European Multi-Grid Conference EMG 2010 - Institut …borzi/ProgAbs...European Multi-Grid Conference...

European Multi-Grid Conference EMG 2010

19 - 23 September, 2010Isola d’Ischia, Italy

Programme and Abstracts

Welcome to EMG2010The organizing committee is delighted to welcome you to the European Multi-Grid conference, EMG2010, in Ischia,Italy! We wish you a very interesting and pleasant participation. We are indeed honored to have you here with us.Our eminent speakers and delegates come from all over the world. There are about 100 participants from 16 countriesgathering here, making this conference a truly international one.

The EMG2010 Conference provides a forum for researchers to present and discuss recent advances on the devel-opment, theory, and application of multigrid, multilevel, and multiscale methodologies. Previous EMG conferenceswere held in Cologne (1981, 1985), Bonn(1990), Amsterdam (1993), Stuttgart (1996), Gent (1999), Hohenwart (2002),Scheveningen (2005), Bad Herrenhalb (2008). With this meeting, the European Multi-Grid Conference celebrates its10th anniversary !

Main topics at EMG2010 are multigrid methods in general (efficiency, robustness, adaptivity, parallelism, appli-cation), multiscale methods, domain decomposition methods, related iterative schemes and discretization methods,software, tools, and high performance computing, applications in classical and emerging areas like analysis of images,financial problems, inverse problems, models related to medicine and biosciences, multiscale and coupled systems,optimization and design.

There is a large number of contributions focusing on the development, analysis, and application of algebraicmultigrid methods (19 talks), on the development and analysis of multigrid strategies for saddle point problems andoptimality systems (16 talks), on nonsymmetric and indefinite problems (10), on multiscale problems (8), on high-performance computing multigrid strategies (7), on new results on the interplay between discretization and multigridschemes, and novel applications; for a total of 75 talks.

The invited talks cover a wide range of recent advances in the field of multigrid methods (in order of appearance):

W. Hackbusch, Hierarchical tensor representation.

V. Schulz, Computational optimization of systems governed by partial differential equations.

S. Vandewalle, Multigrid methods for partial differential equations with random coefficients.

J. van der Vegt, Optimizing multigrid performance for higher order accurate space-time discontinuous Galerkindiscretizations of advection dominated flows.

S. Serra-Capizzano, Multigrid methods for structured matrices and a regularized version in imaging.

M. Giles, Multilevel Monte Carlo method.

G. Rainer, Powertrain engineering needs mathematics.

R. Basri, Multiscale methods for edge detection and image segmentation.

M.W. Gee, Algebraic multigrid in multiphysics and multiscale.

We encourage delegates to participate actively in interesting discussions during the conference.

Acknowledgement

The EMG2010 builds on the success of the former European Multi-Grid Conferences. At this juncture, we would liketo take the opportunity to thank the local organizing committee, and the scientific committee, for their work andeffort in planning and coordinating this event. The CEREBRA Company is acknowledged for the invaluable Webdesign and management support. The AESSE Company is acknowledged for the design and printing work.

We would also like to thank the CWI - Center for Mathematics and Computer Science, Amsterdam, the DelftUniversity of Technology, in particular the Delft Center for Computational Science and Engineering (DCSE), and theUniversita degli Studi del Sannio, for their support in making this conference possible. In particular, the Universitadegli Studi del Sannio at Benevento is gratefully acknowledged for all the administrative work related to this conference.

Essential for the realization of this conference is the generous support of these universities and also of the companyAVL GmbH Austria and of the German Fraunhofer Institute SCAI (Institute for Scientific computing and Algorithms).We thank them all for their support, and we wish everyone a successful and fruitful conference.

With our best regards and thanks

Alfio Borzi and Kees Oosterlee

Sunday, September 19, at 19:00 – Welcome Reception

EMG2010’s SupportersCWI - Center for Mathematics and Computer Science, Amsterdam.

The Delft University of Technology.

The Delft Center for Computational Science and Engineering (DCSE).

La Universita degli Studi del Sannio.

The AVL company.

The German Fraunhofer Institute SCAI.

Contents

I Programme 11

II Collection of Abstracts 29

III Information 55

Part I

Programme

11

Monday, Sept. 20 Monday, Sept. 20

Monday,September 20

Registration8:00 - 16:00

Welcome09:00 - 9:15

Schedule

Time Talk Room

09:15 - 10:00 IS1 Pithecusa


10:45 - 11:15 Coffee Break

11:15 - 12:30 A1 PithecusaB1 Primavera

12:30 - 14:00 Lunch Break

14:00 - 15:15 C1 PithecusaD1 Primavera

15:15 - 15:45 Tea Break

15:45 - 17:00 E1 PithecusaF1 Primavera


IS1Hierarchical tensorrepresentation9:15 - 10:00

We are not only interested in theefficient storage of high-dimensionaltensors, but also in the efficient eval-uation and possibly approximationof the tensor operations. Since alltypes of representation tend to in-crease a certain rank under the op-erations, an efficient tensor calculusmust contain an efficient method forthe rank truncation of the intermedi-ate results. Here, the traditional rep-resentations are less attractive, sincerank truncation leads to nonlinearoptimisation problems.In the lecture we describe a newrepresentation scheme with the fol-lowing properties: 1) tensors rep-resented in the n-term representa-tion, in the Tucker format, or inthe sparse-grid form are exactly rep-resentable in the new scheme withsimilar storage cost as in the origi-nal form. 2) The truncation can beperformed (non-iteratively) only onthe basis of singular value decompo-sitions.The operations count for the basictensor operations can be described.In particular, they are linear in theorder (dimension) of the tensor.

W. HackbuschMax-Planck-Institut fuerMathematik in den Naturwis-senschaften, Germany

IS2Computational opti-mization of systemsgoverned by partialdifferential equations10:00 – 10:45

This talk tries to present a survey onthe computational challenges as wellas solution approaches in the field ofPDE constrained optimization. Thisvivid field of research combines tech-niques from optimization as well asnumerical analysis. Naturally, multi-grid optimization methods play animportant role but also so-called one-shot methods which iterate simul-taneously over the state and opti-mization variables. The key to ef-ficient optimization methods is theexploitation of problem structures.Examples of these structures in thearea of PDE constrained optimiza-tion are multigrid hierarchies, itera-tive solvers and resulting adjoint iter-ations, optimization preconditioners,shape calculus etc. From the abun-dance of application problems withinthis discipline, some examples areselected to highlight important fea-tures and to demonstrate the compu-tational potential of gradient basedmethods dovetailed to the respectivePDE constraints.

V. SchulzDepartment of Mathematics, Uni-versity of Trier, Germany


A1AMG Coarsening11:15 – 12:30

Chair: J. Wan

A1-1: 11:15 - 11:40An AMG method for the graph Laplacian usingmatching of graphsJ. Brannick

A1-2: 11:40 - 12:05A non-hermitian coarsening strategy for alge-braic multigridS. MacLachlan

A1-3: 12:05 - 12:30Stopping and restarting criteria for Krylov-accelerated AMG solversT. Clees

B1Optimization and Control11:15 – 12:30

Chair: S. Vandewalle

B1-1: 11:15 - 11:40Convergence and descent properties for a classof multilevel optimization algorithmsS. Nash

B1-2: 11:40 - 12:05Multigrid methods for state-constrained ellipticoptimal control problemsM. Vallejos

B1-3: 12:05 - 12:30A space-time multigrid solver for optimal dis-tributed control of incompressible fluid flowM. Koster



C1Helmholtz, Waves14:00 – 15:15

Chair: L. Grasedyck

C1-1: 14:00 - 14:25Analysis of a perturbed two-grid preconditionerfor indefinite three-dimensional HelmholtzproblemsX. Pinel

C1-2: 14:25 - 14:50Multigrid preconditioners for the Helmholtzequation on complex stretched gridsB. Reps

C1-3: 14:50 - 15:15A frequency-robust solver for eddy currentproblemsU. Langer

D1Control Problems14:00 – 15:15

Chair: F. Troeltzsch

D1-1: 14:00 - 14:25Multigrid of the second kind for the optimal con-trol of time-periodic, parabolic, partial differen-tial equationsD. Abbeloos

D1-2: 14:25 - 14:50Multigrid second-order accurate solution ofparabolic control-constrained problemsS. Gonzalez-Andrade

D1-3: 14:50 - 15:15Fast numerical schemes for optimal controlproblems with Fredholm constraintsM. Annunziato

15:15 - 15:45 Tea Break


E1Waves, Signals15:45 – 17:00

Chair: U. Langer

E1-1: 15:45 - 16:10AMG solver for Helmholtz equationsI. Livshits

E1-2: 16:10 - 16:35Multigrid achievements in thin layer flow ofrolling element bearingsC.H. Venner

E1-3: 16:35 - 17:00Multigrid algorithms for sparse representationof signalsI. Yavneh

F1GPU Computing15:45 – 17:00

Chair: P. Thum

F1-1: 15:45 - 16:10Local volatility estimation using GP-GPU accel-erationC.C. Douglas

F1-2: 16:10 - 16:35Multigrid algorithms on multi-GPU architec-turesH. Kostler

F1-3: 16:35 - 17:00Multigrid in quantum chemistry on multipleGPUsD. Ritter

Tuesday, Sept. 21 Tuesday, Sept. 21

Tuesday,September 21


Schedule

Time Talk Room







15:15 - 15:45 Tea Break



IS3Multigrid methodsfor partial differen-tial equations withrandom coefficients9:15 - 10:00

Mathematical models of real lifeengineering and scientific processestypically depend on parameterswhich are known only approximatelyor which are inherently variable andstochastic. These models often takethe form of a stochastic partial differ-ential equation with coefficients thatare random variables, fields, or pro-cesses.A popular method to determine thestochastic characteristics of the PDEsolution is the so-called stochas-tic finite element method. Thismethod approximates the solution ofthe PDE by a generalized polyno-mial chaos expansion. By using aGalerkin projection in the stochas-tic dimension, the original problemis transformed into a coupled set ofdeterministic PDEs. A finite ele-ment discretization converts this de-terministic PDE system into a highdimensional algebraic system.In this talk, we shall present anoverview of iterative solution ap-proaches. We start from iterativemethods based on a block splittingof the system matrices. Next, we ex-tend these methods for use as precon-ditioner for a Krylov method, and foruse as smoother in a multilevel con-text. Then, the various solvers willbe compared based on their conver-gence properties, computational costand implementation effort. Our find-ings are illustrated by means of twonumerical problems. The first one isa steady-state diffusion problem witha discontinuous random field as dif-fusion coefficient. The second is a de-terministic diffusion problem definedon a random domain.

S. VandewalleDepartment of Computerscience,Katholieke Universiteit Leuven,Belgium

IS4Optimizing multi-grid performance forhigher order accu-rate space-time dis-continuous Galerkindiscretizations of ad-vection dominatedflows10:00 – 10:45

Higher order accurate space-timediscontinuous Galerkin finite elementmethods are well suited for time-dependent problems requiring mov-ing and deforming meshes, such asfluid-structure interaction and non-linear water waves. Space-time DGmethods share many of the benefitsof DG methods, such as their suit-ability for hp-adaptivity and paral-lel computing. The algorithm re-sults, however, in a large systemof (non)linear algebraic equationswhich need to be solved each timestep. In this presentation we will dis-cuss various methods to improve theefficiency of multigrid algorithms forspace-time DG methods, with spe-cial emphasis on higher order ac-curate discretizations of advection-dominated flows.

J. van der VegtDepartment of Applied Mathe-matics, University of Twente,The Netherlands


A2Nonsymmetric Problems11:15 – 12:30

Chair: I. Yavneh

A2-1: 11:15 - 11:40A multigrid method for systems of hyperbolicconservation lawsJ. Wan

A2-2: 11:40 - 12:05Defect correction with algebraic multigrid forlinearized problems in compressible aerodynam-icsA. Naumovich

A2-3: 12:05 - 12:30A reformulation approach for the multigrid so-lution of the Helmholtz equationS. MacLachlan

B2Applications11:15 – 12:30

Chair: G. Wittum

B2-1: 11:15 - 11:40A two-scale iterative method for computingflows in highly porous mediaJ. Willems

B2-2: 11:40 - 12:05Multigrid preconditioned Newton-Krylovmethod for solving the 2-D 3-T energy equa-tionsHeng-Bin An

B2-3: 12:05 - 12:30Nonlinearly preconditioned globalization strate-gies - Theory and applicationsC. Groß



C2Saddlepoint Problems14:00 – 15:15

Chair: F. Gaspar

C2-1: 14:00 - 14:25Nonstandard norms and robust estimates forsaddle point problemsW. Zulehner

C2-2: 14:25 - 14:50Towards AMG for saddle point problemsB. Metsch

C2-3: 14:50 - 15:15An efficiency-based multigrid solver for incom-pressible resistive magnetohydrodynamicsJ. Adler

D2Novel Applications14:00 – 15:15

Chair: J. Toivanen

D2-1: 14:00 - 14:25Modelling of signal processing in neuronsG. Wittum

D2-2: 14:25 - 14:50Multigrid methods for zero-sum stochasticgamesS. Detournay

D2-3: 14:50 - 15:15A multilevel algorithm to compute steady statesof lattice Boltzmann modelsG. Samaey

15:15 - 15:45 Tea Break


E2Parallel and Adaptive Solvers15:45 – 17:25

Chair: C.C. Douglas

E2-1: 15:45 - 16:10Parallel multilevel solvers for 3D fluid dynamicsimulationsL. Carracciuolo

E2-2: 16:10 - 16:35MLD2P4: parallel algebraic multilevel precon-ditioners for large-scale linear systemsD. di Serafino

E2-3: 16:35 - 17:00Adaptive numerical methods for an hydrody-namic problem arising in magnetic reading de-vicesI. Arregui

E2-4: 17:00 - 17:25An adaptive domain decomposition Techniquefor parallelisation of the fast marching methodE. Cristiani

F2AMG15:45 – 17:25

Chair: S. Nash

F2-1: 15:45 - 16:10An algebraic multigrid method with proved con-vergence rate I: two-grid analysisA. Napov

F2-2: 16:10 - 16:35Algebraic interface in sparse matrix and its ap-plication in AMGX. Xu

F2-3: 16:35 - 17:00Algebraic multigrid based on clusteringL. Grasedyck

F2-4: 17:00 - 17:25Towards robust algebraic multigrid for nonsym-metric problemsJ. Lottes

Wednesday, Sept. 22 Wednesday, Sept. 22

Wednesday,September 22


Schedule

Time Talk Room







15:15 - 15:45 Tea Break



IS5Symbol approach forstructured matrices inmultigrid with appli-cations in imaging9:15 - 10:00

We consider the deblurring problemof noisy and blurred images in thecase of space invariant point spreadfunctions. The use of appropriateboundary conditions leads to lin-ear systems with structured coeffi-cient matrices related to space in-variant operators like Toeplitz, cir-culants, trigonometric matrix alge-bras etc. We propose to combinethe an algebraic multigrid (which istypical for structured matrices) withthe low-pass filtering properties ofthe classical geometrical multigridused in a PDEs context or with theTikhonov regularization. In the firstcase, using an appropriate smoother,we obtain an iterative regularizingmethod, while in the second case weobtain an optimal technique whichseems to be robust with respect tothe regularization parameter.In the talk we first review the theo-retical ground given by the conver-gence theory in the case of struc-tured matrices and then we presentthe adaptation in the case of imagerestoration problems. More in de-tail, we will emphasize (both in thetheoretical study and in the applica-tions) the role of the generating func-tion in two directions: A) in orderto minimize the storage requirementsand the complexity of every single it-eration, and B) in order to obtainan optimal method, that is an it-erative technique whose convergencerate is independent of the size of theinvolved matrices and depending onsome analytical features of the sym-bol.

M. DonatelliDipartimento di Fisica e Matem-atica, Universita dell’Insubria,Italy

IS6Multilevel MonteCarlo method10:00 – 10:45

The multilevel Monte Carlo methodtakes the multigrid philosophy ofworking with different levels of res-olution to achieve fine grid accuracyat a coarse grid cost, and applies itto the simulation of stochastic differ-ential equations (SDEs) and partialdifferential equations (SPDEs).Unlike standard multigrid methodswhich are concerned with the effi-cient iterative solution of large sys-tems of equations, the objective inthe multilevel Monte Carlo methodis the accurate estimation of an ex-pected value which is a functional ofthe solution of the SDE or SPDE.The key observation is that the ex-pected value on the finest level of res-olution can be expressed as the ex-pectation on the coarsest level of res-olution, plus a sum of expected cor-rections between neighbouring levelsof resolution.This talk will introduce the basicideas, motivated by applications incomputational finance. It will alsodiscuss new research with R. Scheichland K.A. Cliffe on elliptic SPDEs inthe simulation of oil reservoirs andnuclear waste repositories.

