Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-12T17:53:10.379Z Has data issue: false hasContentIssue false
  • English
  • Français

Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs*

Published online by Cambridge University Press:  28 April 2011

Christoph Schwab and
Claude Jeffrey Gittelson
Christoph Schwab
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland E-mail: schwab@sam.math.ethz.ch
Claude Jeffrey Gittelson
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland E-mail: claude.gittelson@sam.math.ethz.ch
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of best N-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficients.

Information

Type
Research Article
Information
Acta Numerica , Volume 20 , May 2011, pp. 291 - 467
Copyright
Copyright © Cambridge University Press 2011

References

REFERENCES

Babuška, I. (1961), ‘On randomised solutions of Laplace's equation’, Časopis Pěest. Mat. 86, 269–276.CrossRefGoogle Scholar
Babuška, I. (1970/1971), ‘Error-bounds for finite element method’, Numer. Math. 16, 322–333.CrossRefGoogle Scholar
Babuška, I., Nobile, F. and Tempone, R. (2005), ‘Worst case scenario analysis for elliptic problems with uncertainty’, Numer. Math. 101, 185–219.Google Scholar
Babuška, I., Nobile, F. and Tempone, R. (2007 a), ‘Reliability of computational science’, Numer. Methods Partial Differential Equations 23, 753–784.CrossRefGoogle Scholar
Babuška, I., Nobile, F. and Tempone, R. (2007 b), ‘A stochastic collocation method for elliptic partial differential equations with random input data’, SIAM J. Numer. Anal. 45, 1005–1034.CrossRefGoogle Scholar
Barth, A. (2010), ‘A finite element method for martingale-driven stochastic partial differential equations’, Comm. Stoch. Anal. 4, 355–375.Google Scholar
Barth, A. and Lang, A. (2009), Almost sure convergence of a Galerkin–Milstein approximation for stochastic partial differential equations. In review.Google Scholar
Barth, A., Schwab, C. and Zollinger, N. (2010), Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Technical Report 2010–18, Seminar for Applied Mathematics, ETH Zürich. To appear In Numer. Math.Google Scholar
Bauer, H. (1996), Probability Theory, Vol. 23 of De Gruyter Studies in Mathematics, Walter de Gruyter. Translation by Burckel, R. B..CrossRefGoogle Scholar
Bauer, H. (2001), Measure and Integration Theory, Vol. 26 of De Gruyter Studies in Mathematics, Walter de Gruyter. Translation by Burckel, R. B..CrossRefGoogle Scholar
Beatson, R. and Greengard, L. (1997), A short course on fast multipole methods, In Wavelets, Multilevel Methods and Elliptic PDEs (Leicester 1996), Numerical Mathematics and Scientific Computation, Oxford University Press, pp. 1–37.Google Scholar
Bebendorf, M. and Hackbusch, W. (2003), ‘Existence of H-matrix approximants to the inverse FE-matrix of elliptic operators with L-coefficients’, Numer. Math. 95, 1–28.CrossRefGoogle Scholar
Bieri, M. (2009a), ‘A sparse composite collocation finite element method for elliptic sPDEs’. Technical Report 2009–8, Seminar for Applied Mathematics, ETH Zürich. To appear in SIAM J. Numer. Anal.Google Scholar
Bieri, M. (2009b), ‘Sparse tensor discretization of elliptic PDEs with random input data’. PhD thesis, ETH Zürich. ETH Dissertation no. 18598.Google Scholar
Bieri, M., Andreev, R. and Schwab, C. (2009), ‘Sparse tensor discretization of elliptic SPDEs’, SIAM J. Sci. Comput. 31, 4281–4304.CrossRefGoogle Scholar
