Comptes Rendus
no. 6
Numerical analysis
Sparse approximate solutions to stochastic Galerkin equations
[Approximations creuses pour la méthode de Galerkin stochastique]

Présenté par : the Editorial Board

Christophe Audouze1 ; Prasanth B. Nair1
1 University of Toronto, Institute for Aerospace Studies, 4925 Dufferin Street, Ontario, M3H 5T6, Canada
Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 561-570.

Dans cette Note, nous formulons une méthode de Krylov creuse pour la résolution de grands systèmes linéaires issus de la discrétisation d'équations aux dérivées partielles elliptiques paramétrées aléatoirement. Nous analysons l'algorithme du gradient conjugué (GC) creux dans le cadre des méthodes de sous-espaces de Krylov inexactes, montrons sa convergence et étudions sa complexité algorithmique. Les études numériques réalisées sur des modèles de diffusion stochastiques montrent que la méthode du GC creux converge plus rapidement que le GC classique lorsque la solution recherchée admet une représentation creuse dans une base de chaos polynomial. Dans ce cas, l'algorithme du GC creux retrouve presque intégralement la structure creuse de la solution exacte, permettant d'accélérer la convergence. Lorsque la solution exacte est dense, l'algorithme du GC creux fournit des résultats de convergence similaires à ceux obtenus avec le GC classique.

In this Note, we formulate a sparse Krylov-based algorithm for solving large-scale linear systems of algebraic equations arising from the discretization of randomly parametrized (or stochastic) elliptic partial differential equations (SPDEs). We analyze the proposed sparse conjugate gradient (CG) algorithm within the framework of inexact Krylov subspace methods, prove its convergence and study its abstract computational cost. Numerical studies conducted on stochastic diffusion models show that the proposed sparse CG algorithm outperforms the classical CG method when the sought solutions admit a sparse representation in a polynomial chaos basis. In such cases, the sparse CG algorithm recovers almost exactly the sparsity pattern of the exact solutions, which enables accelerated convergence. In the case when the SPDE solution does not admit a sparse representation, the convergence of the proposed algorithm is very similar to the classical CG method.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.05.009
Christophe Audouze 1 ; Prasanth B. Nair 1

1 University of Toronto, Institute for Aerospace Studies, 4925 Dufferin Street, Ontario, M3H 5T6, Canada 
@article{CRMATH_2019__357_6_561_0,
     author = {Christophe Audouze and Prasanth B. Nair},
     title = {Sparse approximate solutions to stochastic {Galerkin} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {561--570},
     publisher = {Elsevier},
     volume = {357},
     number = {6},
     year = {2019},
     doi = {10.1016/j.crma.2019.05.009},
     language = {en},
}
TY  - JOUR
AU  - Christophe Audouze
AU  - Prasanth B. Nair
TI  - Sparse approximate solutions to stochastic Galerkin equations
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 561
EP  - 570
VL  - 357
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2019.05.009
LA  - en
ID  - CRMATH_2019__357_6_561_0
ER  - 
%0 Journal Article
%A Christophe Audouze
%A Prasanth B. Nair
%T Sparse approximate solutions to stochastic Galerkin equations
%J Comptes Rendus. Mathématique
%D 2019
%P 561-570
%V 357
%N 6
%I Elsevier
%R 10.1016/j.crma.2019.05.009
%G en
%F CRMATH_2019__357_6_561_0
Christophe Audouze; Prasanth B. Nair. Sparse approximate solutions to stochastic Galerkin equations. Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 561-570. doi : 10.1016/j.crma.2019.05.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.05.009/

[1] C. Audouze; P. Håkansson; P.B. Nair A stopping criterion for iterative solution of stochastic Galerkin matrix equations, Int. J. Uncertain. Quantificat., Volume 6 (2016) no. 3, pp. 245-269

[2] I. Babuška; R. Tempone; G.E. Zouraris Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., Volume 42 (2004), pp. 800-825

[3] J. Bäck; F. Nobile; L. Tamellini; R. Tempone Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients, Comput. Math. Appl., Volume 67 (2014) no. 4, pp. 732-751

[4] A. Bespalov; C.E. Powell; D. Silvester Energy norm a posteriori estimation for parametric operator equations, SIAM J. Sci. Comput., Volume 36 (2014), pp. 339-363

[5] M. Bieri; C. Schwab Sparse High Order FEM for Elliptic SPDEs, ETH, Zürich, Switzerland, 2008 (Research Report No. 2008-22)

[6] T. Blumensath Accelerated iterative hard thresholding, Signal Process., Volume 92 (2012), pp. 752-756