M. GilesOxford-Man Institute of Quanti-tative Finance, Oxford Univer-sity, United Kingdom


A3Optimization and Control11:15 – 12:30

Chair: M. Hinze

A3-1: 11:15 - 11:40Some results in the optimal control of electro-magnetic fieldsF. Troltzsch

A3-2: 11:40 - 12:05Algebraic multigrid for nonsymmetric con-strained problemsT. Wiesner

A3-3: 12:05 - 12:30Efficient preconditioning of optimality systemsK.-A. Mardal

B3Elliptic Problems11:15 – 12:30

Chair: T. Clees

B3-1: 11:15 - 11:40An adaptive parallel geometric space-timemultigrid algorithm for the heat equationT. Koppl

B3-2: 11:40 - 12:05A Multigrid approach for Poisson equation withmixed boundary conditions in arbitrary domainA. Coco

B3-3: 12:05 - 12:30Tau-extrapolation on 3D semi-structured finiteelement meshesB. Gmeiner



C3Toeplitz and LFA14:00 – 15:15

Chair: X. Xu

C3-1: 14:00 - 14:25An iterative multilevel regularization methodfor deblurring problemsM. Donatelli

C3-2: 14:25 - 14:50Aggregation and smoothed aggregation-basedmultigrid methods for circulant and ToeplitzmatricesM. Bolten

C3-3: 14:50 - 15:15Accuracy measures and Fourier analysis for thefull multigrid methodC. Rodrigo

D3AMG Application14:00 – 15:15

Chair: D. di Serafino

D3-1: 14:00 - 14:25On the use of AMG for electrochemical platingprocessesP. Thum

D3-2: 14:25 - 14:50Efficient structured AMG preconditioners forthe cardiac Bidomain model in 3DM. Pennacchio

D3-3: 14:50 - 15:15AMG and micromechanicsM. Kabel

15:15 - 15:45 Tea Break


E3FEM/FVM15:45 – 17:25

Chair: J. van der Vegt

E3-1: 15:45 - 16:10Auxiliary space preconditioner for a locking-freefinite element approximation of the linear elas-ticity equationsE. Karer

E3-2: 16:10 - 16:35Subspace correction method for discontinuousGalerkin discretizations of linear elasticity equa-tionsJ. Kraus

E3-3: 16:35 - 17:00Local Fourier analysis for quadratic finite ele-ment methodsF. Gaspar

E3-4: 17:00 - 17:25On finite volume multigrid methodK.S. Kang

F3AMG15:45 – 17:25

Chair: M.W. Gee

F3-1: 15:45 - 16:10Point-based AMG for coupled circuit and devicesimulationN. Mannig

F3-2: 16:10 - 16:35Adaptive algebraic multigrid: A bootstrap ap-proachK. Kahl

F3-3: 16:35 - 17:00A projected AMG for LCPsJ. Toivanen

F3-4: 17:00 - 17:25Algebraic multigrid with proved convergence II:automatic coarsening, multilevel convergenceY. Notay

Wednesday, Sept. 22

Wednesday,September 22

Aperitivo19:30 - 20:00

Conference Dinner20:00 - 22:00

ISPowertrain engineering needsmathematics17:40 - 18:25

With increasing interest in the development of hy-brid (HEV) and electrical (PEV) vehicles, the de-mand for comprehensive system design and anal-ysis to support the powertrain development pro-cess (PDP) is rising. The development of highlysophisticated modern powertrains as e.g. todaysHEVs is even not possible without applying math-ematical simulation quite from the beginning ofthe development process. In that respect moreadvanced mathematical models of great variety intheir scope, complexity and sophistication to rep-resent propulsion systems and components (bothsteady state and dynamic) are requested. Basedon the flexibility of mathematical models it is pos-sible to adjust them to the requirements of allphases of the product development process fromsimple (fast) models to very complex (CPU inten-sive) models.Mathematical models are capable to represent thereal- world phenomena and by applying them withnumerical or analytical techniques, both qualita-tive and quantitative predictions can be made.Additionally mathematical, physical and chemi-cal models are capable to be re- used for simulat-ing a great variety of phenomena occurring in dif-ferent fields of applications, e.g. Finite Elements(FEM), Computational Fluid Dynamcs (CFD) orMulti Body Dynamics (MBD) have become firmlyestablished tools supporting the product develop-ment process, complementing and more and morereducing the traditional experimental approaches.Based on various case studies this presentationshall give an insight on the one hand into a) thevertical applications of mathematical methods forpowertrain system and component simulation in-cluding optimization, control development and de-sign along the development process; and on theother hand into b) the horizontal deployment ofmathematical methods showing the derivation ofapplications, e.g. for the pharmaceutical industryor the process industry from methods originallycreated for engine and powertrain development.

G. RainerVice President, Advanced Simulation Tech-nologies, AVL List, Austria, Graz

Thursday, Sept. 23 Thursday, Sept. 23

Thursday,September 23


Schedule

Time Talk Room







IS7Multiscale methodsfor edge detection andimage segmentation9:15 - 10:00

Detecting object boundaries in im-ages is essential for their accurateinterpretation. Multiscale methodscan take an important role in thistask, as they provide means to adap-tively overcome noise and to incor-porate object properties of varyingcomplexities. In this talk I will de-scribe multiscale methods for edgedetection and image segmentation.For edge detection we consider meth-ods that overcome noise by applyingfilters whose shape is adapted to theshape of the sought edges. I willfurther present results on the lim-its of detectability, as a function ofthe lengths of edges and their combi-natorics. For image segmentation Iwill describe an efficient method in-spired by Algebraic Multigrid. Thismethod utilizes features whose ex-pressive power increases with the sizeof the sought segments.

R. BasriDepartment of Computer Scienceand Applied Mathematics, Weiz-mann Institute of Science, Israel

IS8Algebraic multigrid inmultiphysics and mul-tiscale10:00 – 10:45

Multifield and multiscale phenom-ena such as fluid-structure inter-action and modeling of turbulentflows are considered. While multi-grid methods and multigrid princi-ples have been excessively studiedfor single field phenomena, less at-tention has been given to the ap-plication of multigrid principles incomplex multiphysics and multiscalesimulations. In the last view yearswe have not only extensively appliedAMG as a solver in various real worldmultiscale and multifield (m&m) ap-plications but also adopted AMGideas as building blocks for the de-sign of new m&m methods. In thistalk, we focus on two recent develop-ments made in the field of algebraicmultigrid (AMG) principles: Thefirst is a monolithic AMG schemefor the implicit solution of fluid-structure interaction (FSI) simula-tions. Therein, an AMG hierar-chy for the nonsymmetric monolithicfluid, structure and mesh movementsystem of equations is constructedthat also considers a coarse represen-tation of interfield coupling in a vari-ationally consistent way. The sec-ond is an algebraic variational mul-tiscale method for convection domi-nated problems and turbulent flow.Therein, simple plain aggregationtransfer operators that possess a pro-jective property are used to constructscale separating operators that allowa purely algebraic scale separationprocess. Opposed to AMG as a so-lution method, here AMG principlesare utilized as modeling tool that in-fluences result behavior. The result-ing method yields an efficient, sta-ble and very accurate scheme thathas been utilized in convection dom-inated problems, in LES simulationof incompressible turbulent and com-pressible lowMach turbulent flow.

M.W. GeeInstitute for Computational Me-chanics, Technische UniversitatMunchen, Germany

Thursday, Sept. 23 Thursday, Sept. 23

A4Multiscale/Multiresolution11:15 – 12:55

Chair: Y. Notay

A4-1: 11:15 - 11:40Lifting in hybrid lattice Boltzmann and PDEmodelsY. Vanderhoydonc

A4-2: 11:40 - 12:05Multigrid preconditioners for adaptive waveletcollocationS. Bertoluzza

A4-3: 12:05 - 12:30Equivalent preconditioning in multiscale prob-lemsW. Vanroose

B4AMG, Optimization and Control11:15 – 12:55

Chair: A. Borzı

B4-1: 11:15 - 11:40Point smoothers for elliptic optimal controlproblemsS. Takacs

B4-2: 11:40 - 12:05Algebraic multigrid for density driven flowA. Nagel

B4-3: 12:05 - 12:303D Multilevel aggregation for segmentation andtracking of live cellsH. De Sterck


Part II

Collection of Abstracts

29

Monday

IS1Hierarchical tensor representationW. Hackbusch

We are not only interested in the efficient storage ofhigh-dimensional tensors, but also in the efficient evalua-tion and possibly approximation of the tensor operations.Since all types of representation tend to increase a cer-tain rank under the operations, an efficient tensor calculusmust contain an efficient method for the rank truncationof the intermediate results. Here, the traditional represen-tations are less attractive, since rank truncation leads tononlinear optimisation problems.

In the lecture we describe a new representation schemewith the following properties: 1) tensors represented inthe n-term representation, in the Tucker format, or inthe sparse-grid form are exactly representable in the newscheme with similar storage cost as in the original form.2) The truncation can be performed (non-iteratively) onlyon the basis of singular value decompositions.

The operations count for the basic tensor operationscan be described. In particular, they are linear in the order(dimension) of the tensor.

IS2Computational optimization of systems governedby partial differential equationsV. Schulz

This talk tries to present a survey on the computa-tional challenges as well as solution approaches in thefield of PDE constrained optimization. This vivid fieldof research combines techniques from optimization as wellas numerical analysis. Naturally, multigrid optimizationmethods play an important role but also so-called one-shot methods which iterate simultaneously over the stateand optimization variables. The key to efficient optimiza-tion methods is the exploitation of problem structures.Examples of these structures in the area of PDE con-strained optimization are multigrid hierarchies, iterativesolvers and resulting adjoint iterations, optimization pre-conditioners, shape calculus etc. From the abundance ofapplication problems within this discipline, some examplesare selected to highlight important features and to demon-strate the computational potential of gradient based meth-ods dovetailed to the respective PDE constraints.

A1-1An AMG method for the graph Laplacian usingmatching of graphsJ. Brannick

We introduce an algebraic multilevel method for theGraph Laplacian based on matching of graphs. We showthat the resulting piecewise constant coarse spaces andan appropriately chosen Algebraic Multilevel Iteration (or

AMLI) method yield an uniformly convergent solver. Inaddition, we introduce a new polynomial smoother andtheory for estimating its performance in practice.

A1-2

A non-hermitian coarsening strategy for algebraicmultigrid

S. MacLachlan

While algebraic multigrid algorithms are well-establishedas effective solvers and preconditioners for Hermitian andpositive-definite matrices, there has been much recent in-terest in extending these algorithms to non-Hermitian prob-lems. Motivated by applications such as fluid dynam-ics, where non- Hermitian operators naturally arise, theseapproaches attempt to retain the AMG setup and cy-cling structure, but replace the relaxation, interpolation,and restriction operators with choices that are more ap-propriate for non- Hermitian problems. In this talk, Iwill outline a strategy for coarse-grid selection that natu-rally addresses non-Hermitian matrices. In particular, theproposed approach requires no assumptions on M-matrixstructure or diagonal dominance of a non-Hermitian ma-trix in order to generate a good coarse-grid structure, incontrast with other approaches that are derived from theHermitian case. The algorithm is closely related to multi-level block factorization algorithms that are known to beeffective for a wide range of problems. Numerical resultsshow that the approach can be nicely integrated into ex-isting AMG codes, with reasonable performance on modelproblems.

A1-3

Stopping and restarting criteria for Krylov accel-erated AMG solvers

T. Clees

Algebraic multigrid methods are efficient solvers orpreconditioners for a broad range of linear systems stem-ming from discretized partial differential (algebraic) equa-tions. Quite often, they are accelerated by means of Krylovmethods. Appropriate stopping criteria are necessary forthe resulting iterative solver. They shall account for thereduction of errors, not only residuals, and prevent themethod to iterate for ever. Additionally, restarts of theiter- ation process might be necessary in order to arriveat more accurate results. Stopping and restarting crite-ria [1, 2] are revisited and a novel strategy is proposed,based on an analysis of residuals and their approximativeanalogs recursively computed during the Krylov run aswell as on the behaviour of the AMG preconditioner em-ployed. Several benchmark cases, taken from industrialsimulation runs, show the efficiency of the novel criteriaas well as the overall strategy proposed.

[1] R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato,J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. Van der Vorst,Templates for the Solution of Linear Systems: Building Blocks forIterative Methods, 2nd ed., SIAM, Philadelphia (PA), 1994.

http://www.netlib.org/templates/templates.pdf.

[2] A. Greenbaum, Estimating the attainable accuracy of recur-sively computed residual methods, SIAM J. Matrix Anal. Appl.,18 (1997), pp. 535551.

B1-1

Convergence and descent properties for a class ofmultilevel optimization algorithms

S. Nash

I present a multilevel optimization approach (termedML/Opt) for the solution of constrained optimizationproblems. The approach assumes that one has a hierar-chy of models, ordered from fine to coarse, of an under-lying optimization problem, and that one is interestedin finding solutions at the finest level of detail. In thishierarchy of models calculations on coarser levels are lessexpensive, but also are of less fidelity, than calculationson finer levels. The intent of ML/Opt is to use calcula-tions on coarser levels to accelerate the progress of theoptimization on the finest level. Global convergence (i.e.,convergence to a Karush-Kuhn-Tucker point from an arbi-trary starting point) is ensured by requiring a single step ofa convergent method on the finest level, plus a line-search(or other globalization technique) for incorporating thecoarse level corrections. The convergence results applyto a broad class of algorithms with minimal assumptionsabout the properties of the coarse models. I also ana-lyze the descent properties of the algorithm, i.e., whetherthe coarse level correction is guaranteed to result in im-provement of the fine level solution. Although additionalassumptions are required in order to guarantee improve-ment, the assumptions required are likely to be satisfiedby a broad range of optimization problems.

B1-2

Multigrid methods for state-constrained elliptic op-timal control problems

M. Vallejos

An elliptic optimal control problem with constraintson the state variable is considered. The Lavrentiev typeregularization is used to treat the constraints on the statevariable. To solve the problem numerically, the multigridfor optimization (MGOPT) technique is implemented. Nu-merical results are reported to illustrate the efficiency ofthe multigrid strategy.

[1] A. Borzı, Smoothers for control- and state-constrained opti-mal control problems , Computing and Visualization in Science, 11(2008), pp. 59–66.

[2] C. Meyer, A. Rosch, and F. Troltzsch, Optimal Control ofPDEs with Regularized Pointwise State Constraints, ComputationalOptimization and Applications, 33 (2006), pp. 209–228.

[3] S. Nash, A multigrid approach to discretized optimizationproblems, Optimization Methods and Software, 14 (2000), pp. 99–116.

B1-3A space-time multigrid solver for optimal distributedcontrol of incompressible fluid flowM. Koster, S. Turek, M. Hinze

We present a multigrid solution concept for the op-timal distributed control of the time-dependent Navier-Stokes equation. This problem is described by a fully cou-pled KKT system in space and time involving primal anddual variables for velocity and pressure. In this talk wepresent basic concepts and ingredients which are necessaryfor setting up a hierarchical solver for such systems. Theunderlying KKT system is discretised in a monolithic wayon the whole space-time domain using finite elements inspace and a one-step-θ-schemes in time. A global Newtonsolver is applied to solve for the nonlinearity, while a space-time multigrid solver is used for the linear subproblems.We obtain a robust solver whose convergence behaviour isquite independent of the number of unknowns of the dis-crete problem and robust with respect to the consideredflow configuration. A set of numerical examples demon-strates the feasibility of this approach.

C1-1Analysis of a perturbed two-grid preconditionerfor indefinite three-dimensional Helmholtz prob-lemsH. Calandra, S .Gratton, X. Pinel, X. Vasseur

We study the three-dimensional Helmholtz equationwritten in the frequency domain modeled by the followingpartial differential equation:

−∆u− k2u = s

with absorbing boundary conditions (PML [1]), where uis the pressure of the wave, k = 2πf

v its wavenumber, fthe frequency, v the propagation velocity in the heteroge-neous media and s a Kronecker function that representsthe wave source. This problem is discretized using second-order finite difference techniques leading to huge linearsystems for large wavenumbers. Following Elman [2], weuse a geometric two-grid preconditioner for a Krylov sub-space method (namely flexible GMRES [4]) as a solutionmethod [3]. Due to the large dimension of the coarse gridproblem we focus on the behavior of a two grid algorithmwhere the coarse problem is not solved exactly. We usea Krylov subspace method to solve this problem only ap-proximately. This leads us to analyze which stopping cri-terion should be used for the coarse grid solver. Since thetwo-grid method is used as a preconditioner, we are inter-ested in the spectrum distribution of the preconditionedHelmholtz operator. A local Fourier analysis enables usto compute the spectrum of this preconditioned operatoraccording to the coarse tolerance ε2h. This analysis em-phasizes the influence of ε2h on the distribution of thespectrum of the preconditioned Helmholtz operator andprovides a tool to choose an appropriate coarse tolerance;considering the difference between the spectra of the pre-conditioned Helmholtz operator for a certain ε2h > 0 and

for ε2h = 0. We also investigate the numerical behaviourof the two-grid method when it is used as a preconditionerfor heterogeneous Helmholtz problems. This will help usunderstanding the effects of the approximate coarse gridsolution on the preconditioner when absorbing boundaryconditions are considered. Parallel numerical experimentswill conclude this talk showing the efficiency of this per-turbed preconditioner at large wavenumbers.

[1] J.-P. Berenger, A perfectly matched layer for absorption ofelectromag- netic waves, J. Comp. Phys., 114 (1994), pp. 185200.

[2] H. R. Elman, O. G. Ernst and D. P. OLeary, A multi-grid method enhanced by Krylov subspace iteration for discreteHelmholtz equations, SIAM J. Sci. Comput., 23 (2001), pp. 1291315.

[3] X. Pinel, A perturbed two-level preconditioner for the solu-tion of three- dimensional heterogeneous Helmholtz problems withapplications to geo- physics, PhD thesis, CERFACS, 2010.

[4] Y. Saad, A flexible inner-outer preconditioned GMRES al-gorithm, SIAM J. Sci. Comput., 14 (1993), pp. 46469.

C1-2

Multigrid preconditioners for the Helmholtz equa-tion on complex stretched grids

B. Reps, W. Vanroose, H. bin Zubair

We make use of exterior complex stretching (ECS), aspecific type of absorbing boundary conditions similar toperfectly matched layers [1], to solve Helmholtz equations.In particular we are interested in problems that origi- natefrom single and multiple ionization of atoms and molecules[2, 3]. The robust ECS method does not need an explicitinput of the wave number in contrast to classical Sommer-feld conditions. The equation is solved numerically usingKrylov methods preconditioned with the complex shiftedLapla- cian (CSL) proposed by Erlangga, Vuik and Oost-erlee [4] and originally designed for Sommerfeld boundaryconditions. We also propose an alternative precondi-tioner based on the same original Helmholtz equation, butsolved on a different complex stretched grid (CSG). Bothpreconditioners are approximately inverted with a multi-grid cycle. The ECS technique allows a straightforwardnumerical eigenvalue analysis of the discretized operatorsin order to comprehend convergence behavior. We presentthe performance of both the CSL and CSG preconditionersapplied to the model problems and compare ECS bound-ary layers to Sommerfeld conditions [5].

[1] J.-P. Berenger, A perfectly matched layer for the absorptionof electromagnetic waves, J. Comput. Phys., 114 (1977), pp. 185200.

[2] L. Tao, W. Vanroose, B. Reps, T. Rescigno, C.W. McCurdy,Long-time solution of the time- dependent Schroedinger equationfor an atom in an electromagnetic field using complex coordinatecontours, Phys. Rev. A, 80 (2009).

[3] W. Vanroose, F. Martin, T. Rescigno, C.W. McCurdy, Com-plete photo-induced breakup of the H2 molecule as a probe of molec-ular electron correlation, Science, 310 (2005), pp. 17871789.

[4] Y.A. Erlangga, C.W. Oosterlee, C. Vuik, A novel multi-grid method for the Helmholtz equation, SIAM J. Sci. Comput., 27(2006), pp. 14711492.

[5] B. Reps, W. Vanroose, H. bin Zubair, On the indefiniteHelmholtz equation: exterior complex stretched absorbing boundarylayers, iterative analysis and preconditioning, under revision for J.Comput. Phys.