Bogachev, V. I. (1998), Gaussian Measures, Vol. 62 of Mathematical Surveys and Monographs, AMS, Providence, RI.CrossRefGoogle Scholar
Bogachev, V. I. (2007), Measure Theory, Vols I and II, Springer.CrossRefGoogle Scholar
Braess, D. (2007), Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory, third edition, Cambridge University Press. Translation by Schumaker, L. L..CrossRefGoogle Scholar
Brenner, S. C. and Scott, L. R. (2002), The Mathematical Theory of Finite Element Methods, Vol. 15 of Texts in Applied Mathematics, second edition, Springer.CrossRefGoogle Scholar
Bungartz, H.-J. and Griebel, M. (2004), Sparse grids. In Acta Numerica, Vol. 13, Cambridge University Press, pp. 147–269.Google Scholar
Cameron, R. H. and Martin, W. T. (1947), ‘The orthogonal development of nonlinear functionals in series of Fourier–Hermite functionals’, Ann. of Math. (2) 48, 385–392.CrossRefGoogle Scholar
Canfield, E. R., Erdös, P. and Pomerance, C. (1983), ‘On a problem of Oppenheim concerning “factorisatio numerorum”’, J. Number Theory 17, 1–28.CrossRefGoogle Scholar
Charrier, J. (2010), Strong and weak error estimates for the solutions of elliptic partial differential equations with random coefficients. Technical Report 7300, INRIGoogle Scholar
Chernov, A. A. and Schwab, C. (2009), ‘Sparse p-version BEM for first kind boundary integral equations with random loading’, Appl. Numer. Math. 59, 2698–2712.CrossRefGoogle Scholar
Christensen, O. (2008), Frames and Bases: An Introductory Course, Applied and Numerical Harmonic Analysis, Birkhäuser.Google Scholar
Christensen, O. (2010), Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering, Applied and Numerical Harmonic Analysis, Birkhäuser.Google Scholar
Ciarlet, P. G. (1978), The Finite Element Method for Elliptic Problems, Vol. 4 of Studies in Mathematics and its Applications, North-Holland.CrossRefGoogle Scholar
Cohen, A. (2003), Numerical Analysis of Wavelet Methods, Vol. 32 of Studies in Mathematics and its Applications, North-Holland.Google Scholar
Cohen, A., DeVore, R. A. and Schwab, C. (2010), Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. J. Found. Comput. Math. 10, 615–646.CrossRefGoogle Scholar
Cohen, A., DeVore, R. A. and Schwab, C. (2011), Analytic regularity and polynomial approximation of parametric stochastic elliptic PDEs. Anal. Appl. 9, 1–37.CrossRefGoogle Scholar
Da Prato, G. (2006), An Introduction to Infinite-Dimensional Analysis, revised and extended edition, Universitext, Springer.CrossRefGoogle Scholar
Da Prato, G. and Zabczyk, J. (1992), Stochastic Equations in Infinite Dimensions, Vol. 44 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.CrossRefGoogle Scholar
Dahmen, W. (1997), Wavelet and multiscale methods for operator equations. In Acta Numerica, Vol. 6, Cambridge University Press, pp. 55–228.Google Scholar
Dahmen, W., Harbrecht, H. and Schneider, R. (2006), ‘Compression techniques for boundary integral equations: Asymptotically optimal complexity estimates’, SIAM J. Numer. Anal. 43, 2251–2271.CrossRefGoogle Scholar
Dalang, R., Khoshnevisan, D., Mueller, C., Nualart, D. and Xiao, Y. (2009), A Minicourse on Stochastic Partial Differential Equations (Salt Lake City 2006; Khoshnevisan, D. and Rassoul-Agha, F., eds), Vol. 1962 of Lecture Notes in Mathematics, Springer.CrossRefGoogle Scholar
Dettinger, M. and Wilson, J. L. (1981), ‘First order analysis of uncertainty in numerical models of groundwater flow 1: Mathematical development’, Water Res. Res. 17, 149–161.CrossRefGoogle Scholar
