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Approximation of high-dimensional parametric PDEs *

  • Albert Cohen (a1) and Ronald DeVore (a2)

Abstract

Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even infinite. Thus, the development of numerical methods for these parametric problems is faced with the possible curse of dimensionality. This article is directed at (i) identifying and understanding which properties of parametric equations allow one to avoid this curse and (ii) developing and analysing effective numerical methods which fully exploit these properties and, in turn, are immune to the growth in dimensionality.

Part I of this article studies the smoothness and approximability of the solution map, that is, the map $a\mapsto u(a)$ , where $a$ is the parameter value and $u(a)$ is the corresponding solution to the PDE. It is shown that for many relevant parametric PDEs, the parametric smoothness of this map is typically holomorphic and also highly anisotropic, in that the relevant parameters are of widely varying importance in describing the solution. These two properties are then exploited to establish convergence rates of $n$ -term approximations to the solution map, for which each term is separable in the parametric and physical variables. These results reveal that, at least on a theoretical level, the solution map can be well approximated by discretizations of moderate complexity, thereby showing how the curse of dimensionality is broken. This theoretical analysis is carried out through concepts of approximation theory such as best $n$ -term approximation, sparsity, and $n$ -widths. These notions determine a priori the best possible performance of numerical methods and thus serve as a benchmark for concrete algorithms.

Part II of this article turns to the development of numerical algorithms based on the theoretically established sparse separable approximations. The numerical methods studied fall into two general categories. The first uses polynomial expansions in terms of the parameters to approximate the solution map. The second one searches for suitable low-dimensional spaces for simultaneously approximating all members of the parametric family. The numerical implementation of these approaches is carried out through adaptive and greedy algorithms. An a priori analysis of the performance of these algorithms establishes how well they meet the theoretical benchmarks.