C1-3

A frequency-robust solver for eddy current prob-lems

U. Langer

In many practical applications, for instance, in compu-tational electromagnetics, the excitation is time-harmonic.Due to the time-harmonic excitation, we can switch fromthe time domain to the frequency domain. Al least in thecase of linear problems this allows us to replace the expen-sive time- integration procedure by the solution of a simplelinear elliptic system for the amplitudes belonging to thesine- and to the cosine-excitation. The fast solution ofthe corresponding linear system of finite element equationsis crucial for the competitiveness of this method. J. Schoe-berl and W. Zulehner (2007) proposed a new parameter-robust MinRes preconditioning technique for saddle pointproblems This method allows us to construct a frequency-robust multigrid-preconditioned MinRes solver. The con-struction of this solver is outlined for a parabolic initialboundary value problem. The generalization of this pre-conditioned MinRes solver to linear time- harmonic eddycurrent problems in electromagnetics is not straight for-ward. Due to the non-trivial kernel of the curl operator wehave to perform an exact regularization of the frequencydomain equations, in order to provide a theoretical basisfor the application of the MinRes preconditioner. Finally,we discuss the application of this solver to linear parabolicinitial boundary value problems with non-harmonic exci-tation and to non-linear problems in the framwork of themultiharmonic technique. The authors acknowledge thesupport by the Austrian Science Fund (FWF) under thegrant P19255.

D1-1

Multigrid of the second kind for the optimal con-trol of time-periodic, parabolic, partial differentialequation

D. Abbeloos, M. Diehl, M. Hinze, S. Vandewalle

We present a multigrid method of the second kindto optimize time-periodic, parabolic, partial differentialequations (PDE). We consider a quadratic tracking ob-jective with a linear or a non-linear PDE constraint. Inboth cases the first order optimality conditions are givenby a coupled system of boundary value problems. Wepresent a derivation of the first and second order opti-mality conditions in the setting of [3]. In the linear casethe first order conditions can be rewritten as an integralequation of the second kind, which is solved by multigridof the second kind [2]. The evaluation of the integral op-erator consists of solving sequentially a boundary valueproblem for respectively the state and the adjoints. Bothare solved efficiently by a space-time multigrid method [1].The nonlinear case is treated by applying a Hessian-freeNewton method in the control space.

[1] G. Horton, S. Vandewalle, A space-time multigrid methodfor parabolic P.D.E.S., SIAM J. Sci. Comp., Vol 16 (1995), pp.

848–864.

[2] P.W. Hemker, H. Schippers, Multiple Grid Methods for thesolution of Fredholm Integral Equations of the Second Kind, Math.Comp., Vol 36 (1981).

[3] M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimizationwith PDE constraints, Mathematical Modelling: Theory and Appli-cations, Vol 23 (2008), Springer

D1-2

Multigrid second-order accurate solution of paraboliccontrol-constrained problems

S. Gonzalez-Andrade, A. Borzı

A mesh-independent and second-order accurate multi-grid framework to solve parabolic optimal control prob-lems is presented. The resulting algorithms appear to berobust with respect to change of values of the control pa-rameters and have the ability to accommodate constraintson the control also in the limit case of bang-bang control.Central to the development of these multigrid schemes isthe design of iterative smoothers which can be formulatedas local or block semismooth Newton methods. The de-sign of distributed controls is considered to drive nonlinearparabolic models to follow optimally a given trajectory orattain a final configuration. In both cases, results of nu-merical experiments and theoretical twogrid local Fourieranalysis estimates demonstrate that the proposed schemesare able to solve parabolic optimality systems with text-book multigrid efficiency. Further results are presentedto validate second-order accuracy and the possibility totrack a trajectory over long time intervals by means of areceding-horizon approach.

[1] A. Borzı and V. Schulz, Multigrid methods for PDE opti-mization, SIAM Review, 51 (2009), 361-395.

[2] S. Gonzalez-Andrade and A. Borzı, Multigrid second-orderaccurate solution of parabolic control-constrained problems, submit-ted.

D1-3

Fast numerical schemes for optimal control prob-lems with Fredholm constraints

M. Annunziato, A. Borzı

The formulation of optimal control problem governedby Fredholm integral equations of the second kind and tworobust and fast numerical schemes for these problem ispresented. Existence and uniqueness of optimal solutionsin the framework of continuous optimization is proved. Anew block Gauss- Seidel method in the context of one-grid iteration and multi-grid scheme are used as solversto the associated discrete optimality system. Robustnessand efficient computational complexity of both schemes isproved by local Fourier analysis. These features of the pro-posed numerical schemes are confirmed by some numericaltests.

[1] A. Borzı and V. Schulz, Multigrid methods for PDE opti-mization, SIAM Review, 51 (2009), 361-395.

[2] M. Annunziato and A. Borzı, Fast solvers of Fredholm op-timal control problems, Numerical Mathematics: Theory, Methodsand Applications (NM-TMA).

E1-1AMG solver for Helmholtz equationsI. Livshits

We discuss an adaptive algebraic multigrid algorithmfor solving Helmholtz equations. Our approach is rem-iniscent of the wave-ray algorithm developed by Brandtand Livshits in the geometric framework - it is too pro-vides a special treatment to the null- and near-null com-ponents of the Helmholtz operator. Unlike it predeces-sor, our algorithm does not have to rely on the knowledgeof differential equations (can use a discrete formulation),analytical solutions to the homogeneous equation (cal-culates and employs numerical approximations to it), andit does not necessarily requires structured grids. It em-ploys algebraic multigrid techniques developed in the lastdecade to achieve an optimal multigrid efficiency, as nu-merical results for one- and two-dimensional Helmholtzequations show.

E1-2Multigrid achievements for the thin layer flow mod-eling in rolling element bearingsC.H. Venner

Rolling element bearings in our cars, motorcycles, bi-cycles, machines, and other daily appliances are often soreliable that one is unaware of the fact that their success-ful operation entirly depends on the formation of a thinlubricant film of some 10-200 nanometers separating thesurfaces of the roller and raceway contacts to prevent wear.Due to the non-conformity of the contacts the pressuresmay rise to 1-3 GPa. At these pressures the deformationof the surfaces is considerable. Therefore this regime oflubrication is generally referred to as Elastohydrodynamiclubrication. Accurate prediction of the film thickness andpressure in the lubricant film is a prerequisite to predic-tion of the service life of bearings. Trends in design aretowards lighter constructions, higher loads, higher tem-peratures, and as a result smaller lubricant films. Ac-curate predictions are more important than ever before.Prediction of the film thickness requires the solution ofa time-varying thin film equation (Reynolds equation) incombination with an exponential viscosity pressure lawand an integral equation describing the elastic deforma-tion of the surfaces. Standard numerical methods requireO(N3) operations. The introduction and further devel-opment of Multigrid techniques for the solution of theequations as well as for the fast evaluation of the con-volution integrals describing the elastic deformation hasled to a major breakthrough in this field. At present thecapabilities to numerically simulate the behaviour of EHLcontacts are so excellent that practically realistic prob-lems can be solved on small scale computers. In additionnew approaches have been developed to predict long termfilm decay in actual bearings. The predictions have beenvalidated using optical interferometry measurements. Inthe lecture an introduction is given to the problem, anoutline of the many considerations crucial to the develop-

ment, of efficient multilevel algorithms, as well as a totallynew approach taken to film decay modelling in bearings.Numerical and experimental results are compared and anoutline is given of future research.

E1-3

Multigrid algorithms for sparse representation ofsignals

I. Yavneh, M. Elad, E. Treister, J. Turek

Algorithms for signal processing tasks necessarily relyon a-priori knowledge about the signal characteristics. Oneof the most exciting related developments in recent yearsis the realization that signals often admit sparse represen-tation over suitable dictionaries that are known or can belearned automatically. Here, a dictionary, D, is a (typi-cally) real-valued matrix, whose columnscalled atomsarethe presumed building blocks of signals of interest. Theunderlying assumption is that signals can be approximatedwell by a linear combination of just a small number ofatoms. This approach has proved to be very effective in awide range of signal processing tasks. For example, in theprocess of denoising a corrupted signal, one may seek asignal (vector) that is as close as possible to the measuredcorrupted one, while being sufficiently sparse, in the sensethat it can be represented as a linear combination of just afew atoms. For a corrupted signal y, the compu- tationaltask might then be to minimize ||Dxy||22 over all represen-tations x containing no more than some given number ofnonzero elements. The number of nonzero elements de-pend on the noise level. In this work we study multigridapproaches for computational problems associated withsuch tasks. In particular, we develop and test monotoneaggregation-based algebraic multigrid algorithms employ-ing the so-called Ex- act Interpolation Scheme.

F1-1

Local volatility estimation using GP-GPU acceler-ation

C.C. Douglas, H. Lee, D. Sheen

We solve an inverse problem for the local volatilitymodel in option pricing using several algorithms. Wecompare the solution times and different solution algo-rithms using traditional multicore CPUs versus GP-GPUaccel- erated parallel solutions. Numerical results confirmthe superiority of the proposed methods on GP-GPU ar-chitectures.

F1-2

Multigrid algorithms on multi-GPU architectures

H. Kostler

Many real world applications involving multigrid al-gorithms require high performance computing, eitherbecause of time constraints like for real-time simulationsor because of a large amount of data to be processed.GPUs offer high computational performance at low costand are therefore an interesting archi- tecture especiallyfor data parallel applications. Recently, more and moreGPU HPC clusters arise1 and thus there is a need foradapting multigrid codes to Multi-GPU environments. Inthis talk we discuss software development for these clus-ters, compare CUDA2 and OpenCL3 , and show perfor-mance results on different GPU platforms and clusters.We use multigrid as a numerical solver for linear and non-linear PDEs and present applications from imaging andcomputational science and engineering. The goal in imag-ing is to achieve maximum single GPU performance toenable real-time computations. For CSE simulations dueto the huge amount of data multi-GPU support becomesmandatory. We will show scaling results for multigrid onGPU clusters.

F1-3

Multigrid in quantum chemistry on multiple GPUs

D. Ritter, H. Kostler, R. Schmid, C. Feichtinger, U. Rude

Graphics processing units (GPUs) provide high com-putational power and are well suited for many high-performancecomputing applications. Espe- cially stencil-based codesgain profit from their enormous floating-point performanceand memory bandwidth when adapted carefully. For ex-ample, a standard V (2, 2)-cycle for the 2D-Poisson equa-tion can be implemented to execute in only one nanosec-ond per unknown on a single graphics board. The com-putation of the electrostatic potential field is of major in-terest in quantum chemistry. Fourier space methods arecommonly used for the solution of the Poisson equation,but they are difficult to apply if different types of bound-ary conditions are mixed. In that case, solving the equa-tion in real space via a fast multigrid solver is a promis-ing approach (previous research, see [1]). We introduce amultigrid algorithm for the potential equation that is im-plemented with the help of waLBerla-core, a massivelyparallel, generic framework for stencil-based codes. Thisframework is primarily developed for computational fluiddynamics using the Lattice Boltzmann method and in-corporates hybrid parallelization functionality (i.e. mul-tiple threads on one NVIDIA R GPU via CUDA R andmultiple GPUs via MPI), as well as support for differ-ent architectures. The code was designed for large-scalesimulations on arbitrary grid sizes with mixed Neumann,Dirichlet and periodic boundary conditions. We apply afinite difference discretization and solve the arising linearsystem by a multigrid method. Besides the implementa-tion within the waLBerla framework, we present resultsof performance and scaling experiments on a multi-GPUcluster. Fur- thermore, we demonstrate how our programcan be integrated into a larger quantum chemistry pro-gram for ab initio molecular dynamics [2].

[1] H. Kostler, R. Schmid, U. Rude, Ch. Scheit: A parallelmultigrid accelerated Poisson solver for ab initio molecular dynamicsapplication. In: Computing and Visualization in Science, 11: 115-122, Heidelberg: Springer, 2008.

[2] R. Schmid, M. Tafipolsky, P. H. Konig, H. Kostler: Car-Parrinello molecular dynamics using real space wavefunctions. Phys.Stat. Sol. (B), 243: 1001-1015, 2006.

Tuesday

IS3

Multigrid methods for partial differential equationswith random coefficients

S. Vandewalle

Mathematical models of real life engineering and sci-entific processes typically depend on parameters which areknown only approximately or which are inherently vari-able and stochastic. These models often take the form ofa stochastic partial differential equation with coefficientsthat are random variables, fields, or processes.

A popular method to determine the stochastic char-acteristics of the PDE solution is the so-called stochas-tic finite element method. This method approximates thesolution of the PDE by a generalized polynomial chaosexpansion. By using a Galerkin projection in the stochas-tic dimension, the original problem is transformed into acoupled set of deterministic PDEs. A finite element dis-cretization converts this deterministic PDE system into ahigh dimensional algebraic system.

In this talk, we shall present an overview of iterativesolution approaches, [1]. We start from iterative meth-ods based on a block splitting of the system matrices.Next, we extend these methods for use as preconditionerfor a Krylov method, and for use as smoother in a multi-level context. Then, the various solvers will be comparedbased on their convergence properties, computational costand implementation effort. Our findings are illustratedby means of two numerical problems. The first one is asteady-state diffusion problem with a discontinuous ran-dom field as diffusion coefficient. The second is a deter-ministic diffusion problem defined on a random domain.

[1] E. Rosseel and S. Vandewalle, Iterative solvers for the stochas-tic finite element method, SIAM J. Sci. Comput., Vol. 32, No. 1(2010),pp. 372–397.

IS4

Optimizing multigrid performance for higher orderaccurate space-time discontinuous Galerkin discretiza-tions of advection dominated flows

J. van der Vegt

Higher order accurate space-time discontinuous Galerkinfinite element methods are well suited for time-dependentproblems requiring moving and deforming meshes, suchas fluid-structure interaction and nonlinear water waves.Space-time DG methods share many of the benefits of DGmethods, such as their suitability for hp-adaptivity andparallel computing. The algorithm results, however, ina large system of (non)linear algebraic equations whichneed to be solved each time step. In this presentationwe will discuss various methods to improve the efficiencyof multigrid algorithms for space-time DG methods, withspecial emphasis on higher order accurate discretizationsof advection-dominated flows.

A2-1A multigrid method for systems of hyperbolic con-servation lawsS. Amarala, J. W.L. Wan

In this talk, we present a multigrid method for com-puting the steady state solution of systems of partial dif-ferential equations (PDEs) of the form:

Ut +

d∑i=1

Fi(U)xi= G(x),

where U(x, t) is an m-dimensional vector of conserved quan-tities, Fi(U) are the flux functions, and G(x) is a sourceterm. Due to the CFL condition, the small time step sizemay lead to a long computational time. Multigrid meth-ods have been exploited extensively for solving scalar con-vection dominated problems, for example, downwindingGauss-Seidel smoothers, and kernel preserving multigridwith restriction operators capturing the kernel of the PDE.The main idea of these approaches is linked to the upwind-ing concept for discretizing hyperbolic PDEs. However, inthe system case, the upwinding or the characteristic di-rections of the unknown quantities are not as apparent asin the scalar case. We propose an interpolation methodwhich is constructed by solving an appropriately definedlocal Riemann problem. We determine the upwinding di-rection for interpolation from the Riemann problem solu-tion which is easy to compute. The restriction operatoris defined similarly. We will discuss the issue of oscilla-tion by analyzing the monotonicity preserving property ofthe multigrid scheme. We demonstrate the effectivenessof the multigrid method by numerical examples includingthe Euler equations.

A2-2Defect correction with algebraic multigrid for lin-earized problems in compressible aerodynamicsA. Naumovich, M. Forster

In the present work the solution of linearized discretiza-tions of compressible Euler and Navier-Stokes equationsis investigated. In particular, the second- order accuratefinite volume discretizations built on unstructured gridsare considered. The goal of this work is to solve the aris-ing linear systems more efficiently than the currently usedlinear solver in the DLR TAU code [4] (linear geo- metricmultigrid accelerated with GMRES) by means of algebraicmultigrid (AMG) [1, 2]. We turn to AMG since it hasthe potential to automatically deal with arbitrary sourcesof stiffness on unstructured grids. In order to solve thecon- sidered problems, proper extensions of AMG algo-rithms suitable for systems of PDEs are necessary (avail-able e.g. in the SAMG package [3], which is employedin the present work). We apply an aggregation-basedAMG which requires some modifications to better suitadvection-dominated nature of the considered problems.The linear systems arising from the considered second-order discretizations are very stiff, and direct application

of the existing AMG solvers is problematic. Our researchis going into two directions, first - development of an AMGsolver which can be directly applied to solve second-orderdis- cretizations, and second - application of AMG withina defect correction (DC) iteration [5]. This work is con-cerned with the latter approach, where at each step ofthe DC algorithm AMG is applied to a first-order accu-rate problem so that the entire algorithm converges to thesolution of the second- order accurate problem. The DCapproach is advantageous since the AMG solver demon-strates very good convergence for the first-order accuratediscretizations and moreover, as the first-order matrix isconstant, only one AMG setup procedure must be per-formed for the entire DC algorithm. The suggested ap-proach was examined for the central Jameson-Schmidt-Turkel scheme [6] and for the upwind Roe scheme [7]. Incase of Euler equations sub-, trans- and supersonic flowregimes were considered, in case of Navier-Stokes equa-tions - laminar subsonic regimes. Compared to the lin-ear TAU solver, significant speed-up in terms of CPUtime with a reasonably increased memory requirement isachieved in all of the considered test cases. Direct solutionby the developed AMG solver shows currently a similarperformance to the defect correction approach presentedhere.

[1] A. Brandt, S.F. McCormick, J. Ruge, Algebraic multigrid(AMG) for sparse matrix equations, Sparsity and its Applications,D.J. Evans (ed.), Cambridge University Press, Cambridge, 1984, pp.257-284.

[2] J.W. Ruge, K. Stuben, Algebraic Multigrid (AMG) , inMultigrid Methods (McCormick, S.F., ed.), SIAM, Frontiers in Appl.Math., 5 (1986), Philadelphia.

[3] K. Stuben, T. Clees, SAMG Users Manual, Fraunhofer In-stitute SCAI, http://www.scai.fraunhofer.de/samg.

[4] T. Gerhold, M. Galle, O. Friedrich, J. Evans, Calculationof complex 3d configurations employing the DLR TAU-Code, AIAAPaper Series, 1997, AIAA-97-0167.

[5] K. Boehmer, P. Hemker, H. Stetter, The defect correctionapproach, Computing Suppl., 5 (1984), pp. 132.

