpolation or reduced-basis approachesand their use in UQ. As a an example application, we will consider electromagneticsimulations for nanosystem design, where inaccuracies in the production process leadto variability in geometry like width of transmission lines, and certain statistical quan-tities about the induced electromagnetic fields are needed to predict the percentage ofwastage.
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Monday, 14:30–15:20
Multilevel Markov chain Monte Carlo with
applications in subsurface flow
Robert Scheichl
Department of Mathematical Sciences, University of Bath,Claverton Down, Bath BA2 7AY, United Kingdom
Joint work with:Christian Ketelsen (Lawrence Livermore National Laboratory) and
Aretha L. Teckentrup (University of Bath)
One of the key tasks in many areas of subsurface flow, most notably in radioactivewaste disposal and oil recovery, is an efficient treatment of data uncertainties andthe quantification of how these uncertainties propagate through the system. Similarquestions arise also in other areas of science and engineering such as weather andclimate prediction or manufacturing. The mathematical challenge associated with thesequestions is the solution of high-dimensional quadrature problems with integrands thatinvolve the solution of PDEs with random coefficients. Due to the heterogeneity of thesubsurface and the complexity of the flow, each realisation of the integrand is very costlyand so it is paramount to make existing uncertainty quantification tools more efficient.Recent advances in Monte Carlo type methods, based on deterministic, Quasi–MonteCarlo sampling rules [2, 3] and multilevel approaches [1, 5], provide unprecedentedopportunities for accurate uncertainty analyses in realistic three-dimensional subsurfaceflow applications. In this talk we show how the multilevel framework can also beextended to Monte Carlo Markov chain (MCMC) methods, allowing for uncertaintyreduction by conditioning on measured data via Bayesian techniques. In particular,we present a multilevel Metropolis-Hastings algorithm including a complete analysis[4]. As in the earlier work, the analysis reduces to classical questions in regularityand finite element approximation error analysis. We can show significant gains overthe standard Metropolis-Hastings estimator for a typical model problem in subsurfaceflow. Numerical experiments confirm the analysis and demonstrate the effectivenessof the method with consistent reductions of a factor of O(10–50) in the ε-cost fortolerances ε around 10−3.
References
[1] K.A. Cliffe, M.B. Giles, R. Scheichl and A.L. Teckentrup Multilevel Monte Carlomethods and applications to elliptic PDEs with random coefficients, Computingand Visualization in Science, 14 (2011), no. 1, 3–15.
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[2] I.G. Graham, F.Y. Kuo, D. Nuyens, R. Scheichl and I.H. Sloan Quasi-Monte Carlomethods for elliptic PDEs with random coefficients and applications, Journal ofComputational Physics, 230 (2011), no. 10, 3668–3694.
[3] I.G. Graham, F.Y. Kuo, J.A. Nichols, R. Scheichl, C. Schwab and I.H. Sloan,Quasi–Monte Carlo finite element methods for elliptic PDEs with lognormal randomcoefficients, in preparation (2013).
[4] C. Ketelsen, R. Scheichl and A.L. Teckentrup, A hierarchical multilevel Markovchain Monte Carlo algorithm with applications to uncertainty quantification in sub-surface flow, submitted March 29 (2013), Preprint arXiv:1303.7343, available atarXiv.org.
[5] A.L. Teckentrup, R. Scheichl, M.B. Giles and E. Ullmann, Further analysis of mul-tilevel Monte Carlo methods for elliptic PDEs with random coefficients, NumerischeMathematik, published online March 12 (2013), DOI 10.1007/s00211-013-0546-4.
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Monday, 15:20–15:50
Model order reduction for linear dynamical
systems with random parameters
Roland Pulch
Lehrstuhl fur Angewandte Mathematik und Numerische Analysis,Bergische Universitat Wuppertal, Gaußstraße 20, D-42119 Wuppertal, Germany.
Mathematical modelling of technical applications often yields large systems of ordinarydifferential equations or differential algebraic equations. Typical examples are mod-els for electric circuits, mechanical multibody systems and chemical reactions. Theinvolved physical parameters often exhibit uncertainties due to measurement errorsor imperfections of an industrial manufacturing. A substitution of the parameters byrandom variables allows for an uncertainty quantification (UQ). Hence the solutionsbecome time-dependent random processes.
We consider linear dynamical systems with random parameters, where both the dimen-sion of the state space and the number of random variables is large. Efficient methodsfor model order reduction (MOR) already exist in case of deterministic linear dynam-ical systems with a huge state space, see [1, 2]. Often these techniques use a transferfunction, which describes the input-output behaviour of the system. We apply andmodify MOR methods to reduce the complexity of the dynamical systems in UQ. Thefocus is on moment-matching approaches and Krylov subspace techniques.
The numerical methods are based on expansions of the state variables and the out-put variables within the generalised polynomial chaos. On the one hand, the originalsystems have to be resolved many times in a non-intrusive technique. Therein, param-eterised model order reduction (PMOR) saves computational effort, which indicatesan efficient application in UQ, see [3] for this idea. On the other hand, an intrusivemethod yields a huge coupled linear dynamical system by a stochastic Galerkin tech-nique. Now the coupled system has to be solved just once. In case of differentialalgebraic equations, the analytical and numerical properties of the system are charac-terised by its index. Hence attention has to be paid to a possible change of the indexin the numerical methods, see [4]. We present numerical simulations of examples forelectric networks.
References
[1] A. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM Publ., 2005.
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[2] P. Benner, M. Hinze, E.J.W. ter Maten (eds.), Model Reduction for Circuit Simu-lation, Lect. Notes in Electr. Engng., vol. 74, Springer, 2011.
[3] E.J.W. ter Maten, R. Pulch, W.H.A. Schilders, and H.H.J.M. Janssen, Efficientcalculation of uncertainty quantification, to appear in: Proceedings ECMI 2012,Springer.
[4] R. Pulch, Polynomial chaos for linear differential algebraic equations with randomparameters, International Journal for Uncertainty Quantification, 1 (2011), no. 3,223–240.
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Monday, 16:20–17:10
A compressive sampling approach to the solution of
PDEs with high-dimensional random inputs
Alireza Doostan
Aerospace Engineering Sciences, University of Colorado, Boulder
Joint work with:Jerrad Hampton (University of Colorado, Boulder),
Ji Peng (University of Colorado, Boulder)
We extend ideas from compressive sampling techniques to Polynomial Chaos (PC)approximation of PDEs with high-dimensional random inputs. This method requiresrandom sampling of the solution of interest, which can be done by using any legacycode for the deterministic problem as a black box.
We primarily focus on the convergence of the method and present bounds on the num-ber of random samples needed for successful reconstruction of solution expanded inLegendre and Hermite PC basis. The convergence is in probability (with probabilis-tic bounds) as a consequence of sparsity of solution and a concentration of measurephenomenon on the empirical correlation between PC basis functions.
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Monday, 17:10–18:00
Solving saddle point formulations of elliptic PDEs
on uncertain parameterised domains
Catherine E. Powell
School of Mathematics, University of Manchester
Joint work with:Andrew Gordon (University of Manchester)
Mixed finite element discretisations of systems of PDEs often lead to saddle pointvariational problems. Of particular interest are problems of the form: find (u, p) ∈Vh ×Wh such that
a(u, v) + b(v, p) = `(v) ∀ v ∈ Vh ⊂ V,
b(u,w) = q(w) ∀w ∈ Wh ⊂ W,
where V and W are Hilbert spaces, a(·, ·) : V × V → R and b(·, ·) : V ×W → R. Aunique solution exists if the bilinear forms are bounded, a(·, ·) is coercive and V andW are compatible. The latter condition is commonly know as an inf-sup conditionand when expressed in discrete form, can inform the design of optimal preconditionersfor the associated linear systems of equations. When the underlying PDE system hasrandom data, analysing well-posedness and designing optimal preconditioners can becomplicated. Various levels of difficulty are encountered, depending on the source ofthe uncertainty.
We focus on the numerical solution of elliptic boundary-value problems on uncertaintwo-dimensional domains via the fictitious domain method [1]. This leads to a saddlepoint problem in which b(·, ·) is affected by uncertainty. Assuming that the domaincan be described by a finite number of independent random variables, discretizationis achieved by a stochastic collocation mixed finite element method. We discuss theefficient iterative solution of the resulting sequence of indefinite linear systems andintroduce a novel and efficient preconditioner. The challenging task is constructing amatrix that provides a robust approximation to a discrete representation of a tracespace norm on an uncertain boundary.
References
[1] C. Canuto and T. Kozubek, A fictitious domain approach to the numerical solutionof PDEs in stochastic domains, Numer. Math., 107 (2007), 257–293.