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References

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Andreev, R., Bieri, M. and Schwab, C. (2006), ‘Sparse tensor discretization of elliptic SPDEs’, SIAM J. Sci. Comput. 31, 42814304.
Babuška, I., Nobile, F. and Tempone, R. (2007), ‘A stochastic collocation method for elliptic partial differential equations with random input data’, SIAM J. Numer. Anal. 45, 10051034.
Babuška, I., Tempone, R. and Zouraris, G. E. (2004), ‘Galerkin finite element approximations of stochastic elliptic partial differential equations’, SIAM J. Numer. Anal. 42, 800825.
Beck, J., Nobile, F., Tamellini, L. and Tempone, R. (2012), ‘On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods’, Math. Models Methods Appl. Sci. 22, 133.
Beck, J., Nobile, F., Tamellini, L. and Tempone, R. (2014), ‘Convergence of quasi-optimal stochastic Galerkin methods for a class of PDEs with random coefficients’, Comput. Math. Appl. 67, 732751.
Bergmann, M. and Cordier, L. (2003), Proper Orthogonal Decomposition: An Overview; Post Processing of Experimental and Numerical Data, Lecture Series 2003/2004, von Karman Institut for Fluid Dynamics.
Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G. and Wojtaszczyk, P. (2011), ‘Convergence rates for greedy algorithms in reduced basis methods’, SIAM J. Math. Anal. 43, 14571472.
Binev, P., Dahmen, W. and DeVore, R. (2004), ‘Adaptive finite element methods with convergence rates’, Numer. Math. 97, 219268.
de Boor, C. and Ron, A. (1992), ‘Computational aspects of polynomial interpolation in several variables’, Math. Comp. 58, 705727.
Brenner, S. and Scott, L. R. (2008), The Mathematical Theory of Finite Elements, second edition, Springer.
Buffa, A., Maday, Y., Patera, A. T., Prud’homme, C. and Turinici, G. (2012), ‘ A priori convergence of the greedy algorithm for the parameterized reduced basis’, Math. Model. Numer. Anal. 46, 595603.
Bungartz, H.-J. and Griebel, M. (2004), Sparse grids. In Acta Numerica, Vol. 13, Cambridge University Press, pp. 1123.
Calvi, J. P. and Phung, V. M. (2011), ‘On the Lebesgue constant of Leja sequences for the unit disk and its applications to multivariate interpolation’, J. Approx. Theory 163, 608622.
Calvi, J. P. and Phung, V. M. (2012), ‘Lagrange interpolation at real projections of Leja sequences for the unit disk’, Proc. Amer. Math. Soc. 140, 42714284.
Canfield, E. R., Erdős, P. and Pomerance, C. (1983), ‘On a problem of Oppenheim concerning “factorisatio numerorum”’, J. Number Theory 17, 128.
Chkifa, A. (2013), ‘On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection’, J. Approx. Theory 166, 176200.
Chkifa, A. (2015), New bounds on the Lebesgue constant of Leja sequences on the unit disc and their projections $\mathfrak{R}$ -Leja sequences. arXiv:1503.01731
Chkifa, A., Cohen, A., DeVore, R. and Schwab, C. (2013), ‘Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs’, Math. Model. Numer. Anal. 47, 253280.
Chkifa, A., Cohen, A., Migliorati, G., Nobile, F. and Tempone, R. (2015a), ‘Discrete least squares polynomial approximation with random evaluations: Application to parametric and stochastic PDEs’, Math. Model. Numer. Anal. doi:10.1051/m2an/2014050
Chkifa, A., Cohen, A. and Schwab, C. (2014), ‘High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs’, Found. Comput. Math. 14, 601633.
Chkifa, A., Cohen, A. and Schwab, C. (2015b), ‘Breaking the curse of dimensionality in parametric PDEs’, J. Math. Pures Appliquées 102, 400428.
Ciarlet, P. G. (1978), The Finite Element Method for Elliptic Problems, Elsevier.
Cohen, A. and DeVore, R. (2015), Kolmogorov widths under holomorphic mappings. IMA J. Numer. Anal. doi:10.1093/imanum/dru066
Cohen, A., Dahmen, W. and DeVore, R. (2000), ‘Adaptive wavelet methods for elliptic operator equations: Convergence rates’, Math. Comp. 70, 2775.
Cohen, A., Dahmen, W. and DeVore, R. (2002), ‘Adaptive wavelet methods for operator equations: Beyond the elliptic case’, Found. Comput. Math. 2, 203245.
Cohen, A., DeVore, R. and Schwab, C. (2010), ‘Convergence rates of best $N$ -term Galerkin approximations for a class of elliptic sPDEs’, Found. Comput. Math. 10, 615646.
Cohen, A., DeVore, R. and Schwab, C. (2011), ‘Analytic regularity and polynomial approximation of parametric and stochastic PDEs’, Anal. Appl. 9, 1147.