[6] A. Jameson, W. Schmidt, E. Turkel, Numerical Solutionsof the Eu- ler Equations by Finite Volume Methods Using Runge-Kutta Time- Stepping Schemes. AIAA Paper Series, 1981, AIAA-1981-1259.

[7] P. L. Roe, Characteristic-based schemes for the Euler equa-tions. Annual Review of Fluid Mechanics, 18 (1986), pp. 337-365.

A2-3A reformulation-based approach for multigrid so-lution of the Helmholtz equationS. MacLachlan

Because of their wide potential applicability in geo-physical and medical imaging algorithms, there has beensustained interest in fast and accurate numerical tech-niques for the solution of the Helmholtz equation in het-erogeneous media. In this talk, I will present a new familyof algorithms for the numerical solution of the Helmholtzequation, using a reformulation-based approach that pro-vides an exact solution algorithm in the continuum. Incontrast to many existing techniques, the proposed algo-rithm does not rely on the application of multigrid princi-ples directly to the indefinite Helmholtz problem. Instead,

we derive an efficient multigrid algorithm for an auxiliaryproblem that appears to be much more amenable to nu-merical solution. Preliminary numerical experiments sug-gest that this approach gives comparable results to theexisting state-of-the-art algorithms, with the potential fora significant improvement. This work is in collaborationwith Eldad Haber.

B2-1A two-scale iterative method for computing flowsin highly porous mediaY. Efendiev, O. Iliev, R. Lazarov, J. Willems

The efficient solution of Brinkmans equations is con-sidered. These equations model fluid flow in highlyporous media, which cannot be described sufficiently wellby Darcys model. A Discontinuous Galerkin discretiza-tion using H(div)-conforming finite elements is employed.For the numerical solution of the arising linear systema two-scale domain decomposition approach is considered,targeting the robust performance of the method with re-spect to mesh parameters and variations in the permeabil-ity field.

B2-2Multigrid preconditioned Newton-Krylov methodfor solving the 2-D 3-T energy equationsHeng-Bin An, Ze-Yao Mo

The 2-D 3-T energy equations is a kind of stronglynonlinear systems that is used to describe the energy dif-fusion and exchanging between electron and photon or ion.In multiphysics simulations, the energy diffusion and ex-changing process is coupled with some other physical pro-cesses, such as fluid dynamics, laser broadcast etc. Conse-quently, the 3-T energy equations should be discretized onthe deforming meshes which is moved with dynamics. Ondeforming meshes, some nine-point scheme must be em-ployed to discretize 3-T energy equations for two dimen-sional case. Because the energy diffusion and swappingcoefficients have a strongly nonlinear dependence on thetemperature, and some physical parameters are discon-tinuous across the materials interfaces, it is a challengeto solve the discretized nonlinear algebraic equations inmultiple physics applications [1]. In this report, a Newton-Krylov method [3] will be used to solve the discretized 3-Tenergy equations, and a Picard type method will be em-ployed to construct a preconditioner. For solving the pre-conditioning system, an algebraic multigrid (AMG) methodin HYPRE [4] is used. Besides, the Newton-Krylov method

is implemented on the KINSOL package [2].

[1] H.-B. An, Z.-Y. Mo et al., On choosing a nonlinear initial it-erate for solving the 2-D 3-T heat conduction equations, J. Comput.Phys., Vol 228 (2009), pp. 32683287.

[2] A.C. Hindmarsh, P.N. Brown et al., SUNDIALS: Suite ofnonlinear and differential/algebraic equation solvers, ACM Transac-tions on Math. Soft., Vol 31 (2005), pp. 363396.

[3] D.A. Knoll, D.A. Keys, Jacobian-free Newton-Krylov method:a survey of approaches and applications, J. Comput. Phys., Vol193 (2004), pp. 357397.

[4] https://computation.llnl.gov/casc/linear solvers/sls hypre.html

B2-3Nonlinearly preconditioned globalization strategies- Theory and applicationsC. Groß, R. Krause

The numerical solution of realistic mechanical prob-lems, such as large-deformation contact between an elas-tic body and a rigid obstacle, often gives rise to nonlinearand possibly non-convex optimization problems. Thus, inorder to succeed in computing a local solution of such op-timization problems their solution is most often carriedout employing globalization strategies, i.e., Trust-Region[1] and Linesearch methods [2].

As is well-known, the paradigm of these solution strate-gies is to compute a search direction and to damp this di-rection to control the descent in the value of the objectivefunction. Usually, search directions are computed as thesolution of constrained quadratic programming problems.Though, even if these quadratic programming problemsare solved exactly, due to the damping of the search direc-tions convergence of the overall scheme might still be slow[3]. Unfortunately, this effect often increases with the sizeof the optimization problem.

In order to bypass (an initial) slow convergence, mul-tilevel globalization strategies were introduced, such asMG/Opt [4], RMTR [5,6] and MLS [7], which can be con-sidered as nonlinear, multiplicatively preconditioned glob-alization strategies. Moreover, additive preconditioningstrategies, such as the APTS [8] and the APLS method[3] were recently introduced, aiming on the parallel so-lution of the optimization problems. The paradigm ofthe nonlinearly preconditioned globalization strategies isto solve related but smaller optimization problems in or-der to compute an improved iterate. In this talk, we willtherefore review the concept of nonlinearly preconditionedglobalization strategies. We will focus on the applicationof such strategies for the solution of large scale nonlinearoptimization problems arising from the discretization oflarge deformation contact problems.

[1] A. R. Conn and N. I. M. Gould and Ph. L. Toint, Trust-region methods (2000).

[2] J. Nocedal and S. Wright, Numerical Optimization (2006).[3] C. Groß, A Unifying Theory for Nonlinear Additively and

Multiplicatively Preconditioned Globalization Strategies: ConvergenceResults and Examples From the Field of Nonlinear Elastostatics andElastodynamics, Online-Publikationen an deutschen Hochschulen,Bonn, Univ., Diss., URN: urn:nbn:de:hbz:5N-18682 (2009).

[4] S. Nash, A multigrid approach to discretized optimizationproblems, JournalJournal of Optimization Methods and Software,

14 (2000), pp. 99–116.[5] S. Gratton and A. Sartenaer and P. L. Toint, Recursive

Trust-Region Methods for Multiscale Nonlinear Optimization, SIAMJournal on Optimization, 19 (2008), pp. 414–444.

[6] C. Groß and R. Krause, On the Convergence of RecursiveTrust–Region Methods for Multiscale Non-linear Optimization andApplications to Non-linear Mechanics, SIAM J. Numer. Anal., 47(2009), pp. 3044-3069.

[7] Z. Wen and D. Goldfarb, Line search Multigrid Methods forLarge-Scale Non-Convex Optimization, IEOR Columbia University(2008).

[8] C. Groß and R. Krause, A new Class of Non–linear Addi-tively Preconditioned Trust–Region Strategies: Convergence Resultsand Applications to Non-linear Mechanics, Institute for NumericalSimulation preprint 904 (2009).

C2-1Nonstandard norms and robust estimates for sad-dle point problemsW. Zulehner

In important issue for constructing and analyzing ef-ficient solvers for a saddle point problem is to understandthe right mapping properties of the problem. We will con-centrate on saddle point problems which result from thedis- cretization of a system of partial differential equa-tions. The mapping properties of the involved differ-ential operators usually suggest the right norms for thediscrete problems leading to mesh-independent estimates.These norms are quite often (discrete versions of) stan-dard norms in Lebesgue or Sobolev spaces. If the saddlepoint problem contains critical parameters (like regulationparameters in optimal control problems) one would like touse norms leading to mesh-independent estimates whichare also robust with respect to these critical parameters.Here standard norms usually do not the job. In this talkwe will discuss the construction of norms for saddle pointproblems which lead to robust estimates: Firstly, on apurely algebraic level, a characterization of such norms isgiven for a general class of symmetric saddle point prob-lems. Then we will apply these results to a family of (time-independent) optimal control problems and show how toderive such norms for this family. Based on these normsrobust solvers will be constructed by using multigrid tech-niques.

C2-2Towards AMG for saddle point problemsB. Metsch

We present an approach to the construction of alge-braic multigrid methods (AMG) for saddle point problemsof the form (

A B

BT −C

)where A is positive definite and C is positive semidefinite.The main challenge to the construction of the coarse grid

and the interpolation operator P in this case is to ensurethat the coarse level system KC = PTKP obtained fromthe Galerkin product is invertible. Our starting point isthe additive Schwarz smoother introduced by Sch berloand Zulehner in [1]. This smoother can be carried outusing information from the matrix K and its partition-ing into A, B and C only and hence is suited for a purealgebraic approach. The smoother is plugged into a com-patible relaxation scheme that ensures the stability of thecoarse level system. Then, interpolation is constructed us-ing the tentative prolongation derived from the compatiblerelaxation process.

[1] Joachim Schoberl and Walter Zulehner, On Schwarz-typeSmoothers for Saddle Point Problems, Numerische Mathematik, 95(2003), pp. 377-399.

C2-3

An efficiency-based multigrid solver for incompress-ible resistive magnetohydrodynamics

J. Adler

Magnetohydrodynamics (MHD) is a fluid theory thatdescribes plasma physics by treating the plasma as a fluidof charged particles. Hence, the equations that describethe plasma form a time-dependent nonlinear system thatcouples Navier- Stokes’ with Maxwell’s equations. Thistalk develops a nested-iteration-multigrid approach withan efficiency-based adaptive local refinement scheme tosolve this type of system. The general frame work is de-scribed with a First-Order Systems Least-Squares (FOSLS)finite element discretization. However, it can be appliedto any discretization with a sharp a posteriori error esti-mator. The goal is to reach a certain error tolerance withthe least amount of computational cost and nearly uni-form distribution of the error over all elements. This talkwill develop theory that supports this argument, as wellas show experiments to confirm that the algorithm can beefficient for MHD problems. These methods are applied toa 2D reduced model of the incompressible, resistive mag-netohydrodynamic (MHD) equations. These equationsare used to simulate instabilities in a large aspect-ratiotokamak as well as instabilities that arise in the magneto-sphere. It is shown that, by using the new nested iterationand adaptive strategies on this system, the physics is re-solved using only 10 percent of the computational costused to approximate the solutions on a uniformly refinedmesh within the same error tolerance. Time permitting,a discussion of how the above methods relate to the ener-getics of the system will be given as well as a discussionof the effects of the time-stepping schemes used.

D2-1

Modelling of signal processing in neurons

G. Wittum

The crucial feature of neuronal ensembles is their high

complexity and vari- ability. This makes modelling andcomputation very difficult, in particular for detailed mod-els based on first principles. The problem starts with mod-elling geometry, which has to extract the essential featuresfrom those highly complex and variable phenotypes and atthe same time has to take in to account the stochastic vari-ability. Moreover, models of the highly complex processeswhich are living on these geometries are far from beingwell established, since those are highly complex too andcouple on a hierarchy of scales in space and time. Simulat-ing such systems always puts the whole approach to test,including modeling, numerical methods and software im-plementations. In combination with validation based onexperimental data, all components have to be enhanced toreach a reliable solving strategy. To handle problems ofthis complexity, new mathematical methods and softwaretools are required. In recent years, new approaches such asparal- lel adaptive multigrid methods and correspondingsoftware tools have been developed allowing to treat prob-lems of huge complexity. In the lecture we present a threedimensional model of signaling in neurons. First we showa method for the reconstruction of the geomety of cells andsubcellular structures as three dimensional objects. Withthis tool, NeuRA, complex geometries of neuron nucleiwere reconstructed. We present the results and discussreasons for the complicated shapes. We further present atool for the automatic generation of realistic networkks ofneurons (Neu- Gen). We then present a model of calciumsignaling to the nucleus and show simulation results onreconstructed nuclear geometries. We discuss the implica-tions of these simulations.

We further show reconstructed cell geometries andsimulations with a three dimensional active model of signaltransduction in the cell which is derived from the Maxwellequations and uses generalized Hodgkin-Huxley fluxes forthe description of the ion channels.

D2-2

Multigrid methods for zero-sum stochastic games

S. Detournay, M. Akian

The value function of a zero-sum two player stochasticgame with perfect information is solution of the dynamicprogramming equation. This nonlinear equation can besolved by policy iteration algorithms, which generalize insome sense the Newton algorithm, and which involve thesolution of linear systems. We develop a fast numericalalgorithm for large scale games, which combines policyiteration and algebraic multigrid (AMG) methods.

Indeed, for a stochastic differential game with infi-nite horizon, the dynamic programming equation is anelliptic partial differential equation of Isaacs type, and adiscretization with monotone scheme yields the dynamicprogramming equation of a finite state space stochasticgame. Then, the above linear systems are equivalent todiscretizations of linear elliptic equations. We solve theselinear systems using the AMG method of Ruge and Stuben,which allows us to apply the resulting algorithm either to

true finite state space zero-sum two player games or todiscretizations of Isaacs equations. Such an association ofmultigrid methods with policy iteration has already beenused and studied in the case of one player games, that isdiscounted stochastic control problems (see Hoppe (86,87)and Akian (88,90) for Hamilton-Jacobi-Bellman eq., Zivand Shimkin (05) for AMG with learning methods). How-ever, it is new in the case of two player games. We shallpresent numerical tests for Isaacs equations or variationalinequalities.

As for the classical Newtons algorithm, the number ofpolicy iterations can be reduced by starting with a goodinitial guess, close to the solution. In this way, we devel-opped a full multi-level policy iteration scheme in a FMGstyle. Numerical examples on variational inequalities showthat the execution time can be much improved using thisfull multi-level scheme.

D2-3

A multilevel algorithm to compute steady statesof lattice Boltzmann models

G. Samaey, C. Vandekerckhove, W. Vanroose

We present a multilevel algorithm to compute steadystates of lattice Boltzmann models directly as fixed pointsof a time-stepper. At the fine scale, we use a Richardsoniteration for the fixed point equation, which amounts totime-stepping towards equilibrium. This fine-scale itera-tion is accelerated by transferring the error to a coarselevel. At this coarse level, one directly solves for the den-sity (the zeroth moment of the lattice Boltzmann distribu-tions), for which a coarse-level equation is known in someappropriate limit. The algorithm closely resembles theclassical multigrid algorithm in spirit, structure and con-vergence behaviour. In this paper, we discuss the formu-lation of this algorithm. We give an intuitive explanationof its convergence behaviour and illustrate with numericalexperiments.

E2-1

Parallel multilevel solvers for 3D fluid dynamicsimulations

L. Carracciuolo, D. Casaburi, L. DAmore, A. Murli

To understand the onset and the evolution of 3D fluiddynamic phenomena a collaboration between computingscientists and chemical engineers was established at Uni-versity of Naples Federico II. The existing code, devel-oped for 2D problems, uses the TFEM (Toolkit for FiniteElement Method) software toolkit equipped with PAR-DISO and MUMPS libraries employed to solve the com-putational kernels. Simulation of 3D processes, becauseof the huge problem size, requires to revise the underlyingnumerical approach in order to exploit advanced comput-ing infrastructures through the development of effectivesolvers. Taking into account that governing equations are

non linear evolutionary PDEs, as computing environmentthat supports the simulation software we choose PETSc(Portable, Extensible Toolkit for Scientific Com- puta-tion). Discretization by using finite elements gives riseto linear systems to solve at each time step. Under suit-able partitioning and reordering, these linear systems havea large scale saddle point structure. We discuss compu-tational efforts towards the employment of parallel multi-level preconditioned iterative methods for numerical solu-tion of such an intensive problem. We thank Prof. P. L.Maffettone and Dr. G. DAvino at Department of Chemi-cal Engineering of University of Naples Federico II, for giv-ing us the opportunity to validate numerical results and todiscuss the simulations. We also thank Prof. M.A. Hulsenof Eindhoven University of Technology for providing thecode TFEM.

E2-2

MLD2P4: parallel algebraic multilevel precondi-tioners for large-scale linear systems

P. D’Ambra, D. di Serafino, S.Filippone

Multilevel domain decomposition methods are widelyrecognized as powerful tools for building parallel precondi-tioners for sparse linear systems arising in large-scale sci-entific and engineering applications. However, careful al-gorithmic and implementation choices must be performedto successfully combine the parallelism of the domain de-composition techniques with the optimality of the mul-tilevel approach, intended as the ability of keeping thenumber of solver iterations independent of the number ofprocessors.

In this talk we present MLD2P4 (MultiLevel DomainDecomposition Parallel Preconditioners Package based onPSBLAS), a package of parallel multilevel precondition-ers that combines additive Schwarz domain decomposi-tion methods with a smoothed aggregation technique tobuild a hierarchy of coarse-level corrections in an alge-braic way. The design of MLD2P4 was guided not onlyby performance issues, but also by objectives such as flex-ibility, extensibility, portability and ease of use. Thesewere achieved by following an object-based approach inthe Fortran 95 language, as well as by employing the Par-allel Sparse BLAS library as a basic framework. The re-sults obtained on various large-scale linear systems, arisingfrom model problems as well as from CFD applications,show the effectiveness of our package.

[1] P. D’Ambra, D. di Serafino, S. Filippone, MLD2P4: a Pack-age of Parallel Algebraic Multilevel Domain Decomposition Precon-ditioners in Fortran 95, ACM Transactions on Mathematical Soft-ware, 37 (3), 2010.

[2] P. D’Ambra, D. di Serafino, S. Filippone, On the Develop-ment of PSBLAS-based Parallel Two-level Schwarz Preconditioners,Applied Numerical Mathematics, 57, 2007.

E2-3

Adaptive numerical methods for an hydrodynamicproblem arising in magnetic reading devices

I. Arregui, J.J. Cendan, C. Vazquez

This work deals with the numerical solution of com-pressible Reynolds equation, which governs the pressureof the air layer between the hard disk and the head in mag-netic reading devices. If a flexible tape is used instead ofa hard disk, its configuration is deformed by the air pres-sure, thus leading to a coupled problem. In previous works[1], we have presented a numerical algorithm, includingcharacteristics and duality methods, for the hydrodynamicand the coupled elastohydrodynamic problems. However,in certain opera- tional conditions and devices we need avery fine mesh in order to capture the large pressure gra-dients. So, we now present a multilevel adaptive strategy:at each level we refine the elements where the pressure gra-dients are larger. Moreover, the system of linear equationsarising at each duality iteration is solved by a multigridalgorithm; in this respect, we have been inspired by Hoppe[2], who proposes a multigrid resolution in the frame of anobstacle problem. Some numerical tests will be presentedin order to illustrate the behaviour of this algorithm.

[1] I. Arregui, J. J. Cendan, C. Vaquez, Numerical simulationof head/tape magnetic reading devices by a new 2-D model, FiniteElements in Analysis and Design, 43 (2007), 4, 311320.