[2] C.E. Powell and A. Gordon, A preconditioner for fictitious domain formulations ofelliptic PDEs on uncertain parameterized domains, In preparation.
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Monday, 18:00–18:30
A parallel multigrid spectral Galerkin solver
for stochastic elliptic problems
Michael Schick
Heidelberg Institute for Theoretical Studies (HITS), Heidelberg, Germany
Joint work with:Vincent Heuveline
(Heidelberg Institute for Theoretical Studies (HITS), Germany / InterdisciplinaryCentre for Scientific Computing (IWR), Heidelberg University, Germany)
We introduce a parallel multigrid method for spectral Galerkin projected systems usingPolynomial Chaos expansions, which utilizes the hierarchical multilevel structure withrespect to the polynomial degree. The smoothing of high frequency errors is carriedout by employing the mean based preconditioner. Furthermore, we develop an efficientload balancing strategy for the parallel computation of the matrix vector product usingdistributed memory, which allows for a decoupled application of the restriction andprolongation operators in the multigrid scheme. A convergence analysis for the twolevel variant is provided and the efficiency and parallel scalability of the numericalmethod is demonstrated on a stochastic Poisson benchmark problem with increasingcomplexity up to one billion degrees of freedom.
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Tuesday, 8:40–9:30
Approximate Guassian filters
Andrew Stuart
Warwick University
The problem of effectively combining data with a mathematical model constitutes a ma-jor challenge in applied mathematics. It is particular challenging for high-dimensionaldynamical systems where data is received sequentially in time and the objective is toestimate the system state in an on-line fashion; this situation arises, for example, inweather forecasting. The sequential particle filter is then impractical and ad hoc filters,which employ some form of Gaussian approximation, are widely used. Prototypical ofthese ad hoc filters is the 3DVAR method, with the extended Kalman filter (ExKF)and ensemble Kalman filter (EnKF) arising as important generalizations. The goal ofthis talk is to describe analysis of these filters when applied to dissipative nonlinear sys-tems with quadratic energy-conserving nonlionearities. The Lorenz ’63 and ’96 models,together with the Navier-Stokes equation on a 2D torus, provide explicit examples.
For further reading see:
http://arxiv.org/abs/1212.4923
http://arxiv.org/abs/1210.1594
http://arxiv.org/abs/1203.5845
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Tuesday, 9:30–10:20
Interpolation through data assimilation:
the generalized empirical interpolation method
Yvon Maday
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris,France, Institut Universitaire de France and Division of Applied Mathematics, Brown
University, Providence RI, USA.
Joint work with:Olga Mula (CEA Saclay, DEN/DANS/DM2S/SERMA/LLPR
91191 Gif-Sur-Yvette CEDEX - Franceand LRC MANON, Laboratoire de Recherche Conventionnee
CEA/DEN/DANS/DM2S and UPMC-CNRS/LJLL.),Gabriel Turinici (CEREMADE, Universite Paris Dauphine
Place du Marechal de Lattre de Tassigny75016 Paris, France)
The extension of the lagrangian interpolation process is an old problem that is stillcurrently subject to active research (see, e.g. [6] and also the activity concerning thekriging [3], [4] in the stochastic community). While this classical method approximatesgeneral functions by finite sums of well chosen, linearly independent interpolating func-tions (e.g. polynomial functions) and the optimal location of the interpolating pointsis well documented (and completely solved in one dimension), the question remains onhow to approximate general functions by finite expansions involving general interpo-lating functions and how to optimally select the interpolation points in this case.
One step in this direction is the Empirical Interpolation Method (EIM, [1], [2], [6])that has been developed in the broad framework where the functions f to approximatebelong to a compact set F of a Banach space X . The structure of F is supposed tomake any f ∈ F be approximable by finite expansions of small size. In particular, thisis the case when the Kolmogorov n−width of F in X is small.
The Empirical Interpolation Method builds simultaneously the set of interpolatingfunctions and the associated interpolating points by a greedy selection procedure (see[1]). A recent generalization of this interpolation process consists in replacing theevaluation at interpolating points by application of a class of interpolating continuouslinear functions chosen in a given dictionary Σ ⊂ L(F ) and this gives rise to the so-called Generalized Empirical Interpolation Method (GEIM, [5]). This newly developedmethod, is in lines with the acquisition of data through “real” measures.
We shall present the basics of this approach, some theoretical results about the con-vergence of the interpolation process compared to the Kolmogorov n−width of F :
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dn(F,L2(Ω)) when it has a polynomial or an exponential decreasing behavior. Weshall also present preliminary work on the use of this approach for data assimilation incase some noise polute the data.
References
[1] Barrault, M. and Maday, Y. and Nguyen, N.C. Y. and Patera, A.T., An empiricalinterpolation method: Application to efficient reduced-basis discretization of partialdifferential equations., C. R. Acad. Sci. Paris, Serie I., 339 (2004), 667–672.
[2] Grepl, M.A. and Maday, Y. and Nguyen, N.C. and Patera, A.T., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations., M2AN(Math. Model. Numer. Anal.), 41(3) (2007), 575-605.
[3] Jack P.C. Kleijnen and Wim C.M. van Beers, Robustness of Kriging when inter-polating in random simulation with heterogeneous variances: Some experiments,European Journal of Operational Research, 165 (2005), no. 3, 826 - 834.
[4] Heping Liu and Saeed Maghsoodloo, Simulation optimization based on TaylorKriging and evolutionary algorithm, Applied Soft Computing, 11 (2011), no. 4,3451 - 3462.
[5] Maday, Y. and Mula, O., A generalized empirical interpolation method: applicationof reduced basis techniques to data assimilation, Analysis and Numerics of PartialDifferential Equations, XIII (2013), 221-236.
[6] Maday, Y. and Nguyen, N.C. and Patera, A.T. and Pau, G.S.H., A general mul-tipurpose interpolation procedure: the magic points, Commun. Pure Appl. Anal.,8(1) (2009), 383–404.
23
Tuesday, 10:50–11:40
Infinite dimensional quadrature approach
to Bayesian inverse problems
Christoph Schwab
SAMETH Zurich, ETH Zentrum, HG G57.1
CH 8092 Zurich, Switzerland
Joint work with:Claudia Schillings (SAM, ETH Zurich)
Markov Chain Monte-Carlo (MCMC) methods are currently the workhorse for thenumerical solution of inverse problems in the presence of uncertain systems and noisydata.
The task is to compute statistical estimates for “most probable” responses of uncer-tain systems conditioned to given, noisy data. Problem classes considered are PDEsand large ODE systems with uncertain coefficients and parameters. The Bayesianframework for response prediction is reduced to a parametric, deterministic infinite-dimensional quadrature problem. The integrand functions (posterior densities condi-tional on given data) are shown to belong to classes of functions with sparse polynomialexpansions.
We present new sparsity results on the density of the Bayesian posterior. We considerseveral quadrature approaches for the numerical evaluation of the Bayesian expecta-tion, given data, which can in principle exploit the sparsity in the density of the pos-terior distribution, and which result in performance that is superior to that of MCMCmethods in terms of the number M of solutions of the forward problem.
Sparse, adaptive tensor quadrature Finite Element algorithms are proposed and ana-lyzed; they are shown to exhibit dimension-independent rates of convergence which aregoverned only by the sparsity models of the prior distributions in the Bayesian estimateand are shown, in particular, to be superior to the MCMC rate of M−1/2. Supportedby ERC AdG 247277.
References
[1] Ch. Schwab and A.M. Stuart, Sparse deterministic approximation of Bayesianinverse problems , Inverse Problems, 28 (2012), 045003.
[2] Cl. Schillings and Ch. Schwab, Sparse, adaptive Smolyak algorithms for Bayesianinverse problems, Report 2012-37, SAM ETH Zurich (in review)
24
Tuesday, 11:40–12:30
Parameter estimation with partial information
Habib N. Najm
Sandia National Laboratories, Livermore, CA, USA
Joint work with:Robert Berry, Kenny Chowdhary, Cosmin Safta, Khachik Sargsyan
and Bert DebusschereSandia National Laboratries, Livermore, CA, USA
Models of physical systems generally involve parameters that are determined from em-pirical measurements. Given such measurement data, Bayesian inference can be used toestimate the posterior density on parameters of interest. In many practical situations,however, relevant raw data is simply not available. Quite commonly, such data, fromlegacy experiments, is discarded after reporting summary statistics on the data itselfor on parameters/quantities of interest. Therefore, subsequent efforts at estimationof the full posterior density are faced with a context where data is not available, butone has to contend with processed data products containing partial information andproviding summary statistics on the parameters of interest. In this talk, I will outlinea Bayesian procedure to deal with this challenge, providing a posterior density thatis consistent with available information. This “Data Free Inference” method [1, 2] isbased on maximum entropy arguments. It provides means for taking available informa-tion, including summary statistics, fit model and other available experimental details,in the absence of raw data, and making it explicit in a joint posterior density on theparameters of interest. I will describe the basics of the method, and its applicationwith different types of available information, including summary statistics on the data,the parameter posterior, and the fit-model output pushed-forward posterior.