[7] T. Butler; P. Constantine; T. Wildey A posteriori error analysis of parameterized linear systems using spectral methods, SIAM J. Matrix Anal. Appl., Volume 33 (2012) no. 1, pp. 195-209

[8] A. Cohen; R. DeVore; C. Schwab Convergence Rates of Best N-Term Galerkin Approximations for a Class of Elliptic SPDEs, University of Zürich, Switzerland, 2009 (Research Report No. 2009-02)

[9] P.G. Constantine; D.F. Gleich; G. Iaccarino Spectral methods for parameterized matrix equations, SIAM J. Matrix Anal. Appl., Volume 31 (2010) no. 5, pp. 2681-2699

[10] O.G. Ernst; E. Ullmann Stochastic Galerkin matrices, SIAM J. Matrix Anal. Appl., Volume 31 (2010) no. 4, pp. 1848-1872

[11] M.A.T. Figueiredo; R.D. Nowak Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems, IEEE J. Sel. Top. Signal Process., Volume 1 (2008) no. 4, pp. 586-597

[12] S. Gazzola; J.G. Nagy Generalized Arnoldi–Tikhonov method for sparse reconstruction, SIAM J. Sci. Comput., Volume 36 (2014) no. 2, p. B225-B247

[13] R. Ghanem; P. Spanos Stochastic Finite Elements: A Spectral Approach, Springer Verlag, New York, 1991

[14] G.H. Golub; C.F. Van Loan Matrix Computations, Johns Hopkins University Press, Baltimore, MD, USA, 2013

[15] T. Hastie; R. Tibshirani; M. Wainwright Statistical Learning with Sparsity: The Lasso and Generalizations, Monographs on Statistics and Applied Probability, vol. 143, Chapman and Hall/CRC, 2015

[16] A. Lanza; S. Morigi; L. Reichel; F. Sgallari A generalized Krylov subspace method for lp–lq minimization, SIAM J. Sci. Comput., Volume 37 (2015) no. 5, p. S30-S50

[17] M. Loève Probability Theory, Springer, 1977

[18] A. Nouy A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations, Comput. Methods Appl. Mech. Eng., Volume 196 (2007), pp. 4521-4537

[19] A. Nouy; O.P. Le Maıître Generalized spectral decomposition for stochastic nonlinear problems, J. Comput. Phys., Volume 228 (2009), pp. 202-235

[20] B. Øksendal Stochastic Differential Equations. An Introduction with Applications, Springer-Verlag, Berlin, 1998

[21] C.E. Powell; H.C. Elman Block-diagonal preconditioning for spectral stochastic finite-element systems, IMA J. Numer. Anal., Volume 29 (2009), pp. 350-375

[22] C.E. Powell; D. Silvester Preconditioning steady-state Navier–Stokes equations with random data, SIAM J. Sci. Comput., Volume 34 (2012) no. 5, p. A2482-A2506

[23] C.E. Powell; D. Silvester; V. Simoncini An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations, University of Manchester, UK, 2015 (MIMS Report 2015.64)

[24] C. Schwab; E. Süli; R.A. Todor Sparse finite element approximation of high-dimensional transport-dominated diffusion problems, ESAIM: Math. Model. Numer. Anal., Volume 42 (2008), pp. 777-819

[25] R. Shu; J. Hu; S. Jin A stochastic Galerkin method for the Boltzmann equation with multi-dimensional random inputs using sparse wavelet bases, Numer. Math., Theory Methods Appl., Volume 10 (2017) no. 2, pp. 1-23

[26] V. Simoncini; D.B. Szyld Theory of inexact Krylov subspace methods and applications to scientific computing, SIAM J. Sci. Comput., Volume 25 (2003) no. 2, pp. 454-477

[27] V. Simoncini; D.B. Szyld Relaxed Krylov subspace approximation, Proc. Appl. Math. Mech., Volume 5 (2005), pp. 797-800

[28] B. Sousedik; R.G. Ghanem; E.T. Phipps Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods, Numer. Linear Algebra Appl., Volume 21 (2014) no. 1, pp. 136-151

[29] R.A. Todor; C. Schwab Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA J. Numer. Anal., Volume 27 (2007), pp. 232-261

[30] E. Ullmann A Kronecker product preconditioner for stochastic Galerkin finite element discretizations, SIAM J. Sci. Comput., Volume 32 (2010) no. 2, pp. 923-946

[31] D. Xiu; J.S. Hesthaven High order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., Volume 27 (2005), pp. 1118-1139

[32] D. Xiu; G.E. Karniadakis Modeling uncertainty in steady-state diffusion problems via generalized polynomial chaos, Comput. Methods Appl. Mech. Eng., Volume 191 (2003), pp. 4927-4948

Cité par Sources :

Commentaires - Politique