DeVore, R. A. (1998), Nonlinear approximation. In Acta Numerica, Vol. 7, Cambridge University Press, pp. 51–150.Google Scholar
Donovan, G. C., Geronimo, J. S. and Hardin, D. P. (1996), ‘Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets’, SIAM J. Math. Anal. 27, 1791–1815.CrossRefGoogle Scholar
Eisenstat, S. C., Elman, H. C. and Schultz, M. H. (1983), ‘Variational iterative methods for nonsymmetric systems of linear equations’, SIAM J. Numer. Anal. 20, 345–357.CrossRefGoogle Scholar
Ernst, O. G., Mugler, A., Starkloff, H.-J. and Ullmann, E. (2010), On the convergence of generalized polynomial chaos expansions. Technical Report 60, DFG Schwerpunktprogramm 1324.Google Scholar
Fishman, G. S. (1996), Monte Carlo: Concepts, Algorithms, and Applications, Springer Series in Operations Research, Springer.CrossRefGoogle Scholar
Galvis, J. and Sarkis, M. (2009), ‘Approximating infinity-dimensional stochastic Darcy's equations without uniform ellipticity’, SIAM J. Numer. Anal. 47, 3624–3651.CrossRefGoogle Scholar
Gautschi, W. (2004), Orthogonal Polynomials: Computation and Approximation, Numerical Mathematics and Scientific Computation, Oxford Science Publications, Oxford University Press.CrossRefGoogle Scholar
Geissert, M., Kovacs, M. and Larsson, S. (2009), ‘Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise’, BIT 49, 343–356.CrossRefGoogle Scholar
Ghanem, R. G. and Spanos, P. D. (2007), Stochastic Finite Elements: A Spectral Approach, second edition, DoverGoogle Scholar
Gittelson, C. J. (2010a), ‘Stochastic Galerkin discretization of the log-normal iso-tropic diffusion problem’, Math. Models Methods Appl. Sci. 20, 237–263.CrossRefGoogle Scholar
Gittelson, C. J. (2010 b), Representation of Gaussian fields in series with independent coefficients. Technical Report 2010–15, Seminar for Applied Mathematics, ETH Zürich. Submitted.Google Scholar
Gittelson, C. J. (2011a) Adaptive Galerkin methods for parametric and stochastic operator equations. ETH Dissertation No. 19533, ETH Zürich.Google Scholar
Gittelson, C. J. (2011b) Stochastic Galerkin approximation of operator equations with infinite dimensional noise. Technical Report 2011–10, Seminar for Applied Mathematics, ETH Zürich.Google Scholar
Gittelson, C. J. (2011c) An adaptive stochastic Galerkin method. Technical Report 2011–11, Seminar for Applied Mathematics, ETH Zürich.Google Scholar
Gittelson, C. J. (2011d) Adaptive stochastic Galerkin methods: Beyond the elliptic case. Technical Report 2011–12, Seminar for Applied Mathematics, ETH Zürich.Google Scholar
Graham, I. G., Kuo, F. Y., Nuyens, D., Scheichl, R. and Sloan, I. H. (2010), Quasi-Monte Carlo methods for computing flow in random porous media. Technical Report 4/10, Bath Institute for Complex Systems.Google Scholar
Griebel, M., Oswald, P. and Schiekofer, T. (1999), ‘Sparse grids for boundary integral equations’, Numer. Math. 83, 279–312.CrossRefGoogle Scholar
Grothendieck, A. (1955), ‘Produits tensoriels topologiques et espaces nucléaires’, Mem. Amer. Math. Soc. 16, 140.Google Scholar
Harbrecht, H. (2001), Wavelet Galerkin schemes for the boundary element method in three dimensions. PhD thesis, Technische Universität Chemnitz.CrossRefGoogle Scholar
Harbrecht, H., Schneider, R. and Schwab, C. (2008), ‘Sparse second moment analysis for elliptic problems in stochastic domains’, Numer. Math. 109, 385–414.CrossRefGoogle Scholar