References
[1] R.D. Berry, H.N. Najm, B.J. Debusschere, H. Adalsteinsson, Y.M. Marzouk, Data-free inference of the joint distribution of uncertain model parameters, Journal ofComputational Physics, 231 (2012), 2180–2198.
[2] H.N. Najm, R.D. Berry, C. Safta, K. Sargsyan, and B.J. Debusschere, Data FreeInference of Uncertain Parameters in Chemical Models, International Journal forUncertainty Quantification, in press (2013).
25
Tuesday, 14:30–15:20
A Fokker-Planck-Kolmogorov control framework
for stochastic processes
Alfio Borzı
Institut fur Mathematik, Universitat Wurzburg, Emil-Fischer-Strasse 30, 97074Wurzburg, Germany
Joint work with:Mario Annunziato (Dipartimento di Matematica, Universita degli Studi di Salerno,
Italia)
An efficient framework for the optimal control of probability density functions (PDF)of multidimensional stochastic processes and piecewise deterministic processes is pre-sented. This framework is based on Kolmogorov-Fokker-Planck-type equations thatgovern the time evolution of the PDF of stochastic processes and piecewise determin-istic processes. In this approach, the control objectives may require to follow a givenPDF trajectory or to minimize an expectation functional. Theoretical results concern-ing the forward and the optimal control problems are provided. In the case of stochastic(Ito) processes, the Fokker-Planck equation is of parabolic type and it is shown thatunder appropriate assumptions the open-loop bilinear control function is unique. In thecase of piecewise deterministic processes (PDP), the Fokker-Planck equation consistsof a first-order hyperbolic system. Discretization schemes are discussed that guaranteepositivity and conservativeness of the forward solution. The proposed control frame-work is validated with multidimensional biological, quantum mechanical, and financialmodels.
This work is in collaboration with Mario Annunziato (U. Salerno) and it is supportedin part by the EU Marie Curie International Training Network Multi-ITN STRIKEProjekt ’Novel Methods in Computational Finance’ and in part by the ESF OPTPDEProgramme.
References
[1] M. Annunziato and A. Borzı, Optimal control of probability density functions ofstochastic processes, Mathematical Modelling and Analysis 15 (2010) 393–407.
[2] M. Annunziato and A. Borzı, A Fokker-Planck control framework for multidimen-sional stochastic processes, Journal of Computational and Applied Mathematics237 (2013) 487–507.
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[3] M. Annunziato and A. Borzı, Fokker-Planck-based control of a two-level open quan-tum system, Mathematical Models and Methods in Applied Sciences (M3AS), toappear.
[4] M. Annunziato and A. Borzı, Optimal control of piecewise deterministic processes,submitted.
27
Tuesday, 15:20–15:50
High-dimensional optimization in the presence of
uncertainty: applications in random
heterogeneous media
P.S. Koutsourelakis
Continuum Mechanics Group, Faculty of Mechanical EngineeringTechnische Universitat Munchen
This talk is concerned with the optimization/design/control of complex systems char-acterized by high-dimensional uncertainties and design variables. Expected utilitymaximization involves maximizing, with respect to design/control parameters, an inte-gral expression with respect to the random variables. The problem is difficult becausethe integration is embedded in the maximization and has to be evaluated many timesfor different design parameters. Furthermore each of these evaluations require severalcalls to the forward model (e.g. discretized system of PDEs). We are especially con-cerned with problems relating to random heterogeneous materials where uncertaintiesarise from the stochastic variability of their properties. In particular, we formulatetopology optimization problems to account for the design of truly random composites.Deterministic optimization tools cannot be directly extended to stochastic settings[1]. Traditional stochastic optimization techniques result in exuberant computationalcost and fail to identify multiple local optima [2]. In this work we reformulate theproblem as one of probabilistic inference [3] and employ sampling tools suitable forhigh-dimensions [4]. The methodological advances proposed in this paper allow theanalyst to identify several local maxima that provide important information with re-gards to the robustness of the design [5]. We further propose statistical learning toolsthat exploit information from approximate models and can ultimately lead to furtherreductions in computational cost.
References
[1] J. Nocedal and S.J. Wright. Numerical Optimization Springer, 1999.
[2] H. Robbins and S. Monro. A stochastic approximation method. The Annals ofMathematical Statistics, 22(3):400–407, 1951.
[3] P. Muller. Simulation based optimal design, in Proceedings of the Sixth ValenciaInternational Meeting, 1998, pp. 323–341.
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[4] H. Kuck, N. de Freitas, and A. Doucet. SMC Samplers for Bayesian OptimalNonlinear Design, in Nonlinear Statistical Signal Processing Workshop (NSSPW),2006.
[5] R. Sternfels and P.S. Koutsourelakis. Stochastic design and control in randomheterogeneous materials. Journal of Multiscale Computational Engineering. 9(4):425-443. 2011
29
Tuesday, 16:50–17:20
Kernel-based multi-GPU parallel uncertainty
quantification with applications in
computational fluid dynamics
Peter Zaspel
Institut fur Numerische Simulation, Universitat Bonn
We propose to use a non-intrusive stochastic collocation method with radial-symmetrickernel basis functions for the application of large-scale uncertainty quantification incomputational fluid dynamics. This approach promises to give high convergence ratesin cases of smooth response surface functions. Furthermore it is very modular thusone can freely choose and combine quadrature methods, collocation points and theunderlying PDE discretization, which gives us great flexibility. Due to its non-intrusivenature, we can easily apply the method to a variety of applications. This includes, butis not limited the application of (two-phase) incompressible Navier-Stokes equations.
A special attention is given to the efficient implementation and application of uncer-tainty quantification with modern highly parallel GPU hardware. Since large-scale UQproblems are hardly tractable with standard parallel and even more single-threadednumerical software, every part of the implemented system is fully (multi-)GPU paral-lelized and will thus run with optimal performance.
In our talk, we will introduce the applied method with applications to a number ofquantities of interest based on full fluid flow fields. The GPU parallelization will bebriefly outlined. Also, we want to discuss the compuation of Karhunen-Love decompo-sitions for large-scale flow problems which involves eigenvalue decompositions of densematrices with 100K and more rows. We finally end up giving some outlook on poten-tial further developments which might include multi-level Monte Carlo methods or thesparse grid combination technique.
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Tuesday, 17:20–18:10
Risk neutral and risk averse approaches to
multistage stochastic programming
Alexander Shapiro
Georgia Institute of Technology, Atlanta, USA
In many practical situations one has to make decisions sequentially based on dataavailable at the time of the decision and facing uncertainty of the future. This leads tooptimization problems which can be formulated in a framework of multistage stochas-tic programming. In this talk we consider risk neutral and risk averse approaches tomultistage stochastic programming. We discuss conceptual and computational issuesinvolved in formulation and solving such problems. As an example we give numer-ical results based on the Stochastic Dual Dynamic Programming method applied toplanning of the Brazilian interconnected power system.
31
Wednesday, 8:40–9:30
Breaking the curse of dimensionality in sparse
polynomial approximation of parametric PDEs
Albert Cohen
Universite Pierre et Marie Curie, Paris
Joint work with:Abdellah Chkifa (Universite Pierre et Marie Curie, Paris),
Christoph Schwab (ETH, Zurich)
The numerical approximation of parametric partial differential equations D(u, y) = 0is a computational challenge when the dimension d of of the parameter vector y islarge, due to the so-called curse of dimensionality. It was recently shown in [1, 2]that, for a certain class of elliptic PDEs with diffusion coefficients depending on theparameters in an affine manner, there exists polynomial approximations to the solutionmap y 7→ u(y) with an algebraic convergence rate that is immune to the growth in theparametric dimension d, in the sense that it holds in the case d =∞. This analysis ishowever heavily tied to the linear nature of the considered diffusion PDE and to theaffine parameter dependence of the operator.