[2] R.W.H. Hoppe, Multigrid Methods for Variational Inequal-ities, SIAM J. Numer. Anal. 24 (1987), 10461065.

E2-4

An adaptive domain decomposition Technique forparallelisation of the fast marching method

M. Breuß, E. Cristiani, P. Gwosdek, O. Vogel

The Fast Marching Method (FMM) is a numericaltechnique introduced by J. A. Sethian in [2] to solve theeikonal equation

f(x)|u(x)| = 1, x ∈ Rn \ Ω, f > 0,u(x) = 0, x ∈ ∂Ω.

This equation appears in a number of different applica-tion fields like image processing, optics, geoscience, frontpropagation and optimal control problems. The paral-lelisation of the FMM is not easy because of its intrinsicse- quential nature. An attempt based on a classical do-main decomposition was proposed in [1], but its efficiencystrictly depends on the choice of the function f and thedomain Ω. In this talk we propose an equation-dependentdomain decomposition technique which allows to paral-lelise the FMM in a simple way. As opposite to [1], thedomain decomposition is not decided a priori, instead itis automatically generated while the solution of the equa-tion is computed, and its shape depends on the choice of fand Ω. Making use of an example from the field of imageprocessing, namely the shape-from-shading problem, weverify that the favourable properties of our approach areuseful for real-world applications.

[1] M. Herrmann, A domain decomposition parallelization of theFast Marching Method, in: Annual Research Briefs-2003, Center forTurbulence Research, Stanford, CA.

[2] J. A. Sethian, A fast marching level set method for mono-tonically advancing fronts, Proc. Natl. Acad. Sci. USA, 93 (1996),pp. 15911595.

F2-1An algebraic multigrid method with proved con-vergence rate I: two-grid analysisA. Napov, Y. Notay

In this and the companion talk (An algebraic multigridmethod with proved convergence rate II: automatic coars-ening and multilevel convergence by Y. Notay) we showhow to derive an algebraic multigrid method with guar-anteed optimal convergence rate. We consider more par-ticularly methods based on coarsening by (unsmoothed)aggregation. In this talk, we develop the convergenceanalysis of the two-grid scheme. For diagonally domi-nant symmetric (M-)matrices, we show that the analysiscan be conducted locally; that is, the convergence factorcan be bounded above by computing separately for eachaggregate a parameter which in some sense measures itsquality. Moreover, the analysis is accurate; for instance,assuming the aggregation pattern sufficiently regular, weshow that the resulting bound is asymptotically sharp fora large class of elliptic boundary value problems, includingproblems with variable and discontinuous coefficients.

F2-2Algebraic interface in sparse matrix and its appli-cation in AMGX.W. Xu, Z.Y. Mo, X. Liu

The linear systems we consider here are those withmulti-scale property, i.e. there are great differences inmagnitude among the off-diagonal elements in the coef-ficient matrix. There are many real world applicationsleading to this kind of linear systems. For many iterativemethods, however, the multi-scale property will greatlydamage their efficiencies. In this work, by exploring thematrix adjacency graph, we introduce the concept of alge-braic interface for sparse matrix based on the idea of con-nection strength in algebraic multigrid method (AMG),and use it to reveal the difficulties on the solution of thesystems. Based on these observations, new AMG compo-nents including special relaxation and coarsening strate-gies are presented. The results of various numerical ex-periments show the effectiveness of the methods presentedin this work.

F2-3Algebraic multigrid based on clusteringL. Grasedyck, J. Xu

In this talk we present recent work on the construc-

tion of coarse grids for geometric multigrid on unstruc-

tured grids. The framework that we use is the auxiliary

space multigrid scheme [1] with local subspace correction.

We present the construction of the grids as well as the

multigrid method itself for the Dirichlet and the Neumann

problem. Our construction allows for a setup and solution

in almost optimal complexity, i.e. optimal up to one addi-

tional logarithm during the setup, cf. the following table

for the Dirichlet problem:

#dof Setup [sec] Solve [sec] Steps

n4 = 737, 933 45.2 12.4 (5)n5 = 2, 970, 149 124 40.2 (5)n6 = 11, 917, 397 414 125.9 (5)n7 = 47, 743, 157 1360 544.9 (5)

We will present numerical examples that underline thepracticability of our approach. There is a strong relationto composite finite elements [2,3]. In contrast to alge-braic multigrid methods [4] we require geometry informa-tion (the given unstructured grid). Knowledge of the dis-cretisation scheme and the underlying partial differentialoperator is beneficial but not necessary.

[1] J. Xu, The auxiliary space method and optimal multigridpreconditioning techniques for unstructured grids, Computing, 55(1996), pp. 215–235.

[2] S. Sauter and W. Hackbusch, Composite finite elements forthe approximation of PDEs on domains with complicated micro-structures, Numerische Mathematik, 75 (1997), pp. 447–472.

[3] D. Feuchter, I. Heppner, S. Sauter and G. Wittum, Bridgingthe gap between geometric and algebraic multi-grid methods, Com-puting and Visualization in Science, 6 (2003), pp. 1–13.

[4] K. Stuben, A review of algebraic multigrid, Journal of Com-putational and Applied Mathematics, 128 (2001), pp. 281–309.

F2-4Towards robust algebraic multigrid for nonsym-metric problemsJ. Lottes

A good algebraic analysis of multigrid methods pro-vides a framework that can be used to derive more robustalgebraic multigrid methods. This approach has provenquite successful. However, the bulk of the attention hasbeen on symmetric positive-definite (SPD) systems, whilecomparatively little has been paid to nonsymmetric sys-tems. Our interest is in systems for which the nonsym-metry plays a prominent role, e.g., discretized convection-diffusion operators with very small diffusion. The energyinner-product and norm are very useful analytical toolsin the SPD case. For example, many successful heuristicsfor interpolation have been based on an energy- minimiz-ing approach. By considering generalizations of the en-ergy inner-product and norm to apply to (coercive) non-symmetric operators, one may derive a two-grid conver-gence bound involving a smoothing property and inde-pendent approximation properties for the coarse trial andtest spaces. Each approximation property is a generalizedenergy norm, one involving the interpolation operator P ,and the other the restriction operator R. The treatmentof P and R in this analysis is completely symmetric and

avoids the Galerkin assumption R = PT , which we feelis inappropriate, or at least limiting, for highly nonsym-metric problems. As an example application, the theorysuggests the correct analogue of energy minimizing inter-polation for nonsymmetric problems.

Wednesday

IS5

Symbol approach for structured matrices in multi-grid with applications in imaging

M. Donatelli

We consider the deblurring problem of noisy and blurredimages in the case of space invariant point spread func-tions. The use of appropriate boundary conditions leadsto linear systems with structured coefficient matrices re-lated to space invariant operators like Toeplitz, circulants,trigonometric matrix algebras etc. We can obtain an ef-fective and fast solver by mixing the algebraic multigriddescribed in [12, 2] with the Tikhonov regularization (see[5]). Moreover we propose to combine the latter algebraicmultigrid (which is typical for structured matrices) withthe low-pass filtering properties of the classical geometri-cal multigrid used in a PDEs context (see e.g. [9]). Thus,using an appropriate smoother, we obtain an iterative reg-ularizing method (see [6]).

The main idea is that any iterative regularizing methodlike conjugate gradient (CG), conjugate gradient for nor-mal equation (CGNE), Landweber etc., can be used assmoother in our multigrid algorithm. The projector ischosen according to [12, 2] in order to maintain the samealgebraic structure at each recursion level and having alow-pass filter property, which is very useful to reduce thenoise effects. In this way we obtain a better restored im-age with a flatter error curve and also in less time thanthe method used as smoother.

Like any multigrid algorithm the resulting technique isparameterized in order to have more degrees of freedom: asimple choice of the parameters, not in charge to the user,allows to devise a powerful regularizing method.

In the talk we first review the theoretical ground givenby the convergence theory in the case of structured matri-ces and then we present the adaptation in the case of imagerestoration problems. More in detail, we will emphasize(both in the theoretical study and in the applications) therole of the generating function in two directions: in orderto minimize the storage requirements and the complexityof every single iteration and in order to obtain an optimalmethod that is an iterative technique whose convergencerate is independent of the size of the involved matrices anddepending on some analytical features of the symbol.

[1] A. Arico and M. Donatelli, A V -cycle multigrid for multilevelmatrix algebras: proof of optimality, Numer. Math. 105-4 (2007),pp. 511–547.

[2] A. Arico, M. Donatelli, and S. Serra Capizzano, V-cycle opti-mal convergence for certain (multilevel) structured matrices, SIAMJ. Matrix Anal. Appl., 26-1 (2004) pp. 186–214.

[3] M. Bertero and P. Boccacci, Introduction to inverse problemsin imaging. Inst. of Physics Publ. Bristol and Philadelphia, London(UK), 1998.

[4] R. Chan, Q. Chang, and H. Sun, Multigrid method for ill-conditioned symmetric Toeplitz systems, SIAM J. Sci. Comput.,19-2 (1998), pp. 516–529.

[5] M. Donatelli, A Multigrid for image deblurring with Tikhonovregularization, Numer. Linear Algebra Appl., 12-8 (2005), pp. 715–729.

[6] M. Donatelli and S. Serra Capizzano, On the regularizingpower of multigrid-type algorithms, SIAM J. Sci. Comput., 27-6(2006), pp. 2053-2076.

[7] G. Fiorentino and S. Serra Capizzano, Multigrid methods forToeplitz matrices, Calcolo 28 (1991), pp. 283–305.

[8] G. Fiorentino and S. Serra Capizzano, Multigrid methods forsymmetric positive definite block Toeplitz matrices with nonnegativegenerating functions, SIAM J. Sci. Comp. 17-4 (1996), pp. 1068–1081.

[9] W. Hackbusch, Multigrid Methods and Applications. SpringerVerlag, Berlin (DE), 1985.

[10] Huckle and J. Staudacher, Multigrid preconditioning andToeplitz matrices, Electr. Trans. Numer. Anal., 13 (2002), pp.81–105.

[11] M. Ng, R. Chan, and W. C. Tang A fast algorithm fordeblurring models with Neumann boundary conditions, SIAM J. Sci.Comput., 21-3 (1999), pp. 851–866.

[12] S. Serra Capizzano, Convergence analysis of two-grid meth-ods for elliptic Toeplitz and PDEs Matrix-sequences, Numer. Math.,92-3 (2002), pp. 433–465.

[13] S. Serra Capizzano, A note on anti-reflective boundary con-ditions and fast deblurring models, SIAM J. Sci. Comput. 25-3(2003), pp. 1307–1325.

IS6

Multilevel Monte Carlo method

M. Giles

The multilevel Monte Carlo method [1,2,3] takes themultigrid philosophy of working with different levels ofresolution to achieve fine grid accuracy at a coarse gridcost, and applies it to the simulation of stochastic differ-ential equations (SDEs) and partial differential equations(SPDEs).

Unlike standard multigrid methods which are concernedwith the efficient iterative solution of large systems ofequations, the objective in the multilevel Monte Carlomethod is the accurate estimation of an expected valuewhich is a functional of the solution of the SDE or SPDE.The key observation is that the expected value on thefinest level of resolution can be expressed as the expec-tation on the coarsest level of resolution, plus a sum ofexpected corrections between neighbouring levels of reso-lution.

This talk will introduce the basic ideas, motivated byapplications in computational finance. It will also discussnew research with R. Scheichl and K.A. Cliffe on ellipticSPDEs in the simulation of oil reservoirs and nuclear wasterepositories.

[1] M.B. Giles, Multilevel Monte Carlo path simulation, Oper-ations Research, 56(3), pp. 607–617, 2008.

[2] M.B. Giles, Improved multilevel Monte Carlo convergenceusing the Milstein scheme, Monte Carlo and Quasi-Monte CarloMethods 2006, pp. 343-358, Springer-Verlag, 2007.

[3] M.B. Giles, D. Higham and X. Mao, Analysing multilevelMonte Carlo for options with non-globally Lipschitz payoff, Financeand Stochastics, 13(3), pp. 403-413, 2009.

A3-1

Some results in the optimal control of electromag-netic fieldsF. Troltzsch

Two optimal control problems are considered, whereelectromagnetic fields are controlled to optimize the un-derlying process. The first, investigated in joint work withK. Altmann, is related to the time- optimal switching be-tween magnetic fields and leads to the minimization of atracking type functional subject to a system of parabolicequations of differential algebraic type. We discuss thesensitivity analysis and present 3D numerical results fora slightly simplified geometry. The second problem, dis-cussed jointly with P.E. Druet, O. Klein, J. Sprekels, andI. Yousept, concerns the optimal control of travelling mag-netic fields in the Czochralski process of crystal growth.Here, a nonlinear system composed of Maxwells equations,Navier-Stokes equations, and a nonlinear heat equationwith nonlocal radiation boundary conditions models theprocess. Certain pointwise state constraints are given,which require sufficient regularity of the state functions.Associated regularity results and a first- and second-ordersensitivity analysis of the optimal control problem are pre-sented.

A3-1Algebraic multigrid for nonsymmetric constrainedproblemsT.A. Wiesner, M.W. Gee, R.S. Tuminaro, W.A. Wall

Even though there are many applications in the area ofengineering and applied sciences where algebraic multi-grid methods are successfully applied, nonsymmetric andconstrained problems are still challenging. While specialsmoothers for constrained systems exist, e.g. [1], construc-tion of appropriate nonsymmetric algebraic transfer oper-ators is ongoing research, see exemplarily [3, 5]. Here, wepresent an overview over several prolongation and restric-tion strategies as a collection of ideas based on energy min-imization principles. We compare aggregate-based strate-gies such as smoothed aggregation (SA-AMG) [4], thePetrov Galerkin approach (PG-AMG) from [5] and energyminimization methods [2, 6, 7] with a new nonsymmetricenergy minimizing method. This new method is based oneither an aggregation strategy or the sparsity pattern of athresholded LU decomposition of coarse/fine nodes. Thenewly proposed method is an extension of energy mini-mization concepts in [2] to the nonsymmetric case thatutilizes a constrained GMRES algorithm for minimization.Special attention is paid to the strict decoupling of primaland constraint variables and the resulting versatile possi-bilities of combining differing approaches for each of thesecomponents. We conclude that exact conservation of con-straints on coarse levels is important but in contrast toaggregation methods is not automatically guaranteed fortransfer operators based on energy minimization princi-ples. We show that our newly proposed nonsymmetric en-ergy minimization guarantees exact representation of con-straints on coarse levels. The investigated AMG hierar-

chies utilize Braess-Sarazin type smoothers and are stud-ied and compared by means of several examples. Separateprolongation and restriction strategies for the primal andconstraint-variable prolongation and restriction are com-bined and examined. The importance of the preservationof constraints is exemplarily demonstrated for the incom-pressible Navier-Stokes equations.

[1] D. Braess and R. Sarazin, An efficient smoother for theStokes problem, Applied Numerical Mathematics, 23 (1997), pp.319.

[2] J. Mandel, M. Brezina and P. Vanek, Energy optimizationof algebraic multigrid bases, Computing, 62 (1999), pp. 205228.

[3] A. Janka, Smoothed aggregation multigrid for a Stokes prob-lem, Com- puting and Visualization in Science, 11 (2008), pp. 169180.

[4] P. Vanek, J. Mandel, and M. Brezina, Algebraic multigrid bysmoothed aggregation for second and fourth order problems, Com-puting, 56 (1996), pp. 179196.

[5] M. Sala and R. S. Tuminaro, A new Petrov-Galerkin smoothedaggregation preconditioner for nonsymmetric linear systems, SIAMJournal on Scientific Computing, 31 (2008), pp. 143166.

[6] J. Brannick, M. Brezina, S. MacLachlan, T. Manteuffel, S.McCormick, J. Ruge, An energy-based AMG coarsening strategy,Num. Lin. Alg. Appl., 13 (2006), pp. 133148

[7] J. Brannick, L. Zikatanov, Algebraic Multigrid MethodsBased on Com- patible Relaxation and Energy Minimization, Lec-ture Notes in Compu- tational Science and Engineering: DomainDecomposition Methods in Science and Engineering XVI, 55 (2007),pp. 1526

A3-3

Efficient preconditioning of optimality systems

B. F. Nielsen, K.-A. Mardal

We propose a rather general preconditioning strategyfor the numerical treatment of linear optimality systems(OS) arising in connection with inverse problems for par-tial differential equations. If this kind of inverse problemsare stabilized with Tikhonov regularization, then it followsfrom classical theory that the associated OS is well-posed,provided that the involved state equation is well-behaved.The purpose of our work is to explain and analyze howcertain mapping properties of the operators appearing inthe OS can be employed to define efficient precondition-ers for finite element (FE) approximations of the involvedsaddle point problem. More specifically, it turns out thatit is possible to define a scheme such that the number ofiterations needed to solve the preconditioned problem isbounded independently of the mesh parameter h, used inthe FE discretization, and only increases moderately asthe regularization parameter tends towards zero. In fact,if the associated energy norm is used to define the stop-ping criterion for the iteration process, then the number ofiterations required (in the severely ill-posed case) cannotgrow faster than O((ln(α))2). This result is obtained bycarefully analyzing the properties of the involved operatorsand thereby revealing the distribution of the eigenvalues ofthe preconditioned OS. One advantage of our approach isthat it only rely on standard elliptic preconditioners basedon e.g. multigrid and domain decomposition. Our theoret-ical results will be illuminated by a number of numericalexperiments.

B3-1

An adaptive parallel geometric space-time multi-grid algorithm for the heat equation

T. Koppl, T. Weinzierl

While almost all approaches to solve partial differen-tial equations (PDEs) are based upon a spatial discreti-sation of the computational domain, few algorithms dis-cretise both time and space in the same manner. Instead,the time is typically meshed different from the space, andthe design of multiscale solvers concentrates on the spa-tial components of the differential operator. In this pa-per, we present a parallel multigrid implementation for theheat equation that discretises both space and time with anoctree-like data structure, i.e. it treats the time as an addi-tional spatial dimension and the differential operator as adegenerated anisotropic elliptic operator. Since we followthe implementation paradigms of [5], we can afford this interms of memory. Such an approach exhibits at least threeimportant advantages: In optimisation and calibration,often a parabolic operator running backward in time hasto be solvedbest on a dynamically refined adaptive grid.To be able to do so, the simulation data of all time stepsof the whole simulation timeframe are of great use. Incomputational steering, often a coarse approximation ofthe solution in the future has to be displayed as soon aspossible. To be able to do so, a simulation run on a verycoarse space-time grid which is refined incrementally lateris of great use. In high performance computing, often thecomputational workload has to be scaled with the num-ber of processing units. To be able to do so, having theworkload of multiple time steps at hand is of great use.