Constantine, P., Eldred, M. and Phipps, E. (2012), ‘Sparse pseudospectral approximation method’, Comput. Methods Appl. Mech. Engrg 229–232, 112.
Dautray, R. and Lions, J.-L. (1992), Mathematical Analysis and Numerical Methods for Science and Technology, Springer.
Davis, P. J. (1963), Interpolation and Approximation, Blaisdell.
DeVore, R. (1998), Nonlinear approximation. In Acta Numerica, Vol. 7, Cambridge University Press, pp. 51150.
DeVore, R., Petrova, G. and Wojtaszczyk, P. (2013), ‘Greedy algorithms for reduced bases in Banach spaces’, Constr. Approx. 37, 455466.
Dieudonné, J. (1969), Treatise on Analysis, Vol. I, Academic Press.
Doostan, A. and Iaccarino, G. (2009), ‘A least-squares approximation of partial differential equations with high-dimensional random inputs’, J. Comput. Phys. 228, 43324345.
Doostan, A. and Owadi, H. (2011), ‘A non-adapted sparse approximation of PDEs with stochastic inputs’, J. Comput. Phys. 230, 30153034.
Dörfler, W. (1996), ‘A convergent adaptive algorithm for Poisson’s equation’, SIAM J. Numer. Anal. 33, 11061124.
Frauenfelder, P., Schwab, C. and Todor, R. A. (2005), ‘Finite elements for elliptic problems with stochastic coefficients’, Comput. Methods Appl. Mech. Engrg 194, 205228.
Gantumur, T., Harbrecht, H. and Stevenson, R. (2007), ‘An optimal adaptive wavelet method without coarsening of the iterands’, Math. Comp. 76, 615629.
Gerstner, T. and Griebel, M. (2003), ‘Dimension-adaptive tensor-product quadrature’, Computing 71, 6587.
Ghanem, R. and Spanos, P. (1991), Stochastic Finite Elements: A Spectral Approach, Springer.
Ghanem, R. and Spanos, P. (1997), ‘Spectral techniques for stochastic finite elements’, Arch. Comput. Methods Engrg 4, 63100.
Gittelson, G. J. (2010), ‘Stochastic Galerkin discretization of the log-normal isotropic diffusion problem’, Math. Models Methods Appl. Sci. 20, 237263..
Gittelson, C. J. (2013), ‘An adaptive stochastic Galerkin method’, Math. Comp. 82, 15151541.
Gittelson, C. J. and Schwab, C. (2011), Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. In Acta Numerica, Vol. 20, Cambridge University Press, pp. 291467.
Graham, I., Kuo, F., Nichols, J., Scheichl, R., Schwab, C. and Sloan, I. (2015), ‘Quasi-Monte Carlo finite element methods for elliptic PDEs with log-normal random coefficient’, Numer. Math. doi:10.1007/s00211-014-0689-y
Gunzburger, M., Webster, C. and Zhang, G. (2014), Stochastic finite element methods for partial differential equations with random input data. In Acta Numerica, Vol. 23, Cambridge University Press, pp. 521650.
Hansen, M. and Schwab, C. (2013a), ‘Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs’, Math. Nachr. 286, 832860.
Hansen, M. and Schwab, C. (2013b), ‘Sparse adaptive approximation of high-dimensional parametric initial value problems’, Vietnam J. Math. 41, 181215.
Hervé, M. (1989), Analyticity in Infinite Dimensional Spaces, De Gruyter.
Hoang, V. H. and Schwab, C. (2014), ‘ $n$ -term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs’, Math. Models Methods Appl. Sci. 24, 797826.
Jerison, D. and Kenig, C. E. (1995), ‘The inhomogeneous Dirichlet problem in Lipschitz domains’, J. Funct. Anal. 130, 161219.
Kahlbacher, M. and Volkwein, S. (2007), ‘Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems’, Discuss. Math. Differ. Incl. Control Optim. 27, 95117.
Karniadakis, G. E. and Xiu, D. B. (2002a), ‘The Wiener–Askey polynomial chaos for stochastic differential equations’, SIAM J. Sci. Comput. 24, 619644.
Karniadakis, G. E. and Xiu, D. (2002b), ‘Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos’, Comput. Methods Appl. Mech. Engrg 191, 49274948.
Kleiber, M. and Hien, T. D. (1992), The Stochastic Finite Element Methods, Wiley.
Knio, O. and Le Maitre, O. (2010), Spectral Methods for Uncertainty Quantication: With Applications to Computational Fluid Dynamics, Springer.
Kolmogorov, A. (1936), ‘Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse’, Ann. of Math. 37, 107110.
Kunoth, A. and Schwab, C. (2013), ‘Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs’, SIAM J. Control Optim. 51, 24422471.
Kuntzman, J. (1959), Méthodes Numériques: Interpolation, Dérivées, Dunod.
Kuo, F. Y., Schwab, C. and Sloan, I. H. (2012), ‘Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients’, SIAM J. Numer. Anal. 50, 33513374.