In this talk, we present a general strategy introduced in [3] in order to establish similarresults for parametric PDEs that do not necessarily fall in this category. Our approachis based on building an analytic extension z 7→ u(z) of the solution map on certaintensor product of ellipses in the complex domain, and using this extension to estimatethe Legendre coefficients of u. The varying radii of the ellipses in each coordinate zjreflect the anisotropy of the solution map with respect to the corresponding paramet-ric variables yj. This allows us to derive algebraic convergence rates for tensorizedLegendre expansions in the case d = ∞. We also show that such rates are preservedwhen using certain interpolation procedures, which is an instance of a non-intrusivemethod. As examples of parametric PDE’s that are covered by this approach, weconsider (i) diffusion equations with uniformly elliptic coefficients that depend on yin a non-affine manner, (ii) nonlinear monotone elliptic PDE’s, (iii) linear, parabolicparametric PDE’s, with coefficients parametrized by y, and (iv) elliptic equations seton a domain that is parametrized by the vector y. While for the first example (i) thevalidity of the analytic extension follows by straightforward arguments, we give gen-eral strategies that allows us to derive it in a simple abstract way for examples (ii) to(iv), in particular based on the holomorphic version of the implicit function theoremin Banach spaces. We expect that this approach can be applied to a large variety ofparametric PDEs, showing that the curse of dimensionality can be overcome undermild assumptions.
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References
[1] A. Cohen, R. DeVore and C. Schwab Convergence rates of best N-term Galerkinapproximations for a class of elliptic sPDEs, J. FoCM, 10 (2010), 615-646.
[2] A. Cohen, R. DeVore and C. Schwab Analytic regularity and polynomial approxi-mation of parametric and stochastic PDE’s, Analysis and Applications, 9 (2011),11–47.
[3] A. Chkifa, A. Cohen and C. Schwab Breaking the curse of dimensionality in sparsepolynomial approximation of parametric PDEs, Preprint Laboratoire J.-L. Lions,UPMC, 2013.
33
Wednesday, 9:30–10:20
Adaptive stochastic Galerkin
finite element methods
Claude J. Gittelson
Purdue University
Joint work with:Martin Eigel (WIAS),
Christoph Schwab (ETH Zurich),Elmar Zander (TU Braunschweig)
Solutions of a class of random elliptic boundary value problems admit efficient ap-proximations by polynomials on the parameter domain, with each coefficient being aspatially-dependent function. Since it is not clear a priori which modes are most signif-icant, we employ an adaptive strategy to construct suitable sets of active polynomialchaos modes along with finite element spaces in which to approximate the correspond-ing coefficients.
Our algorithm [1] follows the structure of adaptive finite element methods and makesuse of standard a posteriori error estimators to guide spatial refinements. Separaterefinements are permitted for each coefficient, resulting in a sparse approximation witha high resolution of the coefficients deemed most important and a sufficient but coarserresolution of secondary coefficients. The use of error indicators to gauge the error onthe parameter domain results in substantial computational savings compared to pre-vious approaches [2, 3] based on adaptive wavelet algorithms, which require additionalrefinements either to approximate the residual or to guide the refinement process.
Recent work [3, 4] suggests that for high-order spatial discretizations, it may sufficeto use a single spatial mesh for all active coefficients, effecting simplifications in theimplementation and theoretical analysis of the adaptive algorithm as well as poten-tial computational savings arising from the avoidance of projections between differentspatial meshes.
References
[1] Martin Eigel, Claude J. Gittelson, Christoph Schwab and Elmar Zander, Adaptivestochastic Galerkin FEM, SAM Report 2013-1, ETH Zurich, submitted.
[2] Claude J. Gittelson, An adaptive stochastic Galerkin method for random ellipticoperators, to appear in Mathematics of Computation.
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[3] Claude J. Gittelson, Convergence rates of multilevel and sparse tensor approxima-tions for a random elliptic PDE, submitted.
[4] Claude J. Gittelson, High-order methods as an alternative to sparse tensor productsfor stochastic Galerkin FEM, submitted.
35
Wednesday, 10:50–11:40
A posteriori error estimation for elliptic PDEs
with random coefficients
David Silvester
University of Manchester
Joint work with:A. Bespalov, C. Powell (University of Manchester, UK)
Stochastic Galerkin finite element methods can be used numerically solve elliptic PDEproblems with correlated random data. Our specific strategy combines conventional (h-) finite element approximation on the spatial domain with spectral (p-) approximationon a finite-dimensional manifold in the (stochastic) parameter space.
For approximations relying on low-dimensional manifolds in the parameter space, stan-dard stochastic Galerkin FEM has superior convergence properties to standard sam-pling techniques. On the other hand, the desire to incorporate more and more param-eters (random variables) together with the need to use high-order polynomial approx-imations in these parameters inevitably generates very high dimensional discretisedsystems. This in turn means that adaptive algorithms are needed to efficiently con-struct approximations and fast and robust linear algebra techniques are needed to solvethe discretised problems.
Both strands will be discussed in the talk. We outline the issues involved in a posteriorierror analysis of computed solutions and present a practical a posteriori estimator forthe approximation error. Our residual-based error estimator is computed by using aparameter-free part of the underlying differential operator and by exploiting the tensor-product structure of the approximation space. We prove that our error estimator isreliable and efficient. We also discuss different strategies for enriching the discrete spaceand prove two-sided estimates of the error reduction for the corresponding enhancedapproximations. These give computable estimates of the error reduction that dependonly on the problem data and the original approximation. The effectiveness of theproposed solution strategy will be demonstrated numerically.
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Wednesday, 11:40–12:30
Polynomial chaos expansions
for the approximation of uncertain
stochastic model solutions
Olivier Le Maıtre
LIMSI-CNRS, Orsay, FranceMechanical Engineering and Materials Science, Duke, North Carolina
Joint work with:Omar Knio (Mechanical Engineering and Materials Science, Duke, Durham, NC)
Recent results concerning the analysis of stochastic systems with parametric uncer-tainty will be reported. The typical situation considered is the case of a stochasticsimulator having a dynamics depending on (few) uncertain parameters. The objectiveis then the characterization of the dependences with respect to the parameters of thestochastic solution. To this end, we focus on functional representations, namely Poly-nomial Chaos (PC) expansions, to account for the parameters dependences.In the first part of the talk, I will consider the PC approximation of the first momentsof the stochastic solution, introducing a robust Bayesian approach for the determina-tion from noisy estimates (sampling noise) of the solution’s moments of their expan-sion coefficients. The preconditioning of the stochastic solution, based on parameter-dependent transformations, is also shown to greatly improve the convergence of thePC-representation in certain situations.The second part of the talk will concern the direct PC expansion of the random so-lution, for the case of simple stochastic ODE driven by Wiener’s processes. In theproposed formulation, the expansion coefficients are stochastic processes (functions ofthe Wiener’s process). A Galerkin method is used to generate trajectories of the expan-sion coefficients, enabling the orthogonal decomposition (Sobol) and detailed analysisof the solution variance. Examples of increasing complexity will be shown for a SODEswith uncertain drift and diffusion terms.Finally, the application of the proposed decomposition to other types of stochasticmodels and simulators will be discussed.
37
Thursday, 8:40–9:30
Inverse uncertainty quantification
Hermann G. Matthies
Technische Universitt Braunschweig, Institute of Scientific ComputingHans-Sommer-Str. 65, D-38106 Braunschweig
Parameter identification involves the observation of a function of the state of somesystem usually described by a PDE - which depends on some unknown parameters.The mapping from parameter to observable is commonly not invertible, which causesthe problem to be ill- posed. In a probabilistic setting the knowledge prior to theobservation is encoded in a probability destribution which is updated according toBayes s rule through the observation, equivalent to a conditional (conditioned on theobservation) expectation. The conditional expectation is equivalent to a minimisationof a functional, hence an optimisation problem. To perform the update one has to solvethe forward problem, propagating the parameter distribution to the forecast observable.The difference with the real observation leads to the update. It is common to changethe underlying measure in the update, but here we update which is the minimiser -directly the random variables describing the parameters, thus changing the parameterdistribution only implicitly. This is achieved by a functional approximation of therandom variables involved. The solution of the forward problem can be addressedmore efficiently through the use of tensor approximations. We show that the same istrue for the inverse problem. Both the computation of the update map a ”filter” andthe update itself can be sped up considerably through the use of tensor approximationmethods. The computations will be demonstrated on some examples from continuummechanics, for linear as well as highly non-linear systems like elasto-plasticity.