These ideas are advantageous if and only if the solveron the space-time discretisation exhibits a behaviour simi-lar to multigrid algorithms for elliptic problems, and if andonly if it can cope with standard time-stepping schemes interms of runtime. In this talk, we present first techniques,comparisons, and insights concerning these two goals forsimple heat equation setups. Here, we follow ideas from [3]and references therein, combine them with well- known al-gorithms for parabolic problems ([2, 4]), e.g.), and realisethem with implementation paradigms proposed in [1, 5].

[1] H.-J. Bungartz, M. Mehl, T. Neckel and T. Weinzierl, ThePDE framework Peano applied to fluid dynamics, ComputationalMechanics, published online (2009).

[2] M. J. Gander, S. Vandewalle, On the Superlinear and LinearConver- gence of the Parareal Algorithm, LNCS 55, pp. 291298(2007).

[3] M. Griebel and D. Oeltz, A Sparse Grid Space-Time Dis-cretization Scheme for Parabolic Problems, Computing, 81(1), pp.134 (2007).

[4] G. Horton and S. Vandewalle, A space-time multigrid methodfor parabolic P.D.E.S., SIAM Journal on Scientific Computing, 16(4),pp. 84886 (1995).

[5] T. Weinzierl, A Framework for Parallel PDE Solvers on Mul-tiscale Adaptive Cartesian Grids, Verlag Dr. Hut (2009).

B3-2

A Multigrid approach for Poisson equation withmixed boundary conditions in arbitrary domain

A. Coco, G. Russo

Poisson equation in arbitrary domain (possibly withmoving boundary) is central to many applications, suchas diffusion phenomena, fluid dynamics, charge transportin semiconductors, crystal growth, electromagnetism andmany others. In this talk we present a rather simple nu-merical method to solve the Poisson equation in arbitarydomain , identified by a level set function in such a way= x Rd : (x) ¡ 0 , and mixed boundary conditions. Themethod is based on finite difference discretization on aregular cartesian grid, and consists of a transformation ofthe Poisson problem and the boundary conditions intoa fictitious-time dependent problem, that leads to an it-erative scheme converging to the solutions of the origi-nal problem. The convergence is speeded-up by a propermultigrid approach. While the discretization of the prob-lem inside the domain is straightforward, a proper dis-cretization of the boundary conditions is performed, mak-ing use of some extra grid points outside the domain calledghost points. If we use the discretized boundary condi-tions to eliminate such extra grid points, the method doesnot converge. The fictitious time-dependence of the prob-lem including boundary conditions leads us to an iterativescheme for the set of all unknowns (internal points andghost points), which is proved to converge, at least for firstorder accurate discretization. The smoothing procedure ofthe multigrid approach in the interior is again Jacobi-like,while the iterations on the boundary are performed inorder to provide smooth errors. Multigrid techniques forghost points are well-studied in literature, but just in thecase of rectangular domain, where a restriction operatoris defined separately for the interior of the domain andfor the boundary, and the restriction of the boundary isperformed using a restriction operator of codimension 1,since ghost points are aligned with the cartesian axis. Inthe case of arbitrary domain, ghost points have an irregu-lar structure and we provide a reasonable definition of therestriction operator for the boundary. We provide also ananalysis of the approach. In particular, we show that aproper treatment of the boundary iterations can improvethe rate of convergence of the multigrid and the cost ofthis extra computational work is negligible, i.e. tends tozero as the dimension of the problem increases.

[1] A. Brandt, Rigorous quantitative analysis of multigrid, I:constant coefficients two-level cycle with L2-norm , SIAM Journalon Numerical Analysis, 31 (1994), pp. 1695-1730.

[2] H. Chen, C Min and F. Gibou, A supra-convergent finitedifference scheme for the Poisson and heat equations on irregulardomains and non-graded adaptive Cartesian grids, Journal of Scien-tific Computing, (2007), pp. 19-60.

[3] F. Gibou and R. Fedkiw, A second-order-accurate symmetricdiscrati- zation of the poisson equation on irregular domains, Journalof Com- putational Physics, 176 (2002), pp. 205-227.

[4] F. Gibou and R. Fedkiw, A fourth order accurate discretiza-tion for thr laplace and heat equations on arbitary domains, with ap-plications to the stefan problem, Journal of Computational Physics,

202 (2005), pp. 577-601.

[5] W. Hackbusch, Mulit-Grid Methods and Applications, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1985).

[6] W. Hackbusch, Elliptic differential equations: Theory andnumerical treatment., Springer-Verlag, Berlin, Heidelberg, (1992).

[7] J. Papac, F. Gibou and C. Ratsch, Efficient Symmetric Dis-cretization for the Poisson, Heat and Stefan-Type Problems withRobin Boundary Conditions, Journal of Computational Physics, 229(2010), pp. 875-889.

[8] U. Trottenberg, C. W. Oosterlee, A. Schuller, Multigrid.,Academic Press, Berlin, Heidelberg, (2001).

B3-3

Tau-extrapolation on 3D semi-structured finite el-ement meshes

B. Gmeiner, U. Rude

τ -extrapolation is a multigrid approach to increase theorder of consistency for e.g. finite difference or finite ele-ment schemes [1]. Since it is an extrapolation method,one advantage is that it does not change the discretiza-tion matrix at all. The computational overhead is com-parable to a Full Approxi- mation Scheme. One step ofmultigrid τ -extrapolation for piecewise linear C0 finite el-ement methods is equivalent to using quadratic elements[2]. Larger discretization errors caused by a lack of reg-ularity at the interfaces between the structured regionswere investigated for semi-structured grids in two dimen-sions [3]. In this paper we apply τ -extrapolation to athree-dimensional finite element discretization. In or-der to obtain an O(h4) order of consistency for triangles,the asymptotic expansion for basic integration methodscould be derived [4]. By numerical experiments, we showthat this also holds for tetrahedron elements. Thereforethe τ -extrapolation is implemented in the semi-structuredmultigrid finite element solver Hierarchical Hybrid Grids.Further we will have a closer look at numerical effects atthe interfaces between structured regions.

[1] A. Brandt, Multigrid Techinques: 1984 Guide with Applica-tions to Fluid Dynamics, GMD-Studien, Bonn (1984)

[2] M. Jung, U. Rude, Implicit Extrapolation Methods for Mul-tilevel Finite Element Computations, SIAM J. Sci. Comput., Vol 17(1996).

[3] H. Blum, Asymptotic Error Expansion and Defect Correc-tion in the Finite Element Method, University of Heidelberg, Institutfur Angewandte Mathematik, Heidelberg

[4] U. Rude, Extrapolation Techniques for Constructing HigherOrder Finite Element Methods, TU Munchen, Institut fur Infor-matik, (1993).

C3-1

An iterative multilevel regularization method fordeblurring problems

M. Donatelli

We consider the de-blurring problem of noisy and blurredsignals/images in the case of space invariant point spreadfunctions. The use of appropriate boundary conditionsleads to linear systems with structured coefficient ma-trices related to space invariant operators like Toeplitz,circulants, trigonometric matrix algebras etc. In [1] itwas proposed an algebraic multigrid (which is designed ad

hoc for structured matrices) with the low-pass grid trans-fer operators typical of the classical geometrical multigridemployed for elliptic partial differential equations (linearand bilinear interpolation). The smoother can be any it-erative regularization method. The main disadvan- tageof the proposed technique is that sometimes it can pro-vide oversmoothed restorations. In this talk we recall suchmultilevel regularization strategy and we show how to cir-cumvent the possible oversmoothing. We consider edgepreserving techniques, in particular we add a threshold tothe high frequen- cies to remove the noise without smoothedges.

[1] M. Donatelli and S. Serra Capizzano, On the regularizingpower of multigrid-type algorithms, SIAM J. Sci. Comput., 276(2006), pp. 20532076.

C3-2

Aggregation and smoothed aggregation-based multi-grid methods for circulant and Toeplitz matrices

M. Bolten, T. Huckle

The convergence theory for multigrid methods for Toeplitzmatrices and circulant matrices that can be found in var-ious articles (c.f. [2, 5, 6]) is based on the convergenceresults for classical AMG as described e.g. in [1]. To ap-ply the classical theory the interpolation operators P aredefined using knowledge about the zeros of the generat-ing symbol of the original system, the restriction opera-tors R are then defined as the transpose of interpolationand the coarse grid operator is defined as the Galerkinoperator AC = PTAP . Besides the classical AMG ap-proach algebraic multigrid methods based on aggregationand smoothed aggregation are widely used. The idea ofag- gregation goes back at least to an article by Braess [3],the idea of smoothed aggregation at least to Vanek, Man-del and Brezina [4]. The key idea in both cases is to usethe near-kernel component to construct the interpolation,in the case of smoothed aggregation this interpolation isfurther improved by applying a smoother to the interpola-tion. We analyzed aggregation and smoothed aggregationin the context of the convergence theory for Toeplitz andcirculant matrices. Both approaches fit well into this con-text and can be analyzed. In the talk we will present thevarious approaches and present the analysis of the result-ing operators in terms of their generating symbols. Wealso address the issue of growing operator complexitiesdue to the use of the Galerkin operator by using appro-priate techniques that reduce the size of the coarse gridoperator.

[1] J. W. Ruge and K. Stuben. Algebraic multigrid. In S. F.McCormick, editor, Multigrid methods, volume 3 of Frontiers Appl.Math., pages 73130. SIAM, Philadelphia, 1987.

[2] G. Fiorentino and S. Serra. Multigrid methods for Toeplitzmatrices. Calcolo, 28:238305, 1991.

[3] D. Braess. Towards algebraic multigrid for elliptic problemsof second order. Computing, 55:379393, 1995.

[4] P. Vanek, J. Mandel, and M. Brezina. Algebraic multigrid bysmoothed aggregation for second and fourth order elliptic problems.Computing, 56:179196, 1996.

[5] S. Serra-Capizzano and C. Tablino-Possio. Multigrid meth-ods for multilevel circulant matrices. SIAM J. Sci. Comput.,26(1):5585, 2004.

[6] A. Arico’ and M. Donatelli. A V-cycle multigrid for multi-level matrix algebras: proof of optimality. Numer. Math., 105:511547,2007.

C3-3Accuracy measures and Fourier analysis for the fullmultigrid methodC. Rodrigo

Nested iteration and multigrid computational tech-niques are combined to yield the so-called full multigrid(FMG) algorithm. In this well-known approach, the it-erative solver is a multigrid cycle which employs the verysame grid hierarchy to greatly accelerate the convergenceof a basic iterative solver (relaxation). The goal of theFMG algorithm should be to yield a numerical solutionwhose error is comparable to the discretization error. Typ-ically, the common lore states that one or two multigridcycles are sufficient to reach such discretization accuracy.However, the key question then is whether the solutionobtained by this algorithm is sufficiently accurate, and inprac- tice, it may be quite difficult to assess whether theFMG solution indeed yields discretization-level accuracy.This notion is formalized by defining a worst-case relativeaccuracy measure, denoted EF M G , which compares thetotal error of the -level FMG solution against the inherentdiscretization error. This measure can be used for tuningalgorithmic components so as to obtain discretization-levelaccuracy. FMG has received relatively little attention interms of analysis, and here local Fourier analysis (LFA)framework for FMG is developed for estimating EF M G. This results in a tool which yields, a-priori, valuable in-sights into the various components of the FMG algorithmand their effect on the final relative accuracy.

D3-1On the use of AMG for electrochemical platingprocessesP. Thum, T. Clees

Electrochemical processes play an increasingly impor-tant role in the manu- facturing process for a broad rangeof industrial activities. They are used for decoration, func-tional coating (wear, corrosion, morphology), electroform-ing, and electrochemical machining. High-end applica-tions are found in electron- ics (e.g. plating on wafersand PCBs), aeronautics (coatings, turbine blades withmicro-cooling channels and moulds), medicine (needles,implants), the automotive industry (motors, injection sys-tems), gas production, etc. We focus on the Multi-ionTransport and Reaction Models (MiTReM) in laminar andturbulent flow. This physically involved system of partialdifferential equations comprises convection, diffusion, mi-gration (drift) and reaction terms. For such applications,point-based AMG (PAMG) methods are promising pre-conditioners [1]. This paper discusses a reordering frame-work for PAMGs smoothing which is oriented on physical

properties of the systems. Level-dependent, alternatingsmoothing is used. Numerical results are presented for arange of electrochemical test cases with scientifc and in-dustrial relevance demonstrating the robustness of the newframework.

[1] P. Thum, T. Clees, Towards physics-oriented smoothing inalgebraic multigrid for systems of partial differential equations aris-ing in multi- ion transport and reaction models, Procs. 14th CopperMountain Con- ference on Multigrid Methods, 2009, March 22-27.Num. Lin. Alg. Appl., NLAA Special Issue. Accepted.

D3-2Efficient structured AMG preconditioners for thecardiac Bidomain model in 3DM. Pennacchio, V. Simoncini

The so called Bidomain model is possibly the mostcomplete model for the cardiac bioelectric activity. Itconsists of a reactiondiffusion system, modeling the in-tra, extracellular and transmembrane potentials, coupledthrough a nonlinear reaction term with a stiff system of or-dinary differential equations describing the ionic currentsthrough the cellular membrane. We address the prob-lem of efficiently solving the large linear system arising inthe finite element discretization of the bidomain model,when a semi-implicit method in time is employed. We an-alyze the use of symmetric and non-symmetric algebraicmultigrid preconditioners on two major formulations ofthe model, and report on our numerical experience underdifferent discretization parameters in a 3D model of leftventricle.

[1] M. Pennacchio and V. Simoncini, Algebraic Multigrid Pre-conditioners for the Bidomain ReactionDiffusion system. Appl. Nu-mer. Math., vol. 59 (2009), pp. 30333050.

[2] M. Pennacchio and V. Simoncini, Non-symmetric AlgebraicMultigrid Preconditioners for the Bidomain Reaction-Diffusion Sys-tem. Proceed- ings ENUMATH 2009, to appear.

D3-3AMG and micromechanicsM. Kabel, H. Andra, O. Iliev, K. Stuben

Nowadays, a wide range of industries, like the aircraftor automotive industry, uses light-weight composite mate-rials to improve their products. For the macroscopic sim-ulation of composite structures a large number of materialparameters are needed and the experimental identificationof these is a defective and expensive process. Furthermore,the values of these properties vary as a function of themicrostructure (e.g. fiber concentration and orientation).An alternative is the use of homogenization techniques [?]to predict the effective anisotropic viscoelastic propertiesof the composite material in terms of the elastic propertiesof the constituents (matrix and fibers).

The numerical homogenization of such heterogeneousmaterials gives rise to some numerical problems. At first,the coefficients of the so called microstructural problemsare discontinuous and, depending on the application, thejump in the coefficients can be quite large. Secondly, thesize of representative volume elements (REVs) is large if

reinforcements of high aspect ratio (fibers) have to be re-solved in volume elements. Such REVs contain a verycomplex microstructure, and therefore adequate finite ele-ment discretizations lead to very large linear systems. Theapplication of standard preconditioners for such problemsis not streightforward, and needs special studies.

In this talk we compare the AMG-solver SAMG [?]with standard solvers, which are used in commercial pack-ages, for the solution of a set of (visco)elastic boundaryvalue problems in complex microstructures. Especially, weshow, that the solution of the linear system becomes moredifficult as the stiffness of the reinforcements (fibers) in-creases. Furthermore, in addition to the robustness we dis-cuss the CPU and memory requirements. Finally, for verylarge linear system arising in discretization of compositematerials, we discuss an alternative to AMG, namely, nu-merical homogenization based multilevel domain decom-position preconditioners.

[1] N. Bakhvalov and G. Panasenko, Homogenisation: Aver-aging Processes in Periodic Media. Mathematical Problems in theMechanics of Composite Materials. Kluwer Academic Publishers,1989.

[2] K. Stuben, A Review of Algebraic Multigrid. A short reporton AMG and the basic solver technology. Journal of Computationaland Applied Mathematics 128 (2001), pp. 281–309.

E3-1

Auxiliary space preconditioner for a locking-free fi-nite element approximation of the linear elasticityequations

E. Karer, J. Kraus, L. Zikatanov

In this talk we consider a finite element discretizationof the equations of linear elasticity, introduced in [1], whichis stable in the case of nearly incompressible materials.This discretization does not suffer from so-called lockingeffects as they are observed when using standard low(est)order conforming methods for the pure displacement for-mulation. In case of pure traction boundary conditionsoptimal order error estimates are available based on an ap-propriate discrete version of Korns second inequality. Thefocus of this work is on constructing uniform precondition-ers for the linear systems arising from this discretizationscheme. We introduce an auxiliary space method whichconsists in solving an auxiliary problem that involves abilinear form on a larger auxiliary space. By defining aproper projection from this larger space to the originalspace a suitable preconditioner for the original problemcan be set up. We discuss the details of the construction,derive spectral equivalence results based on the fictitiousspace lemma and present numerical experiments.

[1] R.S. Falk, Nonconforming finite element methods for theequations of linear elasticity, Mathematics of Computation, 57(196),(1991), pp. 529550.

E3-2

Subspace correction method for discontinuous Galerkindiscretizations of linear elasticity equations

B. Ayuso, I. Georgiev, J. Kraus, L. Zikatanov

In the first part of this talk we discuss certain classesof discontinuous Galerkin (DG) methods, so-called Inte-rior Penalty (IP) Finite Element (FE) methods, for linearelasticity problems in primal (displacement) formulation,see e.g., [2, 4, 5]. Here we recall some of their stabilityand approximation properties and comment on their suit-ability as a discretization tool for problems with nearlyincompressible materials. Next we propose a natural split-ting of the DG space, which gives rise to uniform pre-conditioners. The presented approach was recently intro-duced in [1] in the context of designing subspace correc-tion methods for scalar elliptic equations and is extendedhere to linear elasticity (vector field) problems. Similar tothe scalar case the solution of the linear algebraic sys-tem corresponding to the IP Galerkin method is reducedto a solution of a problem arising from discretization bynonconforming Crouzeix-Raviart elements plus the solu-tion of a well-conditioned problem on the complementaryspace. Regarding the sub-problem on the nonconformingFE space and considering the case of Dirichlet boundaryconditions on the entire boundarythe so-called pure dis-placement problemit is known how to construct optimalorder multilevel preconditioners that are robust with re-spect to the Poisson ratio , i.e., when approaches 1/2 inthe incompressible limit, see e.g., [3]. However, for mixedboundary conditions or pure Neumann boundary condi-tions (the traction free case), it is much more difficult todevise a robust optimal order method. Nevertheless, thepresented subspace correction method is robust and re-duces the efficient solution of the original problem on theDG space to the solution of a problem on a much smallernonconforming FE space.