Kuo, F. Y., Schwab, C. and Sloan, I. H. (2015), Multi-level quasi-Monte Carlo finite element methods for a class of elliptic PDEs with random coefficients’, Found. Comput. Math. doi:10.1007/s10208-014-9237-5
Kuo, F. Y., Sloan, I. H., Wasilkowski, G. W. and Wozniakowski, H. (2010), ‘Liberating the dimension’, J. Complexity 26, 422454.
Lorentz, G. G. and Lorentz, R. (1986), ‘Solvability problems of bivariate interpolation I’, Constr. Approx. 2, 153169.
Lorentz, G. G., von Golitschek, M. and Makovoz, Y. (1996), Constructive Approximation: Advanced Problems, Springer.
Luca, F., Mukhopadhyay, A. and Srinivas, K. (2010), ‘On the Oppenheim’s “factorisatio numerorum” function’, Acta Arithmetica 142, 4150.
Machiels, L., Maday, Y, Oliveira, I. B., Patera, A. T. and Rovas, D. V. (2000), ‘Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems’, C. R. Acad. Sci. Paris, Sér. I 331, 153158.
Maday, Y., Patera, A. T. and Turinici, G. (2002a), ‘ A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations’, J. Sci. Comput. 17, 437446.
Maday, Y., Patera, A. T. and Turinici, G. (2002b), ‘Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations’, C. R. Acad. Sci. Paris, Sér. I 335, 289294.
Migliorati, G., Nobile, F., Tempone, R. and Von Schwerin, E. (2013), ‘Approximation of quantities of interest in stochastic PDEs by the random discrete $L^{2}$ projection on polynomial spaces’, SIAM J. Sci. Comput. 35, 14401460.
Morin, P., Nochetto, R. H. and Siebert, K. G. (2000), ‘Data oscillation and convergence of adaptive FEM’, SIAM J. Numer. Anal. 38, 466488.
Nobile, F., Tempone, R. and Webster, C. G. (2008a), ‘A sparse grid stochastic collocation method for elliptic partial differential equations with random input data’, SIAM J. Numer. Anal. 46, 23092345.
Nobile, F., Tempone, R. and Webster, C. G. (2008b), ‘An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data’, SIAM J. Numer. Anal. 46, 24112442.
Noor, A. K. and Peters, J. M. (1980), ‘Reduced basis technique for nonlinear analysis of structures’, AIAA J. 18, 455462.
Pinkus, A. (1985), n-Widths in Approximation Theory, Springer.
Pisier, G. (1989), The Volume of Convex Bodies and Banach Space Geometry, Cambridge University Press.
Rozza, G., Huynh, D. B. P. and Patera, A. T. (2008), ‘Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics’, Arch. Comput. Methods Engrg 15, 229275.
Runst, T. and Sickel, W. (1996), Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter.
Schillings, C. and Schwab, C. (2013), ‘Sparse, adaptive Smolyak quadratures for Bayesian inverse problems’, Inverse Problems 29, 065011.
Schillings, C. and Schwab, C. (2014), ‘Sparsity in Bayesian inversion of parametric operator equations’, Inverse Problems 30, 065007.
Sen, S. (2008), ‘Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems’, Numerical Heat Transfer B 54, 369389.
Schwab, C. and Stevenson, R. (2009), ‘Space–time adaptive wavelet methods for parabolic evolution equations’, Math. Comp. 78, 12931318.
Schwab, C. and Stuart, A. M. (2012), ‘Sparse deterministic approximation of Bayesian inverse problems’, Inverse Problems 28, 045003.
Schwab, C. and Todor, R. (2003), ‘Sparse finite elements for elliptic problems with stochastic loading’, Numer. Math. 95, 707734.
Schwab, C. and Todor, R. (2007), ‘Convergence rates of sparse chaos approximations of elliptic problems with stochastic coefficients’, IMA J. Numer. Anal. 44, 232261.
Smolyak, S. (1963), ‘Quadrature and interpolation formulas for tensor products of certain classes of functions’, Doklady Akademii Nauk SSSR 4, 240243.
Stevenson, R. (2007), ‘Optimality of a standard adaptive finite element method’, Found. Comput. Math. 7, 245269.
Stuart, A. M. (2010), Inverse problems: A Bayesian perspective. In Acta Numerica, Vol. 19, Cambridge University Press, pp. 451559.
Veroy, K., Prud’homme, C., Rovas, D. V. and Patera, A. T. (2003), A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In Proc. 16th AIAA Computational Fluid Dynamics Conference, paper 2003-3847.
Xiu, D. (2007), ‘Efficient collocational approach for parametric uncertainty analysis’, Commun. Comput. Phys. 2, 293309.
Xiu, D. (2010), Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press.

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