38
Thursday, 9:30–10:20
Quantification of uncertainty for
large-scale inverse wave propagation problems
Omar Ghattas
Institute for Computational Engineering and SciencesDepartments of Geological Sciences and Mechanical Engineering
The University of Texas at Austin
Joint work with:Tan Bui-Thanh (ICES, The University of Texas at Austin)
Carsten Burstedde (Institute for Numerical Simulation, University of Bonn)James Martin (ICES, The University of Texas at Austin)Georg Stadler (ICES, The University of Texas at Austin)Hari Sundar (ICES, The University of Texas at Austin)
Lucas Wilcox (Applied Mathematics, Naval Postgraduate School)
Inverse problems governed by acoustic, elastic, or electromagnetic wave propagation—in which we seek to reconstruct the unknown shape of a scatterer, or the unknownproperties of a medium, from observations of waves that are scattered by the shapeor medium—play an important role in a number of engineered or natural systems.The deterministic solution of large-scale inverse wave propagation problems presentssignificant challenges even in the deterministic setting [2, 1, 13]. Our aim is to addressthe quantification of uncertainty in the solution of the inverse problem by casting theinverse problem as one in Bayesian inference. This provides a systematic and coherenttreatment of uncertainties in all components of the inverse problem, from observa-tions to prior knowledge to the wave propagation model, yielding the uncertainty inthe inferred medium/shape in a systematic and consistent manner. Unfortunately,state-of-the-art MCMC methods for characterizing the solution of Bayesian inverseproblems are prohibitive when the forward problem is expensive (as in our 100–1000wavelength target problems) and a high-dimensional parametrization is employed todescribe the unknown medium (as in our target problems involving infinite-dimensionalmedium/shape fields, which result in millions of parameters when discretized).
We report on recent work aimed at overcoming mathematical and computational barri-ers associated with large-scale Bayesian inverse wave propagation problems, including:
• a high order, parallel, adaptive hp-non-conforming discontinuous Galerkin (DG)method for wave propagation [19, 4]
• parallel adaptive mesh refinement/coarsening algorithms based on forests of oc-trees [11, 12] to tailor mesh to the local wave speed and adapt to the inversionparameter field and its uncertainty;
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• an infinite-dimensional formulation of Bayesian inverse problems based on theframework of Stuart [17] and its consistent finite-dimensional discretization [10,16, 5];
• parallel scalable hybrid geometric-algebraic multigrid methods based on forest ofoctree meshes for the treatment of regularizing prior operators that are posed asinverses of elliptic operators [18];
• a stochastic Newton MCMC method for solution of the statistical inverse prob-lem that reduces the number of samples needed by several orders of magnitude,relative to conventional MCMC [15, 9, 16];
• fast low rank randomized SVD approximation of the Hessian [3, 14] based on itscompactness properties [6, 7, 8]; and
• prototype applications to solutions of Bayesian inverse wave propagation modelproblem in global seismology with up to one million earth model parameters, 630million state variables, on up to 100,000 processors [3, 10].
References
[1] Volkan Akcelik, Jacobo Bielak, George Biros, Ioannis Epanomeritakis, AntonioFernandez, Omar Ghattas, Eui J. Kim, Julio Lopez, David R. O’Hallaron, TiankaiTu, and John Urbanic. High resolution forward and inverse earthquake modelingon terascale computers. In SC03: Proceedings of the International Conference forHigh Performance Computing, Networking, Storage, and Analysis. ACM/IEEE,2003. Gordon Bell Prize for Special Achievement.
[2] Volkan Akcelik, George Biros, and Omar Ghattas. Parallel multiscale Gauss-Newton-Krylov methods for inverse wave propagation. In Proceedings ofIEEE/ACM SC2002 Conference, Baltimore, MD, Nov. 2002. SC2002 Best Techni-cal Paper Award.
[3] Tan Bui-Thanh, Carsten Burstedde, Omar Ghattas, James Martin, Georg Stadler,and Lucas C. Wilcox. Extreme-scale UQ for Bayesian inverse problems governed byPDEs. In SC12: Proceedings of the International Conference for High PerformanceComputing, Networking, Storage and Analysis, 2012.
[4] Tan Bui-Thanh and Omar Ghattas. Analysis of an hp-non-conforming discontin-uous Galerkin spectral element method for wave propagation. SIAM Journal onNumerical Analysis, 50(3):1801–1826, 2012.
[5] Tan Bui-Thanh and Omar Ghattas. An analysis of infinite dimensional Bayesianinverse shape acoustic scattering and its numerical approximation. Submitted toSIAM Journal of Uncertainty Quantification, 2012. ICES Report ICES-12-31.
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[6] Tan Bui-Thanh and Omar Ghattas. Analysis of the Hessian for inverse scatteringproblems. Part I: Inverse shape scattering of acoustic waves. Inverse Problems,28(5):055001, 2012.
[7] Tan Bui-Thanh and Omar Ghattas. Analysis of the Hessian for inverse scatteringproblems. Part II: Inverse medium scattering of acoustic waves. Inverse Problems,28(5):055002, 2012.
[8] Tan Bui-Thanh and Omar Ghattas. Analysis of the Hessian for inverse scatteringproblems. Part III: Inverse medium scattering of electromagnetic waves. Submittedto Inverse Problems, 2012.
[9] Tan Bui-Thanh and Omar Ghattas. A scaled stochastic Newton algorithm forMarkov chain Monte Carlo simulations. Submitted to SIAM Journal of UncertaintyQuantification, 2012. Submitted to SIAM Journal of Uncertainty Quantification.
[10] Tan Bui-Thanh, Omar Ghattas, James Martin, and Georg Stadler. A computa-tional framework for infinite-dimensional Bayesian inverse problems. Part I: Thelinearized case, with applications to global seismic inversion, 2012. Submitted.
[11] Carsten Burstedde, Omar Ghattas, Michael Gurnis, Tobin Isaac, Georg Stadler,Tim Warburton, and Lucas C. Wilcox. Extreme-scale AMR. In SC10: Proceed-ings of the International Conference for High Performance Computing, Networking,Storage and Analysis. ACM/IEEE, 2010.
[12] Carsten Burstedde, Lucas C. Wilcox, and Omar Ghattas. p4est: Scalable algo-rithms for parallel adaptive mesh refinement on forests of octrees. SIAM Journalon Scientific Computing, 33(3):1103–1133, 2011.
[13] Ioannis Epanomeritakis, Volkan Akcelik, Omar Ghattas, and Jacobo Bielak. ANewton-CG method for large-scale three-dimensional elastic full-waveform seismicinversion. Inverse Problems, 24(3):034015 (26pp), 2008.
[14] H. Pearl Flath, Lucas C. Wilcox, Volkan Akcelik, Judy Hill, Bart van Bloe-men Waanders, and Omar Ghattas. Fast algorithms for Bayesian uncertainty quan-tification in large-scale linear inverse problems based on low-rank partial Hessianapproximations. SIAM Journal on Scientific Computing, 33(1):407–432, 2011.
[15] James Martin, Lucas C. Wilcox, Carsten Burstedde, and Omar Ghattas. Astochastic Newton MCMC method for large-scale statistical inverse problemswith application to seismic inversion. SIAM Journal on Scientific Computing,34(3):A1460–A1487, 2012.
[16] N. Petra, J. Martin, G. Stadler, and O. Ghattas. A computational framework forinfinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMCwith application to ice sheet inverse problems. In preparation, 2013.
[17] Andrew M. Stuart. Inverse problems: A Bayesian perspective. Acta Numerica,19:451–559, 2010.
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[18] Hari Sundar, George Biros, Carsten Burstedde, Johann Rudi, Omar Ghattas, andGeorg Stadler. Parallel geometric-algebraic multigrid on unstructured forests ofoctrees. In SC12: Proceedings of the International Conference for High PerformanceComputing, Networking, Storage and Analysis. ACM/IEEE, 2012.
[19] Lucas C. Wilcox, Georg Stadler, Carsten Burstedde, and Omar Ghattas. A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media. Journal of Computational Physics, 229(24):9373–9396, 2010.
42
Thursday, 10:50–11:40
A probabilistic graphical model approach to
uncertainty quantification
Nicholas Zabaras
Cornell University, Ithaca, NY 14853, USA
We present a probabilistic graphical model based methodology to efficiently performuncertainty quantification in the presence of both stochastic input and multiple scales.Both the stochastic input and model responses are treated as random variables in thisframework. Their relationships are modeled by graphical models which give explicitfactorization of a high-dimensional joint probability distribution.The hyperparametersin the probabilistic model are learned using sequential Monte Carlo (SMC) method.We make predictions from the probabilistic graphical model using belief propagationalgorithms. Numerical examples are presented to show the accuracy and efficiency ofthe predictive capability of the developed graphical model.
References
[1] Jiang Wan, and Nicholas Zabaras, A probabilistic graphical model approachto stochastic multiscale partial differential equations, Journal of ComputationalPhysics, under review (2013).
