Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs

Chkifa, Abdellah; Cohen, Albert; DeVore, Ronald; Schwab, Christoph

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 253-280.

The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated in the Hilbert space V = H₀¹(D) by multivariate sparse polynomials in the parameter vector y with a controlled number N of terms. The convergence rate in terms of N does not depend on the number of parameters in V, which may be arbitrarily large or countably infinite, thereby breaking the curse of dimensionality. However, these approximation results do not describe the concrete construction of these polynomial expansions, and should therefore rather be viewed as benchmark for the convergence analysis of numerical methods. The present paper presents an adaptive numerical algorithm for constructing a sequence of sparse polynomials that is proved to converge toward the solution with the optimal benchmark rate. Numerical experiments are presented in large parameter dimension, which confirm the effectiveness of the adaptive approach.

Classification : 65N35, 65L10, 35J25
Keywords: parametric and stochastic PDE's, sparse polynomial approximation, high dimensional problems, adaptive algorithms

[1] I. Babuska, R. Tempone and G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800−825. | MR | Zbl

[2] I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45 (2007) 1005−1034. | MR | Zbl

[3] C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579−608. | MR | Zbl

[4] P. Binev, W. Dahmen, and R. Devore, Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219−268. | MR | Zbl

[5] A. Buffa, Y. Maday, A.T. Patera, C. Prudhomme and G. Turinici, A priori convergence of the greedy algorithm for the parameterized reduced basis. ESAIM : M2AN 3 (2012) 595-603. | Numdam | MR | Zbl

[6] A. Cohen, Numerical analysis of wavelet methods. Elsevier, Amsterdam (2003). | MR | Zbl

[7] A. Cohen, W. Dahmen and R. Devore, Adaptive wavelet methods for elliptic operator equations - Convergence rates. Math. Comput. 70 (2000) 27−75. | MR | Zbl

[8] A. Cohen, W. Dahmen and R. Devore, Adaptive wavelet methods for operator equations - Beyond the elliptic case. J. FoCM 2 (2002) 203−245. | MR | Zbl

[9] A. Cohen, R. Devore and C. Schwab, Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10 (2010) 615-646. | MR | Zbl

[10] A. Cohen, R. Devore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic PDE's. Anal. Appl. 9 (2011) 11-47. | MR | Zbl

[11] R. Devore, Nonlinear approximation. Acta Numer. 7 (1998) 51-150. | MR | Zbl

[12] W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR | Zbl

[13] Ph. Frauenfelder, Ch. Schwab and R.A. Todor, Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng. 194 (2005) 205-228. | MR | Zbl

[14] T. Gantumur, H. Harbrecht and R. Stevenson, An optimal adaptive wavelet method without coarsening of the iterands. Math. Comput. 76 (2007) 615-629. | MR | Zbl

[15] R. Ghanem and P. Spanos, Spectral techniques for stochastic finite elements. Arch. Comput. Methods Eng. 4 (1997) 63-100. | MR | Zbl

[16] P. Grisvard, Elliptic problems on non-smooth domains. Pitman (1983). | Zbl

[17] V.H. Hoang and Ch. Schwab, Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs I : Analytic regularity and gpc-approximation. Report 2010-11, Seminar for Applied Mathematics, ETH Zürich (in review). | Zbl

[18] V.H. Hoang and Ch. Schwab, Analytic regularity and gpc approximation for parametric and random 2nd order hyperbolic PDEs. Report 2010-19, Seminar for Applied Mathematics, ETH Zürich (to appear in Anal. Appl. (2011)).

[19] M. Kleiber and T.D. Hien, The stochastic finite element methods. John Wiley & Sons, Chichester (1992). | MR | Zbl

[20] R. Milani, A. Quarteroni and G. Rozza, Reduced basis methods in linear elasticity with many parameters. Comput. Methods Appl. Mech. Eng. 197 (2008) 4812-4829. | MR | Zbl

[21] P. Morin, R.H. Nochetto and K.G. Siebert, Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000) 466-488. | MR | Zbl

[22] F. Nobile, R. Tempone and C.G. Webster, A sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2309-2345. | MR | Zbl

[23] F. Nobile, R. Tempone and C.G. Webster, An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2411-2442. | MR | Zbl

[24] Ch. Schwab and A.M. Stuart Sparse deterministic approximation of Bayesian inverse problems. Report 2011-16, Seminar for Applied Mathematics, ETH Zürich (to appear in Inverse Probl.). | MR | Zbl

[25] Ch. Schwab and R.A. Todor, Karhúnen-Loève, Approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217 (2000) 100-122. | MR | Zbl

[26] R. Stevenson, Optimality of a standard adaptive finite element method. Found. Comput. Math. 7 (2007) 245-269. | MR | Zbl