Hervé, M. (1989), Analyticity in Infinite-Dimensional Spaces, Vol. 10 of De Gruyter Studies in Mathematics, Walter de Gruyter.CrossRefGoogle Scholar
Hildebrandt, S. and Wienholtz, E. (1964), ‘Constructive proofs of representation theorems in separable Hilbert space’, Comm. Pure Appl. Math. 17, 369–373.CrossRefGoogle Scholar
Hoang, V. H. and Schwab, C. (2004/2005), ‘High-dimensional finite elements for elliptic problems with multiple scales’, Multiscale Model. Simul. 3, 168–194.CrossRefGoogle Scholar
Hoang, V.-H. and Schwab, C. (2010a), Analytic regularity and gpc approximation for parametric and random 2nd order hyperbolic PDEs. Technical Report 2010–19, Seminar for Applied Mathematics, ETH Zürich. To appear In Anal. Appl.Google Scholar
Hoang, V.-H. and Schwab, C. (2010b), Sparse tensor Galerkin discretization for parametric and random parabolic PDEs I: Analytic regularity and gpc-approximation. Technical Report 2010–11, Seminar for Applied Mathematics, ETH Zürich. Submitted.Google Scholar
Holden, H., Oksendal, B., Uboe, J. and Zhang, T. (1996), Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, Probability and its Applications, Birkhäuser.CrossRefGoogle Scholar
Hsiao, G. C. and Wendland, W. L. (1977), ‘A finite element method for some integral equations of the first kind’, J. Math. Anal. Appl. 58, 449–481.CrossRefGoogle Scholar
Janson, S. (1997), Gaussian Hilbert spaces, Vol. 129 of Cambridge Tracts in Mathematics, Cambridge University Press.CrossRefGoogle Scholar
Kadison, R. V. and Ringrose, J. R. (1997), Fundamentals of the Theory of Operator Algebras I: Elementary Theory, Vol. 15 of Graduate Studies in Mathematics, AMS, Providence, RI.Google Scholar
Kakutani, S. (1948), ‘On equivalence of infinite product measures’, Ann. of Math. (2) 49, 214–224.CrossRefGoogle Scholar
Kalton, N. (2003), Quasi-Banach spaces. In Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, pp. 1099–1130.CrossRefGoogle Scholar
Khoromskij, B. N. and Schwab, C. (2011), ‘Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs’, SIAM J. Sci. Comput. 33, 364–385.CrossRefGoogle Scholar
Kovács, M., Larsson, S. and Lindgren, F. (2010 a), ‘Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise’, Numer. Algorithms 53, 309–320.CrossRefGoogle Scholar
Kováacs, M., Larsson, S. and Saedpanah, F. (2010 b), ‘Finite element approximation of the linear stochastic wave equation with additive noise’, SIAM J. Numer. Anal. 48, 408–427.CrossRefGoogle Scholar
Kressner, D. and Tobler, C. (2010), Low-rank tensor Krylov subspace methods for parametrized linear systems. Technical Report 2010–16, Seminar for Applied Mathematics, ETH Zürich. Submitted.Google Scholar
Ledoux, M. and Talagrand, M. (1991), Probability in Banach Spaces: Isoperime-try and Processes, Vol. 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer.CrossRefGoogle Scholar
Light, W. A. and Cheney, E. W. (1985), Approximation Theory in Tensor Product Spaces, Vol. 1169 of Lecture Notes in Mathematics, Springer.CrossRefGoogle Scholar
Lototsky, S. and Rozovskii, B. (2006), Stochastic differential equations: A Wiener chaos approach. In From Stochastic Calculus to Mathematical Finance, Springer, pp. 433–506.CrossRefGoogle Scholar
McLean, W. (2000), Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press.Google Scholar
Matthies, H. G. and Keese, A. (2005), ‘Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations’, Comput. Methods Appl. Mech. Engrg 194, 1295–1331.CrossRefGoogle Scholar