43
Thursday, 11:40–12:30
UQ for groundwater flow
Oliver Ernst
TU Bergakademie Freiberg
Joint work with:Bjorn Sprungk (TU Bergakademie Freiberg),K. Andrew Cliffe (University of Nottingham)
We consider the problem of Darcy flow through a porous medium with lognormally dis-tributed random conductivity field as a model of groundwater flow with uncertainty.We review our recent work on uncertainty propagation using sparse grid stochasticcollocation based on a statistical model of the input random field derived from con-ductivity observations for a specific radioactive waste storage application. We will alsopresent preliminary work on solving the stochastic inverse problem of estimating theinput random field from measurements of hydraulic head.
References
[1] O. G. Ernst and B. Sprungk, Stochastic Collocation for Elliptic PDEs with randomdata - the lognormal case, (submitted) 2013.
[2] K. A. Cliffe, O. G. Ernst, B. Sprungk, E. Ullmann, and K. G. van den Boogaart.Probabilistic uncertainty quantification using PDEs with random data: A casestudy. (in preparation).
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Thursday, 14:30–15:20
Sparse polynomial chaos expansions
in engineering applications
Bruno Sudret
ETH Zurich, Institute of Structural MechanicsChair of Risk, Safety & Uncertainty Quantification
Uncertainty quantification has become a hot topic in many fields of science and en-gineering. The related research is rich since the problems may be addressed throughdifferent viewpoints, namely the study of particular stochastic partial differential equa-tions, statistical estimation problems or from engineering-oriented goals such as struc-tural robust design, sensitivity- and reliability analysis.
From the engineering perspective, uncertainty quantification methods preferably fea-ture the following characteristics:
• they are compatible with existing computational models such as finite elementcodes in civil- and mechanical engineering: thus non intrusive methods that relyupon a set of evaluations of the pre-existing model considered as a black-boxprogram are preferred ;
• the practitioners are usually interested only in a limited set of output quantities,so that the approaches should be able to focus on so-called quantities of interest ;
• non linearities in the constitutive equations should be easily handled ;
• large-dimensional problems featuring tens to hundreds of variables should beaffordable.
According to these constraints, we focus on the so-called regression approach to poly-nomial chaos expansions which relies upon considering the computation of polynomialchaos coefficients as a mean-square minimization problem [1]. This statistical settingallows us to define a global mean-square error estimator which is easy to compute byleave-one-out cross validation.
The use of the least angle regression algorithm proposed by Efron et. al [4] allows usto build sparse expansions [2], i.e. estimate the optimal truncation scheme and theassociated coefficients at the same time. This also opens the path to adaptive methodsin which the analysis is simply driven by a tolerated mean-square error on the quantityof interest.
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In order to cope with the curse of dimensionality so-called hyperbolic truncation schemesare introduced so as to drastically decrease the size of the pre-selected basis in whichan optimal sparse representation is searched for.
In this talk we discuss the various tools associated with these sparse polynomial chaosexpansions, namely experimental designs, leave-one-out cross validation and adaptivity.The case of vector-valued quantities of interest that correspond to discretized solutionfields is addressed by coupling the above machinery with principal component analysis[3]. This eventually leads to a versatile, fully non intrusive framework, which has closerelationships with the proper generalized decomposition techniques [5].
Several engineering applications are shown in the field of structural and geotechnicalengineering, including the assessment of the vulnerability of structures in earthquakeengineering.
References
[1] M. Berveiller, B. Sudret, and M. Lemaire. Stochastic finite elements: a non intrusiveapproach by regression. Eur. J. Comput. Mech., 15(1-3):81–92, 2006.
[2] G. Blatman and B. Sudret. Adaptive sparse polynomial chaos expansion based onLeast Angle Regression. J. Comput. Phys, 230:2345–2367, 2011.
[3] G. Blatman and B. Sudret. Principal component analysis and Least Angle Re-gression in spectral stochastic finite element analysis. In M. Faber, editor, Proc.11th Int. Conf. on Applications of Stat. and Prob. in Civil Engineering (ICASP11),Zurich, Switzerland, 2011.
[4] B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani. Least angle regression. Annalsof Statistics, 32:407–499, 2004.
[5] A. Nouy. Proper generalized decompositions and separated representations for thenumerical solution of high dimensional stochastic problems. Archives of Computa-tional Methods in Engineering, 17:403–434, 2010.
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Thursday, 15:20–15:50
Measurement uncertainty of
ensemble-averaged flow data
Stephan Bansmer
Institute of Fluid Mechanics on behalf of Technische Universitat Braunschweig,Hermann-Blenk-Str. 37, 38108 Braunschweig
Turbulent flows are characterized by three-dimensional, stochastic fluid motions in-cluding the phenomena of diffusion and dissipation [1, 2]. One possibility to quantifythe properties of turbulent flow is to introduce the Reynolds-Stresses that arise froma momentum balance when averaging the Navier-Stokes equations. There are normaland shear stresses given by
〈u′2〉 =1
N
N∑j
[u′
2j
]=
1
N
N∑j
[uj − 〈u〉]2
〈u′w′〉 =1
N
N∑j
[u′j · w′j
]=
1
N
N∑j
[(uj − 〈u〉) (wj − 〈w〉)] ,
where 〈· · ·〉 is the ensemble averaging operator, u and w are flow velocities and Nrepresents the number of averaged samples. In essence, the Reynolds-Stresses describethe viscous momentum transport as a result of the turbulent motion. From a math-ematical point of view, these stresses represent the correlation between the velocityfluctuations of flow and can be understood as a stochastic property. A precise mea-surement of the Reynolds-Stresses is far reaching. Fundamental turbulence research,reliable turbulence-model design for computational fluid dynamics, optimization of in-dustrial mixing processes are just a few examples.
As one example, the Reynolds shear-stress distribution 〈u′w′〉 of the flow around aSD7003 airfoil at Reynolds number 66.000 is presented by measuring 5000 sampleswith particle image velocimetry, see figure 1. In the wall near area, the shear-stresshas high values indicating a strong viscous momentum transport.
The evolution of the Reynolds shear-stress 〈u′w′〉 over the number of data samples Nat data point B of figure 1 is presented in figure 2. Interestingly, 〈u′w′〉 seems to reacha steady state for N > 500, however there is still a large fluctuation of more than 10%for N > 500. The questions that arise from figure 2 are:
• How large is the uncertainty for ensemble averaged measurement data, like 〈u〉and 〈u′w′〉?
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• Can the uncertainty be approximated by some simple formulas?
• Does an uncertainty of more than 10% for the Reynolds stress 〈u′w′〉 indicate anunreliable measurement?
• Is a database of 5000 samples better than 500 samples?
0 0.2 0.4 0.6 0.8 1
0
0.2 0.002
0.003
0.004
0.005
0.006
0.007
0.008
U
X
Z
M
A
B
Figure 1: Distribution of the Reynoldsstress 〈u′w′〉 for the flow along a SD7003 airfoilat Reynolds number 66000, measured with standard PIV.
100
101
102
103
0.008
0.006
0.004
0.002
0
U
R
N
Figure 2: Reynoldsstress 〈u′w′〉 as a function of the number of measurement samplesN . Although 5000 samples were taken, the Reynoldsstress has an uncertainty of morethan 10%.
The underlying uncertainty of velocity U (u), which is needed to determine the uncer-tainty of the Reynolds stresses, depends of course of the specific measurement tech-nique. For the widely used PIV technique, estimates for U (u) can be found for instancein Raffel et al. [3] which were determined by Monte-Carlo simulations with artificialparticle images.
The idea of a presentation at the Hausdorff-Center is to introduce the audience tothe PIV measurement technique that is widely used in modern experimental fluid
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mechanics. Further, the author like to present a relatively simple approach based on thestochastic theory of probability distribution functions to determine an approximation ofthe uncertainties for ensemble averaged data, like U (〈u〉) and U (〈u′w′〉). The approachshall then be discussed with the audience in order to get an exchange between the ’twoworlds’ of engineers and mathematicians.