[1] B. Ayuso de Dios and L. Zikatanov, Uniformly convergentiterative methods for discontinuous Galerkin discretizations, Journalof Scientific Computing, 40 (2009), pp. 436.

[2] P. Hansbo and M. Larson, Discontinuous Galerkin and theCrouzeix- Raviart element: application to elasticity, M2AN. Mathe-matical Mod- elling and Numerical Analysis, 37 (2003), pp. 6372.

[3] I. Georgiev and J. Kraus and S. Margenov, Multilevel precon-ditioning of Crouzeix-Raviart 3D pure displacement elasticity prob-lems, Lecture Notes in Computer Science (LNCS), 5910 (2010), toappear.

[4] T. Wihler, Locking-free DGFEM for elasticity problems inpolygons, IMA Journal of Numerical Analysis, 24 (2004), pp. 4575.

[5] T. Wihler, Locking-free adaptive discontinuous Galerkin FEMfor linear elasticity problems, Mathematics of Computation, 75 (2006),pp. 1087 1102.

E3-3Local Fourier analysis for quadratic finite elementmethodsF.J. Gaspar, F.J. Lisbona, C. Rodrigo

Quadratic finite element methods offer some advan-tages for the numerical solution of partial differential equa-tions, due to their improved approximation properties incomparison to linear approaches. The algebraic linear sys-tems arising from the discretization of PDEs by this kindof methods require an efficient resolution, and multigridmethods provide a good way to solve this problem. To de-sign geometric multigrid methods, a Local Fourier Anal-

ysis (LFA) is a very useful tool. This analysis, based onan expression of the Fourier transform, permits to choosesuitable components for an efficient multigrid method. LFAfor quadratic finite element discretizations can not be per-formed in a standard way, since the discrete operator is de-fined by different stencils depending on the different typesof points in the grid. In this work, a technique to overcomethis difficulty is developed, and some results showing thegood correspondence between the two-grid convergencefactors predicted by the analysis and the experimentallycomputed asymptotic convergence factors are presented.

E3-4

On finite volume multigrid method

K.S. Kang

In this talk, we consider the finite volume multigridmethod on triangular meshes. We use primal-dual meshesand function spaces to define and anlyze intergrid transferoperators of the multigrid method. Also, we show thescalability of the multigrid method on massively parallelcomputer.

F3-1

Point-based AMG for coupled circuit and devicesimulation

N. Mannig, B. Klaassen, T. Clees

In the development of integrated circuits, so-calledcompact models are used to describe the transistors in cir-cuit simulations. Compact models are small circuits thevoltage-current characteristics of which are fitted to theones of the real device by means of parameter tuning. Un-fortunately, not even expensive compact models with upto several hundreds of parameters are able to fully capturethe switching behavior of advanced field effect transistors,especially in the high-frequency range. This motivatesthe employment of refined semiconductor device modelswithin circuit simulation. In order to set up the cou-pled system to be solved, lumped circuit models are com-bined with distributed device models (drift-diffusion sys-tems), leading to coupled systems of differential-algebraicequations and partial differential equations (PDAEs). Wepropose and discuss an adaptive framework of solvers (α-SAMG) employing point-based AMG preconditioners forsolving the arising linear systems of equations efficientlyand robustly. First benchmarks with α-SAMG integratedinto the coupled circuit and device simulator MECS [1]show the potential of the resulting package. In particular,α-SAMG outperforms the direct linear solvers PARDISOand MUMPS for problems of relevant physical complexityand size.

[1] M. Selva Soto, C. Tischendorf, Numerical analysis of DAEsfrom coupled circuit and semiconductor simulation, Appl. Numer.Math., 53 (2005), pp. 471488.

F3-2

Adaptive algebraic multigrid: A bootstrap approach

K. Kahl, J. Brannick

By the time of its development Algebraic Multigrid(AMG) was thought of as a black box solver for systemsof linear equations [1]. The classical formulation of AMGturned out to lack the robustness to overcome challengesencountered in many of today’s computational simula-tions. In recent years several methods have been proposedthat try to overcome such difficulties by means of adaptivetechniques, such as the framework of smoothed aggrega-tion or adaptive algebraic multigrid [2,3].

Initially proposed by A. Brandt in [4,5] we developa Bootstrap approach to adaptive AMG to achieve thesought for optimal multigrid complexity in a purely alge-braic fashion. We introduce the so-called “Least SquaresInterpolation” which allows us to define interpolation op-erators in an algebraic multigrid framework solely basedon prototypes of algebraically smooth error. Furthermore,we introduce a “Bootstrap Setup” which enables us tocompute accurate Least Squares Interpolation operatorsin linear, i.e., optimal, complexity by using an observationthat links the eigenvectors and eigenvalues of the operatorsin the multigrid hierarchy. This leads to an efficient mul-tiscale computation of prototypes of algebraically smootherror [6].

Besides introducing the techniques of Bootstrap Alge-braic Multigrid, we demonstrate its efficiency in the ap-plication to a variety of problems, each illustrating a com-ponent of the method.

[1] Ruge, J. W. and Stuben, K., Algebraic Multigrid in Multigridmethods, Frontiers Appl. Math., Vol 3, 1987, pp. 73–130.

[2] Brezina, M. and Falgout, R. and MacLachlan, S. and Man-teuffel, T. and McCormick, S. and Ruge, J., Adaptive smoothed ag-gregation (αSA) multigrid, SIAM Rev., Vol 47(2), 2005, pp. 317–346.

[3] Brezina, M. and Falgout, R. and Manteuffel, T. A. andMacLachlan, S. and McCormick, S. and Ruge, J., Adaptive algebraicmultigrid, SIAM J. Sci. Comput., Vol 27(4), 2006, pp. 1261–1286.

[4] A. Brandt, General Highly Accurate Algebraic Coarsening,Electron. Trans. Numer. Anal., Vol 10, 2000, pp. 1–20.

[5] Brandt, A., Multiscale scientific computation: Review 2001in Multiscale and multiresolution methods. Theory and applica-tions., Lecture Notes in Computational Science and Engineering,Vol 20, 2002, pp. 1–96.

[6] K. Kahl, Adaptive Algebraic Multigrid for Lattice QCD Com-putations, Ph. D. Thesis, Bergische Universitat Wuppertal, Fach-bereich Mathematik und Naturwissenschaften, 2009.

F3-3

A projected AMG for LCPs

J. Toivanen, C.W. Oosterlee

We adapt an algebraic multigrid (AMG) method [4]for solving linear complementarity problems (LCPs)

L ≥ 0, u ≥ g, L(u− g) = 0,

where L is a partial differential operator. Here we assumethat the discretization of L yields an M-matrix. Insteadof the PFAS approach [1], we base our method on the

projected multigrid method [3]. As applications we con-sider pricing American options under Hestons stochasticvolatility model and elasto-plastic torsion problems; see[2].

[1] A. Brandt and C. W. Cryer, Multigrid algorithms for the so-lution of linear complementarity problems arising from free boundaryproblems, SIAM J. Sci. Statist. Comput., 4 (1983), pp. 655684.

[2] C. W. Oosterlee, On multigrid for linear complementar-ity problems with application to American-style options, Electron.Trans. Numer. Anal., 15 (2003), pp. 165185.

[3] C. Reisinger and G. Wittum, On multigrid for anisotropicequations and variational inequalities: pricing multi-dimensional Eu-ropean and American options, Comput. Vis. Sci., 7 (2004), pp.189197.

[4] J. W. Ruge and K. Stuben, Algebraic multigrid, in Multigridmethods, SIAM, 1987, pp. 73130.

F3-4Algebraic multigrid with proved convergence II:automatic coarsening, multilevel convergenceY. Notay, A. Napov

In a companion talk (An algebraic multigrid methodwith proved convergence rate I: two-grid analysis, byA. Napov) we develop the convergence analysis of (un-smoothed) aggregation-based two-grid methods. In par-ticular, we show that the convergence is characterized bya quantity which can be assessed considering only a singleaggregate at a time, measuring in some sense its qual-ity. Here we show how this allows to derive an algebraicmultigrid method with guaranteed optimal convergencerate. Actually, two more ingredients are needed. Firstly,we propose an aggregation algorithm which builds aggre-gates ensuring that the quality measure is above a chosenthreshold. Next, we propose a cycling strategy that, giventhe known bound on the two-grid convergence rate, en-sures level-independent convergence in multilevel setting.

ISPowertrain engineering needs mathematicsG. Rainer

The strong demand on the OEM’s to reduce fleet fuelconsumption (CO2 emission) enforce the technology de-velopment of fuel efficient vehicle concepts. To achievethese goals R&D-programmes were launched to bring newpowertrain concepts to market, e.g. hybrid powertrains asa combination of conventional and electrical systems, fullelectrical powertrains with range extenders as well as IC-engine equipped powertrains with downsized combustionengines. The market success of these concepts will be de-cided by the realizable compromise of fuel consumption,driving dynamics and customer acceptance. The question,which optimum powertrain concept can be delivered to thecustomer, must take into account the cost/benefit ratio.

With increasing interest in the development of hy-brid (HEV) and electrical (PEV) vehicles, the demandfor comprehensive system design and analysis to supportthe powertrain development process (PDP) is rising. Thedevelopment of highly sophisticated modern powertrainsas e.g. todays HEVs is even not possible without applying

mathematical simulation quite from the beginning of thedevelopment process.

In that respect more advanced mathematical modelsof great variety in their scope, complexity and sophisti-cation to represent propulsion systems and components(both steady state and dynamic) are requested. Basedon the flexibility of mathematical models it is possible toadjust them to the requirements of all phases of the prod-uct development process from simple (fast) models to verycomplex (CPU intensive) models.

But there are additional aspects why mathematicalmodels are essential for the industry if a connection be-tween its abstract concepts and techniques and the realworld is established:

Mathematical models are capable to represent the real-world phenomena and by applying them with numericalor analytical techniques, both qualitative and quantita-tive predictions can be made. Additionally mathematical,physical and chemical models are capable to be re- used forsimulating a great variety of phenomena occurring in dif-ferent fields of applications, e.g. Finite Elements (FEM),Computational Fluid Dynamcs (CFD) or Multi Body Dy-namics (MBD) have become firmly established tools sup-porting the product development process, complementingand more and more reducing the traditional experimentalapproaches.

Based on various case studies this presentation shallgive an insight on the one hand into

a) the vertical applications of mathematical methodsfor powertrain system and component simulation includ-ing optimization, control development and design alongthe development process; and on the other hand into

b) the horizontal deployment of mathematical meth-ods showing the derivation of applications e.g. for thepharmaceutical industry or the process industry from meth-ods originally created for engine and powertrain develop-ment.

Thursday

IS7

Multiscale methods for edge detection and imagesegmentation

R. Basri

Detecting object boundaries in images is essential fortheir accurate interpretation. Multiscale methods can takean important role in this task, as they provide means toadaptively overcome noise and to incorporate object prop-erties of varying complexities. In this talk I will describemultiscale methods for edge detection and image segmen-tation. For edge detection we consider methods that over-come noise by applying filters whose shape is adapted tothe shape of the sought edges. I will further present resultson the limits of detectability, as a function of the lengthsof edges and their combinatorics. For image segmentationI will describe an efficient method inspired by AlgebraicMultigrid. This method utilizes features whose expressivepower increases with the size of the sought segments.

IS8

Algebraic multigrid in multiphysics and multiscale

M.W. Gee, W.A. Wall

Multifield and multiscale phenomena arise in manysituations in engineering and applied sciences. Two exam-ples are fluid-structure interaction and modeling of turbu-lent flows. While multigrid methods and multigrid prin-ciples have been excessively studied for single field phe-nomena, less attention has been given to the applicationof multigrid principles in complex multiphysics and mul-tiscale simulations. In the last view years we have notonly extensively applied AMG as a solver in various realworld multiscale and multifield (m&m) applications butalso adopted AMG ideas as building blocks for the designof new m&m methods. In this talk, we focus on two re-cent developments made in the field of algebraic multigrid(AMG) principles:

The first is a new monolithic AMG scheme for the im-plicit solution of fluid-structure interaction (FSI) simula-tions [1]. Therein, an AMG hierarchy for the nonsymmet-ric monolithic fluid, structure and mesh movement systemof equations is constructed that also considers a coarserepresentation of interfield off-diagonal coupling blocks ina variationally consistent way. This FSI multigrid pre-conditioner is based on a mixed smoothed aggregation[5,7,2] and nonsymmetric smoothed aggregation approach[6] that accounts for the hybrid nature of the monolithicFSI problem. It limits the amount of artificial smearing ofindividual fields on coarse levels by construction and al-lows for field specific smoothers within its block orientedsmoothing procedure. It is mainly utilized in the very chal-lenging regime of soft tissue biomechanics of the vascularand respiratory system where disadvantageous density ra-tios and large deformations appear.

The second recent development is an algebraic varia-tional multiscale method for convection dominated prob-lems and turbulent flow. Therein, simple plain aggrega-tion transfer operators that possess a projective propertyare used to construct scale separating operators that al-low a purely algebraic scale separation process. Opposedto AMG as a solution method, here AMG principles areutilized as modeling tool that influences result behavior.The algebraic variational scale separation is also utilized toimplement a discontinuity capturing term that affects onlyfine scales while coarse scales remain unmodified. The re-sulting method yields an efficient, stable and very accuratescheme that has been utilized in convection dominatedconvection-diffusion problems [3], in LES simulation ofincompressible turbulent [4] and compressible low–Machturbulent flow.

[1] M.W. Gee, U. Kuttler, W.A. Wall, Truly Monolithic Alge-braic Multigrid for Fluid-Structure Interation, International Journalfor Numerical Methods in Engineering, (2010), submitted.

[2] Gee, M.W., Hu, J.J., Tuminaro, R.S., A New SmoothedAggregation Multigrid Method for Anisotropic Problems, NumericalLinear Algebra With Applications, (2009), 16, pp. 19-37.

[3] Gravemeier, V., Gee, M.W., Wall, W.A., An algebraic vari-ational multiscale-multigrid method based on plain aggregation forconvection-diffusion problems, Computer Methods in Applied Me-chanics and Engineering, (2009), 198, 3821-3835.

[4] Gravemeier, V., Gee, M.W., Kronbichler, M., Wall, W.A.,An Algebraic Variational Multiscale-Multigrid Method for Large EddySimulation of Turbulent Flow, Computer Methods in Applied Me-chanics and Engineering, (2009), 199, 853-864.

[5] J Mandel, Marian Brezina, and Petr Vanek, Energy opti-mization of algebraic multigrid bases, Computing, (1999), 62, 205–228.

[6] M. Sala and R.S. Tuminaro, A new petrov-galerkin smoothedaggregation preconditioner for nonsymmetric linear systems, SiamJ. Scientific Comp, (2008), 31, 143–166.

[7] P. Vanek, J. Mandel, and M. Brezina. Algebraic multigridbased on smoothed aggregation for second and fourth order problems,Computing, (1996), 56, 179–196.

A4-1

Lifting in hybrid lattice Boltzmann and PDE mod-els

Y. Vanderhoydonc, W. Vanroose

Mathematical models based on kinetic equations areubiquitous in the modeling of granular media, populationdynamics of biological colonies, chemical reactions andmany other scientific problems. These individual basedmodels are computational very expensive because the evo-lution takes place in the phase space. Hybrid simulationscan bring down this huge computational cost. These cal-culations replace locally the kinetic models with a partialdifferential equation in the regions where it is justified.Then the computational domain is split into subdomains.The mathematical question is how to couple these mod-els in a correct way. At each interface we are faced with amissing data problem and this requires an appropriate lift-ing operator. Indeed, a kinetic model has typically morevariables than a PDE model and at each interface we needto fill in this missing data. In this contribution we reporton different lifting operators for a hybrid simulation that

combines a Lattice Boltzmann model with a reaction- dif-fusion PDE. We extend the work on smooth initializationof [1] and [2]. We also give numerical illustrations.

[1] P. Van Leemput, C. Vandekerckhove, W. Vanroose and D.Roose, Accuracy of hybrid Lattice Boltzmann/Finite Difference schemesfor reaction-diffusion systems, Multiscale Model. Sim, 6 (2007) pp.838857.

[2] P. Van Leemput, M. Rheinlander and M. Junk, Smooth ini-tialization of lattice Boltzmann schemes, Computers & Mathematicswith Applications, 58 (2009) pp. 867882.

A4-2Multigrid preconditioners for adaptive wavelet col-locationS. Bertoluzza, M. Pennacchio, L. Castro

The adaptive wavelet collocation method based on in-terpolationg wavelets ([1]) has been successfully appliedfor the solution of problems arising in various applica-tion fields (for instance in fluid dynamics and in elas-ticity). Unlike what happens for the classical waveletGalerkin method, where the wavelet structure itself pro-vides a multigrid type optimal preconditioner, for suchmethod the intrinsic wavelet preconditioner is estimatedto be only sub- optimal (with, in the case of a secondorder problem, a loss of a logarithmic factor in two di-mensions, and of a factor 1/h in three dimensions). Wewill compare the performaces of such preconditioner withthe performances of an algebraic multigrid preconditioneron matrices arising from the application of the (adaptive)wavelet collocation method to some elasticity problems [2].

[1] S. Bertoluzza, An adaptive wavelet collocation method basedon interpo- lating wavelets, in Multiscale Wavelet Methods for Par-tial Differential Equations, W. Dahmen, A. Kurdila and P. Oswaldeds., Academic Press, (1997), pp. 109135.

[2] S. Bertoluzza, L. Castro, A. J. M. Ferreira, An high ordercollocation method for the static and vibration analysis of compositeplates using a first-order theory, Composite Structures 89 (2009).

A4-3Equivalent preconditioning in multiscale problemsG. Samaey, W. Vanroose

We discuss preconditioning techniques for the compu-tation of a fixed points of a coarse-scale time-stepper as itappears in multiscale problems. These time-steppers con-sist of a set of computational routines wrapped around ap-propriately initialized fine-scale simulations. Fixed pointproblems are then solved using Newton-Krylov iterations[1]. The inner Krylov iteration in these problems sufferfrom slow convergence due to ill-conditioned spectrum.For a linear, scalar advection-reactiondiffusion equation,we investigate in detail how the convergence rate dependson the choice of preconditioner parameters and on the timediscretization. Both analytical results and numerical ex-periments are presented, showing that one can speed upthe convergence of iterative methods significantly for awide range of parameter values in the preconditioner [2].