Mishra, S. and Schwab, C. (2010), Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random intitial data. Technical Report 2010–24, Seminar for Applied Mathematics, ETH Zürich. Submitted.Google Scholar
Nédélec, J.-C. and Planchard, J. (1973), ‘Une méthode variationnelle d'éléments finis pour la résolution numérique d'un problème extérieur dans R 3’, Rev. Française Autom. Inform. Recherche Opérationnelle Sér. Rouge 7, 105–129.Google Scholar
Nguyen, H. and Stevenson, R. (2009), ‘Finite element wavelets with improved quantitative properties’, J. Comput. Appl. Math. 230, 706–727.CrossRefGoogle Scholar
Nobile, F. and Tempone, R. (2009), ‘Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients’, Internat. J. Numer. Methods Engrg 80, 979–1006.CrossRefGoogle Scholar
Nobile, F., Tempone, R. and Webster, C. G. (2008 a), ‘An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data’, SIAM J. Numer. Anal. 46, 2411–2442.CrossRefGoogle Scholar
Nobile, F., Tempone, R. and Webster, C. G. (2008 b), ‘A sparse grid stochastic collocation method for partial differential equations with random input data’, SIAM J. Numer. Anal. 46, 2309–2345.CrossRefGoogle Scholar
Oden, J. T., Babuška, I., Nobile, F., Feng, Y. and Tempone, R. (2005), ‘Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty’, Comput. Methods Appl. Mech. Engrg 194, 195–204.CrossRefGoogle Scholar
Oppenheim, A. (1927), ‘On an arithmetic function’, J. London Math. Soc.> s1–2, 123–130.CrossRefGoogle Scholar
Peszat, S. and Zabczyk, J. (2007), Stochastic Partial Differential Equations with L évy Noise: An Evolution Equation Approach, Vol. 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.CrossRefGoogle Scholar
von Petersdorff, T. and Schwab, C. (1996), ‘Wavelet approximations for first kind boundary integral equations on polygons’, Numer. Math. 74, 479–516.CrossRefGoogle Scholar
von Petersdorff, T. and Schwab, C. (2004), ‘Numerical solution of parabolic equations in high dimensions’, M2AN Math. Model. Numer. Anal. 38, 93–127.CrossRefGoogle Scholar
von Petersdorff, T. and Schwab, C. (2006), ‘Sparse finite element methods for operator equations with stochastic data’, Appl. Math. 51, 145–180.CrossRefGoogle Scholar
Prévôt, C. and Röckner, M. (2007), A Concise Course on Stochastic Partial Differential Equations, Vol. 1905 of Lecture Notes in Mathematics, Springer.Google Scholar
Protter, P. E. (2005), Stochastic Integration and Differential Equations, Vol. 21 of Stochastic Modelling and Applied Probability, second edition, version 2.1, Springer.CrossRefGoogle Scholar
Reed, M. and Simon, B. (1980), Functional Analysis, Vol. 1 of Methods of Modern Mathematical Physics, second edition, Academic Press (Harcourt Brace Jovanovich).Google Scholar
Ryan, R. A. (2002), Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, Springer.CrossRefGoogle Scholar
Sauter, S. and Schwab, C. (2010), Boundary Element Methods, Springer.CrossRefGoogle Scholar
Schatten, R. (1943), ‘On the direct product of Banach spaces’, Trans. Amer. Math. Soc. 53, 195–217.CrossRefGoogle Scholar
Schmidlin, G., Lage, C. and Schwab, C. (2003), ‘Rapid solution of first kind boundary integral equations In R 3’, Engrg Anal. Boundary Elem. (special issue on solving large scale problems using BEM) 27, 469–490.CrossRefGoogle Scholar
Schneider, R. (1998), Multiskalen- und Wavelet-Matrixkompression, Advances in Numerical Mathematics, Teubner. Analysisbasierte Methoden zur effizienten Lösung groβer vollbesetzter Gleichungssysteme. [Analysis-based methods for the efficient solution of large nonsparse systems of equations].CrossRefGoogle Scholar