References
[1] S. Pope, Turbulent Flows. Cambridge University Press (2000)
[2] J.C. Rotta, Turbulente Stromungen. Teubner, Stuttgart (1972)
[3] M. Raffel et al., Particle Image Velocimetry. Springer (2006)
49
Thursday, 16:20–16:50
Multilevel estimation of rare events
Elisabeth Ullmann
Department of Mathematical Sciences, University of Bath
Joint work with:Iason Papaioannou (Engineering Risk Analysis Group, TU Munchen)
Estimation of failure probabilities is a fundamental problem in reliability analysis andrisk management of engineering systems with uncertain inputs. We focus on systemsdescribed by partial differential equations (PDEs) with random coefficients. Failureevents occur if some output process exceeds a given threshold. Monte Carlo simulationmethods estimate the probability of failure events by sampling the random inputs andsolving the associated PDE to obtain samples of the output process. However, in mod-ern applications with highly resolved physical models and a possibly high-dimensionalsample space the cost to obtain only a single output sample is nontrivial. If a largenumber of samples is required then the total computational cost might become pro-hibitively high.
Typically, we are interested in estimating small failure probabilities Pf associated withrare events. It is well known that the standard Monte Carlo method is very inefficientfor the estimation of small probabilities (the number of samples required is of the orderof 1/Pf which is extremely large for small Pf ). More sophisticated sampling methodsreduce the computational cost by reducing the number of samples and/or the cost persample.
We employ subset simulation [1] which has been developed for the estimation of smallprobabilities in high-dimensional sample spaces (in the mathematical literature, thistechnique is termed [generalized] splitting [2]). Subset simulation reduces the totalnumber of samples by decomposition of the sample space into a sequence of nested,partial failure sets. However, the physical discretization of the engineering system -typically done by finite elements - is fixed in each failure set and sampling is stillcomputing-intensive.
Multilevel Monte Carlo (MLMC) methods [3, 4] have been applied recently for PDE-based simulations with random input data, see the pioneering works [5] and [6]. Fordiscretized PDEs with random coefficients MLMC estimators reduce the cost per sam-ple by decomposition of the physical space using a hierarchy of finite element meshes.Most of the computational workload is shifted to coarse finite element meshes wheresimulations are inexpensive.
We discuss a novel multilevel approach to subset simulation where the failure regionsare computed on a hierarchy of finite element meshes. We report numerical experimentswith a simple 1D displacement problem and illustrate properties of the new method.
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References
[1] S.-K. Au and J. L. Beck, Estimation of small failure probabilities in high dimensionsby subset simulation, Prob. Eng. Mech., 16 (2001), 263–277.
[2] Z. I. Botev and D. P. Kroese, Efficient Monte Carlo simulation via the generalizedsplitting method, Stat. Comput., 22 (2012), 1–16.
[3] S. Heinrich, Multilevel Monte Carlo methods, Lecture Notes in Large Scale ScientificComputing, 2179 (2001), 58–67.
[4] M. B. Giles, Multilevel Monte Carlo path simulation, Operations Research, 56(2008), no. 3, 607–617.
[5] A. Barth, Ch. Schwab, and N. Zollinger, Multi-level Monte Carlo finite elementmethod for elliptic PDE’s with stochastic coefficients, Numer. Math., 119 (2011),no. 1, 123–161.
[6] K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, Multilevel MonteCarlo methods and applications to elliptic PDEs with random coefficients, Comput.Visual. Sci., 14 (2011), no. 1, 3–15.
51
Thursday, 16:50–17:20
Representation and treatment of random functions
with divergent integer moments
Giulio Cottone
Engineering Risk Analysis Group Technische Universitat Munchen
Joint work with:Iason Papaioannou (Engineering Risk Analysis Group Technische Universitat
Munchen)
Polynomial chaos expansion (PCE) is a standard approach for representation of func-tions of random variables. It is based on performing a spectral expansion of the functionin terms of orthogonal polynomials with respect to the probability measure of the in-put random variables. A fundamental requirement of the PCE is that the randomfunction has finite mean square. However, in some applications this might not be thecase. For example, it is well known that the reciprocal of a Gaussian random vari-able has divergent integer moments of any order k ≥ 1, which is caused by the slowinverse power-law decay of the density. Distributions of this type are encountered ina wide variety of contexts such as in the framework of anomalous diffusion, in thetravel length distribution in human motion patterns and animal search processes, influctuations in plasma devices, in scaling laws for polymer models and its applicationsto gene regulation models.
In this talk we investigate on strategies to overcome this limitation of PCE. One wayto achieve this, is within the framework of the fractional calculus [1]. In particular,starting from the considerations that random variables with infinite variance may havefinite fractional absolute moments of order p < 2, it shown that they can be used todescribe other statistics by fractional integration and derivation. The results on therepresentation of random variables and processes in terms of fractional moments isextended to functions of random variables.
References
[1] G. Cottone, M. Di Paola, and R. Metzler. Fractional calculus approach to thestatistical characterization of random variables and vectors. Physica A: StatisticalMechanics and its Applications, 389(5):909 – 920, 2010.
52
Thursday, 17:20–18:10
Bayesian uncertainty quantification and
propagation in molecular dynamics simulations
of nanoscale flows
Petros Koumoutsakos
Professorship for Computational Science, ETH Zurich, CH-8092, Switzerland
Joint work with:Panos Angelikopoulos (Professorship for Computational Science, ETH Zurich,
CH-8092, Switzerland),Costas Papadimitriou (Department of Mechanical Engineering, University of
Thessaly, Volos 38334, Greece)
Molecular Dynamics (MD) simulations are broadly used to study nanoscale phenomenaacross many fields of Engineering and Science. Albeit the large number of parametersinvolved in MD simulations, and the uncertainty associated with related experiments,the issue of uncertainty quantification and propagation (UQ+P) in MD simulationshas received little attention. Our results indicate that this is a major issue, challengingthe value of MD simulations. We propose a comprehensive Bayesian probabilisticframework for quantifying and calibrating the uncertainties in MD simulations [1].
We consider MD simulations of nanoscale flows. High Performance Computing (HPC)issues encountered in Bayesian UQ+P for MD simulations are addressed. A Transi-tional MCMC (TMCMC) approach is shown to have the best performance in populat-ing the posterior probability distribution of the model parameters. Parallel computingalgorithms are proposed to efficiently handle the large number of MD model runs anddistribute the computations in available GPUs and multi-core CPUs in heterogeneousarchitecture clusters. Surrogate models are adopted to drastically reduce the numberof MD model runs. We illustrate the effectiveness of the framework via large scale MDsimulations to assess the uncertainties in simulations of graphitic nano-structures inaqueous environments. In particular, we investigate the reliability in the predictions oftwo representative systems namely the wetting of graphene by nanoscale water dropletsand the aggregation dynamics and thermodynamics of fullerenes in aqueous solution.
References
[1] Angelikopoulos Panos, Papadimitriou Costas, and Koumoutsakos Petros, Bayesianuncertainty quantification and propagation in molecular dynamics simulations:A high performance computing framework, Journal of Chemical Physics, 2012,37(14):144103
53
Friday, 8:40–9:30
QMC lattice methods for PDE
with random coefficients
Ian H Sloan
University of New South Wales, Australia
This talk presents recent developments in applying Quasi-Monte Carlo methods (specif-ically lattice methods) to elliptic PDE with random coefficients. A guiding exampleis the flow of liquid through a porous material, with the permeability modeled as ahigh-dimensional random field with lognormal Gaussian random field. In this work weapply the theory of lattice methods in weighted spaces to the design of lattice rules withgood convergence properties for expected values of functionals of the solution of thePDE. The talk describes joint work with Frances Kuo and James Nichols (Universityof New South Wales), Christoph Schwab (ETH), Ivan Graham and Robert Scheichl(University of Bath).
References
[1] I.G. Graham, F.Y. Kuo, D. Nuyens, R. Scheichl, and I.H. Sloan. Quasi-MonteCarlo methods for elliptic PDEs with random coefficients and applications. Journalof Computational Physics, 230 (2011), 3668-3694.
[2] F.Y. Kuo, C. Schwab, and I.H. Sloan. Quasi-Monte carlo for very high dimemnsi-noal integration: the standard setting and beyond, ANZIAM Journal 53 (2011),1–37.
[3] F.Y. Kuo, C. Schwab, and I.H. Sloan. Quasi-Monte Carlo finite element methodsfor a class of elliptic partial differential equations with random coefficients, SIAMJournal on Numerical Analysis, 50 (2012) 3351–3374 .
[4] I.G. Graham, F.Y. Kuo, J.A. Nichols, R. Scheichl, C. Schwab, and I.H. Sloan.Quasi-Monte Carlo finite element methods for elliptic PDEs with log-normal ran-dom coefficients In preparation.