[1] G. Samaey, W. Vanroose, D. Roose and I. G. Kevrekidis,Newton- Krylov solvers for the equation-free computation of coarse

traveling waves, Comp. Meth. Appl. Mech. Eng., 197 (2008) pp3480-3491.

[2] G. Samaey and W. Vanroose, An analysis of equivalentoperator preconditioning for equation-free NewtonKrylov methods,SIAM Num. Analysis, To appear, 2010.

B4-1Point smoothers for elliptic optimal control prob-lemsS. Takacs, W. Zulehner

In this talk we will discuss multigrid methods for solv-ing discretized optimality systems for elliptic optimal con-trol problems of tracking type, like the model problem

Minimize J(y, u) = 12 ||y − yD||

2L2(Ω) + γ

2 ||u||2L2(Ω)

subject to −∆y + y = u in Ω,∂y∂n

= 0 on ∂Ω.

The proposed approach is based on the Karush-Kuhn-Tucker system (KKT-system). In many cases the state ycan be expressed as analytic term containing the adjoinedstate p, like u = γ−1p for the model problem. Using sucha relation we can reduce the system to the variables y andp only.

We propose and discuss multigrid methods for solvingthe discretized system. In this talk we want to concentrateon point smoothers which result from constructing precon-ditioners by collecting the matrix entries corresponding toone node. Smoothers, that are well understood for sym-metric positive definite matrices, can be carried over to thecase of saddle point problems, which leads mainly to col-lective Richardson-, Jacobi- and Gauss-Seidel-smoothers.

Numerical examples show good behavior of such smoothers.We will discuss convergence proofs of the proposed meth-ods applied to the distributed control problem. In partic-ular, we will show how to use techniques from computeralgebra for analyzing the smoothing property.

B4-2Algebraic multigrid for density driven flowA. Nagel, G. Wittum

Algebraic multigrid methods, as, e.g., introduced inthe classic work by Ruge and Stuben [1], provide a usefultool for the solution of large sparse linear systems withalmost optimal complexity. One variant is the FilteringAlgebraic Multigrid approach, introduced in [2,3]. Thekey idea of this method is to construct the interpolationoperator P , such that the norm of the two-grid operatoris minimised in an approximate sense. At the same timeconstraints are imposed to guarantee filter conditions forcertain test vectors.

In this study we focus on the application for densitydriven flow in porous media. In [4] Newton’s method isemployed to solve both the Boussinesq-approximation aswell as the full equations. Challenges in the resulting lin-ear systems of equations arise from anisotropies, discontin-uous coefficients and fluctuations in the velocity profiles.We present a strategy for a robust preconditioner within a

point-block-setting. This includes the choice of appropri-ate smoothers, a preconditioner for the setup phase andmodifications for the choice of strong connections.

[1] J.W. Ruge and K. Stuben, Algebraic Multigrid (AMG). InS.F. McCormick (ed.), Multigrid Methods, Volume 3 of Frontiers inApplied Mathematics SIAM, Philadelphia, PA, (1987), pp. 73–130.

[2] Ch. Wagner, On the algebraic construction of multileveltransfer operators, Computing 65 (2000), pp. 73–95

[3] A. Nagel, R.D. Falgout, G. Wittum, Filtering algebraic multi-grid and adaptive strategies, Comput Visual Sci 11 (2008), pp. 159–167

[4] E. Fein. d3f - Ein Programmpaket zur Modellierung von

Dichte-stromungen, GRS, Braunschweig (1998).

B4-33D Multilevel aggregation for segmentation andtracking of live cellsH. De Sterck

We apply the so-called Segmentation by Weighted Ag-gregation technique by Sharon, Brandt, Basri et al. tothe problem of segmenting live cell bright field micro-scope images. The multilevel aggregation algorithm isbased on Algebraic Multigrid methods. The variant ofthe method used is described, and it is explained how it istailored to the application at hand. In particular, a newscale-invariant saliency measure is proposed for decidingwhen aggregates of pixels constitute salient segments thatshould not be grouped further. It is shown how segmenta-tion based on multilevel intensity similarity alone does notlead to satisfactory results for bright field cells. However,as expected, the addition of multilevel intensity variance(as a measure of texture) to the feature vector of eachaggregate leads to improved cell segmentation. Prelimi-nary results are presented for applying the multilevel ag-gregation algorithm in space-time to temporal sequencesof microscope images, with the goal of obtaining space-time segments (object tunnels) that track individual cells.Application of the algorithm to segmentation and roadfinding for satellite images is also briefly discussed.

Part III

Information

55

InformationIschia (pronunciation: [iskia]) is a volcanic island in the Tyrrhenian Sea, at the northern end of the Gulf of Naples.The roughly trapezoidal island lies about 30 km from Naples and measures around 10 km east to west and 7 km northto south with a 34 kilometers (21 mi) coastline. It is almost entirely mountainous, with the highest peak being MountEpomeo at 788 meters. The island has a population of over 60,000 people. Ischia Porto is the name of the maincommune of the island, where the EMG2010 Conference takes place. Other community areas include Barano d’Ischia,Casamicciola Terme, Forio, Lacco Ameno and Serrara Fontana; see Figure 1.

The British classical composer William Walton settled in Ischia in 1949 and lived on the island for the remainderof his life, dying there in 1983. In 1948, American author Truman Capote stayed in room number 3 in the PensioneLustro in the town of Forio on the island. He wrote an essay about his stay there, which later appeared in LocalColor, published in 1950 by Random House. Parts of the Hollywood film The Talented Mr Ripley were filmed on theisland. Norwegian playwright Henrik Ibsen lived on the island for a short period, and is said to have finished PeerGynt there in 1867. The Hollywood Hit The Crimson Pirate was also filmed on the island. French novelist PascalQuignard set much of his book Villa Amalia on the island. Cleopatra with Elizabeth Taylor was also filmed on theisland. Herg’s The Adventures of Tintin ends in Ischia, which serves as the location of Endaddine Akass’s villa in theunfinished book Tintin and Alph-Art. W.H. Auden wrote his poem ”In Praise of Limestone” here. A sentence: Ischiahas seized all my senses. by Friedrich Nietzsche.

You can easily reach all places of Ischia by Bus:

Linea CD (Driving clockwise) Ischia Porto - Pilastri - Piedimonte - Barano - Buonopane - Fontana - Serrara -Ciglio - Panza (Forio) - Succhivo (Serrara) - Cava Grado (Sant’Angelo) - Succhivo (Serrara) - Panza - Forio - LaccoAmeno - Casamicciola Terme - Ischia Porto.

Linea CS (Driving counter clockwise) Ischia Porto - Casamicciola Terme - Lacco Ameno - Forio - Panza - Succhivo(Serrara) - Cava Grado (Sant’Angelo) - Succhivo (Serrara) - Panza - Ciglio (Serrara) - Serrara - Fontana - Buonopane- Barano - Piedimonte - Pilastri - Ischia Porto.

In addition there are 16 different Bus lines driving to all places; see Figure 2.

Linea 1 Ischia Porto - Casamicciola Terme - Lacco Ameno - Forio - Panza (Forio) - Succhivo (Serrara) - CavaGrado (Sant’Angelo) e back.

Linea 2 Ischia Porto - Casamicciola Terme - Lacco Ameno - Forio - Citara (Giardini Poseidon) e back.Linea 3 Ischia Porto - Piazza Marina (Casamicciola Terme) - Piazza Bagni (Casamicciola Terme) - La Rita

(Casamicciola Terme) - Piazza Maio (Casamicciola Terme) - Fango (Lacco Ameno) - Piazza Maio (CasamicciolaTerme) - Piazza Marina (Casamicciola Terme) - Ischia Porto.

Linea 4 Piazza Marina (Casamicciola Terme) - Piazza Bagni (Casamicciola Terme) - La Rita (Casamicciola Terme)- Piazza Maio (Casamicciola Terme) - Via Principessa Margherita (Casamicciola Terme) - Piazza Marina (CasamicciolaTerme).

Linea 5 Ischia Porto - Pilastri (Ischia) - Piedimonte (Barano) - Testaccio (Barano) - Spiaggia dei Maronti (Barano)e back.

Linea 6 Ischia Porto - Pilastri (Ischia) - Piedimonte (Barano) - Fiaiano (Barano) e back.Linea 7 Ischia Porto - Ischia Ponte e back.Linea 8 Ischia Porto - via delle Terme (Ischia) - Palazzetto (Ischia) - S. Michele (Ischia) - S. Antuono (Ischia) -

Campagnano (Ischia) e back.Linea C12 Ischia Porto - Piazza degli Eroi (Ischia) - Cartaromana (Ischia) - S. Michele (Ischia) - S. Antuono

(Ischia) - Campagnano (Ischia) - S. Antuono (Ischia) - Pilastri (Ischia) - Ischia Porto.Linea C13 Ischia Porto - Via delle Terme (Ischia) - Pilastri (Ischia) - S. Antuono (Ischia) - Campagnano (Ischia)

- S. Antuono (Ischia) - S. Michele (Ischia) - Cartaromana (Ischia) - Piazza degli Eroi (Ischia) - Ischia Porto.Linea 14 Piazza Marina (Casamicciola Terme) - Piazza Bagni (Casamicciola Terme) - La Rita (Casamicciola

Terme) - Piazza Maio (Casamicciola Terme) - Fango (Lacco Ameno) - Forio - Giardini Poseidon (Forio) e back.Linea 15 Ischia Porto - Palazzetto (Ischia) - Ischia Ponte - Via Pontano (Ischia) - Palazzetto (Ischia) - S. Antuono

(Ischia) - Pilastri (Ischia) - Ischia Porto - Via Quercia (Ischia) - Ischia Porto.Linea 16 Piazza Marina (Casamicciola Terme) - Salita S. Pasquale (Casamicciola Terme) - Cretaio (Casamicciola

Terme) - Via Principessa Margherita (Casamicciola Terme) - Piazza Marina (Casamicciola Terme) - Lacco Ameno -167 (Lacco Ameno) - Fango (Lacco Ameno) - 167 (Lacco Ameno) - - Lacco Ameno - Piazza Marina (CasamicciolaTerme).

Conference venue

The EMG2010 Conference’s venue is the 4-star Hotel Continental Terme in Ischia. It is characterized by an elegantlyfurnished, thoroughly conceived setting with an exclusive and refined allure while creating a warm and comfortableambient. Several building units resembling Mediterranean style villas are embedded in a wide, lush green area. Withits 5 thermal swimming pools with different sizes and temperatures, harmoniously set along an ideal relaxation path,it can be considered a real wellness park set in midst of a green scenery. The thermal and wellness center offers a widerange of treatments satisfying every taste and need, with individually tailored programs aimed at conveying a feelingof complete relaxation.

The park area includes a well equipped playground for children, who will have a lot of fun while you will beenjoying the sunshine and the wonderful benefits of thermal water. Thanks to its both central and quiet position, thehotel is ideally suited also for your wellness and relaxation holidays with the whole family. Moreover, the hotel has aspecial agreement with a beach, easily to be reached by a comfortable shuttle bus service (to and from the beach).

The rooms of the hotel, each designed as a small villa embedded in a large park and displaying an elegant andtasteful furniture, offer a comfort which is always up to the ever increasing demands of our international guests. In therestaurants, serving carefully created, refined dishes, you will be able to taste all the specialties of the local cuisine.The Hotel Continental Terme is also the ideal choice for those wishing to organize business trips, meetings, workshops,conventions and conferences. It boasts a wide choice of meeting rooms with cutting-edge technological equipment from15 to 350 seats.

For its organization and structure, the Hotel Continental Terme has all the features of a small thermal park inthe center of Ischia Porto, boasting 5 thermal swimming pools with different temperatures and a first-class SPA andBeauty Center, complemented by the beneficial effect of the surrounding lush park.

Hotel Continental Terme in Ischia (4****)Via Michele Mazzella 7480077 Isola d’Ischia (Naples) ItalyPhone: +39 0813336111Fax: +39 0813336276Mobile: +39 3384819436www.continentalterme.it

To reach the Hotel Continental Term from Ischia harbor, take the bus lines 5,6,CD. There are also taxi and minitaxiavailable.

The EMG2010 Lectures will be given in the Pithecusa room (Main Meeting Room)

and the Primavera room.

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Participants

Alfio Borzi [email protected] Universita del Sannio ItalyCornelis Oosterlee [email protected] CWI- Amsterdam NetherlandsLuisa D’Amore [email protected] University of Naples Federico II ItalyMarianne AKIAN [email protected] INRIA and Ecole Polytechnique FranceEmmanuel Math-ioudakis

[email protected] Technical University of Crete Greece

Irad Yavneh [email protected] Technion IsraelXiaowen Xu [email protected] IAPCM - HPCC P. R. ChinaFredi Troeltzsch [email protected] Technische Universitaet Berlin GermanyHans De Sterck [email protected] University of Waterloo CanadaJoerg Willems [email protected] RICAM AustriaPeter Thum [email protected] Fraunhofer SCAI GermanyHuidong Yang huidong [email protected] JK University Linz AustriaSylvie Detournay c Inria Saclay and CMAP FranceTobias Wiesner [email protected] Technische Universitaet Muenchen GermanyChristian Groß [email protected] Universita della Svizzera italiana SwitzerlandMicol Pennacchio [email protected] CNR Pavia ItalyLars Grasedyck [email protected] RWTH Aachen - IGPM GermanyJames Adler [email protected] Penn State University United StatesArmando Coco [email protected] University of Catania ItalyKarolina Burzynska [email protected] GridwiseTech PolandYvan Notay [email protected] Universite Libre de Bruxelles BelgiumInigo Arregui [email protected] University of La Coruna SpainStephen Nash [email protected] George Mason University United StatesJustin Wan [email protected] University of Waterloo CanadaJames Brannick [email protected] Penn State United StatesBram Reps [email protected] University of Antwerp BelgiumKab Seok Kang [email protected] MPI Plasmaphysik HLST GermanyMichael Koster [email protected]

dortmund.deTechnische Universitaet Dortmund Germany

Sergio Gonzalez [email protected] Escuela Politecnica Nacional EcuadorHarald Koestler [email protected]

erlangen.deUniversity of Erlangen-Nuremberg Germany

Mario Annunziato [email protected] Universita degli Studi di Salerno ItalyGiovanni Samaey [email protected] K.U. Leuven BelgiumDaniel Ritter [email protected]

erlangen.deUniversity of Erlangen-Nuernberg Germany

Artem Napov [email protected] Universitaet Libre de Bruxelles BelgiumVadim Lisitsa [email protected] Inst. of Petrol. Geology RussiaBjorn Gmeiner [email protected]

erlangen.deUniversity of Erlangen-Nuernberg Germany

Kent-Andre Mardal [email protected] Simula Research Laboratory NorwayMarco Donatelli [email protected] Universita dell’Insubria ItalyMichelle Vallejos [email protected] University of the Philippines PhilippinesHeng-Bin An an [email protected] Institute of Applied Physics and

Computational MathematicsP.R. China

Alessandro Celona [email protected] IRCCS ItalyKarsten Kahl [email protected] University of Wuppertal GermanyGalina Muratova [email protected] Southern Federal University RussiaStefan Takacs [email protected] JK University Linz AustriaTobias Weinzierl [email protected] Technische Universitaet Muenchen GermanyXavier Pinel [email protected] CERFACS FranceAnna Naumovich [email protected] DLR Germany

Wim Vanroose [email protected] Universiteit Antwerpen BelgiumFrancisco Gaspar [email protected] University of Zaragoza SpainCarmen Rodrigo [email protected] University of Zaragoza SpainEmiliano Cristiani [email protected] INdAM ItalySilvia Bertoluzza [email protected] CNR Pavia ItalyScott MacLachlan [email protected] Tufts University United StatesBram Metsch [email protected] Universitaet Bonn GermanyJari Toivanen [email protected] Stanford University ICME United StatesMatthias Bolten [email protected] University of Wuppertal GermanyDirk Abbeloos [email protected] K.U. Leuven BelgiumErwin Karer [email protected] RICAM AustriaTanja Clees [email protected] Fraunhofer SCAI GermanyJohannes Kraus [email protected] RICAM AustriaWalter Zulehner [email protected] JK University Linz AustriaTania Firijulina [email protected] Universidad Catolica de Temuco ChileMichael Hinze [email protected] Universitaet Hamburg GermanyRolf Krause [email protected] University of Lugano SwitzerlandGabriel Wittum [email protected] Universitaet Frankfurt GermanyArne Naegel [email protected] Universitaet Frankfurt GermanyUlrich Langer [email protected] JK University Linz AustriaRonen Basri [email protected] Weizmann Institute of Science IsraelMichael Gee [email protected] Technische Universitaet Muenchen GermanyMike Giles [email protected] University of Oxford United KingdomWolfgang Hackbusch [email protected] MPI Mathematik in den Naturwis-

senschaftenGermany

Volker Schulz [email protected] University of Trier GermanyStefano Serra Capiz-zano

[email protected] University of Insubria Italy

Stefan Vandewalle [email protected] K.U. Leuven BelgiumCraig C. Douglas [email protected] University of Wyoming United StatesJaap van der Vegt [email protected] University of Twente NetherlandsOleg Iliev [email protected] Fraunhofer ITWM GermanyMatthias Kabel [email protected] Fraunhofer ITWM GermanyGiovanna Carcano [email protected] Universita degli Studi Milano Bic-

occaItaly

Yanbin Shen [email protected] Delft University of Technology NetherlandsGotthard Rainer [email protected] AVL List GmbH AustriaIra Livshits [email protected] Ball State University United StatesElena Zhebel [email protected] Shell International Exploration and

Production B.V.Netherlands

James Lottes [email protected] Oxford Centre for CollaborativeApplied Mathematics

United Kingdom

Tobias Koppl [email protected] Technische Universitaet Muenchen GermanyYnte Vanderhoydonc [email protected] Universiteit Antwerpen BelgiumValeria Mele [email protected] University of Naples Federico II ItalyDaniela Casaburi [email protected] University of Naples Federico II ItalyDirk van Eijkeren [email protected] University of Twente NetherlandsBabett Lemke [email protected]

frankfurt.deUniversitaet Frankfurt Germany

Kees Venner [email protected] University of Twente The NetherlandVan Emden Henson [email protected] Lawrence Livermore Nat. Lab. USAGotthard Rainer [email protected] AVL List Austria

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