Schoutens, W. (2000), Stochastic Processes and Orthogonal Polynomials, Vol. 146 of Lecture Notes in Statistics, Springer.CrossRefGoogle Scholar
Schwab, C. (2002), High dimensional finite elements for elliptic problems with multiple scales and stochastic data, In Proc. International Congress of Mathematicians, Vol. III (Beijing 2002), Higher Education Press, Beijing, pp. 727–734.Google Scholar
Schwab, C. and Stevenson, R. (2008), ‘Adaptive wavelet algorithms for elliptic PDEs on product domains’, Math. Comp. 77, 71–92.CrossRefGoogle Scholar
Schwab, C. and Stevenson, R. (2009), ‘Space–time adaptive wavelet methods for parabolic evolution problems’, Math. Comp. 78, 1293–1318.CrossRefGoogle Scholar
Schwab, C. and Stuart, A. M. (2011), Sparse deterministic approximation of Bayesian inverse problems. Technical Report 2011–16, Seminar for Applied Mathematics, ETH Zürich.Google Scholar
Schwab, C. and Todor, R. A. (2003 a), ‘Sparse finite elements for elliptic problems with stochastic loading’, Numer. Math. 95, 707–734.CrossRefGoogle Scholar
Schwab, C. and Todor, R. A. (2003 b), ‘Sparse finite elements for stochastic elliptic problems: Higher order moments’, Computing 71, 43–63.CrossRefGoogle Scholar
Smolyak, S. (1963), ‘Quadrature and interpolation formulas for tensor products of certain classes of functions’, Sov. Math. Dokl. 4, 240–243.Google Scholar
Strassen, V. (1964), ‘An invariance principle for the law of the iterated logarithm’, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3, 211–226 (1964).CrossRefGoogle Scholar
Szegő, G. (1975), Orthogonal Polynomials, fourth edition, Colloquium Publications, Vol. XXIII, AMS, Providence, RI.Google Scholar
Szekeres, G. and Turán, P. (1933), ‘Uber das zweite Hauptproblem der “factorisatio numerorum”’, Acta Litt. Sci. Szeged 6, 143–154.Google Scholar
Temlyakov, V. N. (1993), Approximation of Periodic Functions, Computational Mathematics and Analysis Series, Nova Science Publishers, Commack, NY.Google Scholar
Todor, R.-A. (2009), ‘A new approach to energy-based sparse finite-element spaces’, IMA J. Numer. Anal. 29, 72–85.CrossRefGoogle Scholar
Todor, R. A. and Schwab, C. (2007), ‘Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients’, IMA J. Numer. Anal. 27, 232–261.CrossRefGoogle Scholar
Vakhania, N. N., Tarieladze, V. I. and Chobanyan, S. A. (1987), Probability Distributions on Banach Spaces, Vol. 14 of Mathematics and its Applications (Soviet Series), Reidel. Translation by Woyczynski, W. A..Google Scholar
Wasilkowskiand, G. W., Woźniakowski, H. (1995), ‘Explicit cost bounds of algorithms for multivariate tensor product problems’, J. Complexity 11, 1–56.CrossRefGoogle Scholar
Wiener, N. (1938), ‘The homogeneous chaos’, Amer. J. Math. 60, 897–936.CrossRefGoogle Scholar
Xiu, D. (2009), ‘Fast numerical methods for stochastic computations: A review’, Commun. Comput. Phys. 5, 242–272.Google Scholar
Xiu, D. and Hesthaven, J. S. (2005), ‘High-order collocation methods for differential equations with random inputs’, SIAM J. Sci. Comput. 27, 1118–1139.CrossRefGoogle Scholar
Xiu, D. and Karniadakis, G. E. (2002 a), ‘Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos’, Comput. Methods Appl. Mech. Engrg 191, 4927–4948.CrossRefGoogle Scholar
Xiu, D. and Karniadakis, G. E. (2002 b), ‘The Wiener–Askey polynomial chaos for stochastic differential equations’, SIAM J. Sci. Comput. 24, 619–644.CrossRefGoogle Scholar