54
Friday, 9:30–10:20
Bayesian data assimilation and
dimension reduction with optimal maps
Youssef Marzouk
Massachusetts Institute of Technology, Cambridge, MA USA
Joint work with:Tarek Moselhy (Massachusetts Institute of Technology)Tiangang Cui (Massachusetts Institute of Technology)
James Martin (University of Texas at Austin)
We present a new approach to Bayesian inference that entirely avoids Markov chainsimulation or sequential importance sampling, by constructing a deterministic map thatpushes forward the prior measure to the posterior measure. Existence and uniquenessof a suitable measure-preserving map is established by formulating the problem in thecontext of optimal transportation. In particular, we present new map-based schemesfor sequential Bayesian data assimilation, i.e., nonlinear filtering, based on the thesuccessive application of maps from a reference measure. Maps are computed efficientlythrough the solution of a stochastic optimization problem, using a sample-averageapproximation approach. Advantages of a map-based representation include analyticalexpressions for posterior moments and the ability to generate arbitrary numbers ofindependent and uniformly-weighted posterior samples without additional evaluationsof the dynamical model. We will show that map-based filters avoid both the limitationsof Gaussian approximations (e.g., in the ensemble Kalman filter) and issues of sampleimpoverishment associated with weighted particle schemes (e.g., sequential importanceresampling).
We then focus on various means of explicitly parameterizing the map in order to takeadvantage of low-dimensional structure present in many problems of parameter infer-ence and state estimation. While a posterior distribution may appear high-dimensional,the intrinsic dimensionality of an inference problem is affected by prior information,limited data, and the smoothing properties of the forward/observation operator. Oftenonly a few directions are needed to capture the change from prior to posterior. Wedescribe a method for identifying these directions through the solution of a generalizedeigenvalue problem, and extend it to nonlinear problems where the data misfit Hessianvaries over parameter space. This approach leads to more efficient Rao-Blackwellizedposterior sampling schemes.
Numerical demonstrations include large-scale inverse problems involving partial differ-ential equations, as well as filtering in nonlinear and chaotic dynamical systems.
55
Friday, 10:20–10:50
Variational inequalities and equilibrium problems
with uncertain data
Joachim Gwinner
Institut fur Mathematik, Fakultat fur Luft- und Raumfahrttechnik, Universitat derBundeswehr Munchen, 85577 Neubiberg/Munchen, Germany
In the first part of the talk we shortly review our paper [4] and our joint work with F.Raciti [1, 2, 3] on random variational inequalities with applications to elliptic unilateralproblems and traffic equilibrium problems, where randomness affects not only loading,but also the constraints and the bilinear form/nonlinear operator. Then we turn tothe expected value formulation and the expected residual minimization formulationas other modeling approaches to stochastic variational inequalities and (non)linearcomplementarity problems. We report on numerical experiments on stochastic trafficequilibrium problems to illustrate the characteristics of the different proposed models.To compare the results some measures like fairness and reliability are used.
References
[1] Gwinner, J. ; Raciti, F.: Some equilibrium problems under uncertainty and randomvariational inequalities. Ann. Oper. Res. 200 (2012), 299–319.
[2] Gwinner, J. ; Raciti, F.: On a class of random variational inequalities on randomsets. Numer. Funct. Anal. Optim. 27 (2006), no. 5-6, 619–636.
[3] Gwinner, J. ; Raciti, F.: Random equilibrium problems on networks. Math. Com-put. Modelling 43 (2006), no. 7-8, 880–891.
[4] Gwinner, J.: A class of random variational inequalities and simple random uni-lateral boundary value problemsexistence, discretization, finite element approxima-tion. Stochastic Anal. Appl. 18 (2000), no. 6, 967–993.
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Friday, 11:20–12:10
On multilevel quadrature for elliptic stochastic
partial differential equations
Helmut Harbrecht
University of Basel
Joint work with:Michael Peters (University of Basel),
Markus Siebenmorgen (University of Basel)
In this talk we show that the multilevel Monte Carlo method for elliptic stochas- ticpartial differential equations can be interpreted as a sparse grid approximation. Byusing this interpretation, the method can straightforwardly be generalized to any givenquadrature rule for high dimensional integrals like the quasi Monte Carlo method orthe polynomial chaos approach. Besides the multilevel quadrature for approximatingthe solutions expectation, a simple and efficient modification of the approach is pro-posed to compute the stochastic solutions variance. Numerical results are provided todemonstrate and quantify the approach.
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Friday, 12:10–13:00
Analysis and computations for linear hyperbolic
PDEs with stochastic coefficients
Raul Tempone
King Abdullah University of Science and Technology, Thuwal
Partial Differential Equations with stochastic coefficients are a suitable tool to describesystems whose parameters are not completely determined, either because of measure-ment errors or intrinsic lack of knowledge on the system. In the case of linear ellipticPDEs with random inputs, an effective strategy to approximate the state variables andtheir statistical moments is to use polynomial based approximations like StochasticGalerkin or Stochastic Collocation method. These approximations exploit the highregularity of the state variables with respect to the input random parameters and fora moderate number of input parameters, are remarkably more effective than classicalsampling methods. In this talk we analyze a Stochastic Collocation method for solvingthe second order wave equation with a random wave speed. The speed is piecewisesmooth in the physical space and depends on a finite number of random variables. Weconsider both full and sparse tensor product spaces of orthogonal polynomials, provid-ing a convergence analysis. In particular, we show that, unlike the solutions of ellipticand parabolic problems, the wave solution to hyperbolic problems is not in general an-alytic with respect to the input random variables. Therefore, the rate of convergencemay only be algebraic. Faster convergence rates are still possible for some quantitiesof interest and for the wave solution with particular types of data.
This is a joint work with F. Nobile and M. Motamed.
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List of Participants
Angelikopoulos Panagiotis ETH Zurich
Bansmer Stephan TU Braunschweig
Benner Peter MPI Magdeburg
Bierig Claudio Universitat Bonn
Borzi Alfio Universitat Wurzburg
Le Bris Claude ENPC, Champs sur Marne
Chen Peng EPF Lausanne
Chernov Alexey Universitat Bonn
Cohen Albert Universite Pierre et Marie Curie, Paris
Cottone Giulio TU Munchen
Doostan Alireza University of Colorado, Boulder
Eigel Martin HU Berlin
Elman Howard University of Maryland
Ernst Oliver TU Bergakademie Freiberg
Franck Isabel TU Munchen
Frauholz Sarah RWTH Aachen
Fuchs Barbara Universitat Bonn
Garcke Jochen Universitat Bonn & Fraunhofer-Institut SCAI
Ghattas Omar University of Texas at Austin
Gittelson Claude Jeffrey Purdue University
Griebel Michael Universitat Bonn & Fraunhofer Institut SCAI
Gwinner Joachim Universitat der Bundeswehr Munchen
Hackbusch Wolfgang MPI Leipzig
Haji-Ali Abdul-Lateef KAUST, Thuwal
Harbrecht Helmut Universitat Basel
Heuveline Vincent Universitat Heidelberg / HITS
Iza-Teran Rodrigo Fraunhofer Institut SCAI
Kostina Ekaterina Universitat Marburg
Koumoutsakos Petros ETH Zurich
Koutsourelakis Phaedon-Stelios TU Munchen
Kumar Dinesh Vrije Universiteit Brussels
Kundu Abhishek University of Swansea
Litvinenko Alexander TU Braunschweig
Maday Yvon Universite Pierre et Marie Curie, Paris
Marzouk Youssef MIT
Mathelin Lionel CNRS Orsay
Matthies Hermann TU Braunschweig
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Le Maıtre Olivier LIMSI-CNRS, Orsay
Najm Habib Sandia National Laboratories, Livermore
Nobile Fabio EPF Lausanne
Papadimitriou Costas University of Thessaly
Papaioannou Iason TU Munchen
Peters Michael Universitat Basel
Pham Duong Universitat Bonn
Polisky Vitali Karlsruher Institut fur Technologie (KIT)
Powell Catherine University of Manchester
Pulch Roland Bergische Universitat Wuppertal
Reinarz Anne Universitat Bonn
Scheichl Robert University of Bath
Schick Michael Heidelberger Institut fur Theoretische Studien (HITS)
Schwab Christoph ETH Zurich
Shapiro Alexander Georgia Tech, Atlanta
Siebenmorgen Markus Universitat Basel
Silvester David University of Manchester
Sinsbeck Michael Universitat Stuttgart
Sloan Ian University of New South Wales, Sydney
Sprungk Bjorn TU Bergakademie Freiberg
Stuart Andrew University of Warwick
Sudret Bruno ETH Zurich
Tamellini Lorenzo EPF Lausanne
Tempone Raul KAUST, Thuwal
Tesei Francesco EPF Lausanne
Ullmann Elisabeth University of Bath
Zabaras Nicholas Cornell University
Zaspel Peter Universitat Bonn
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