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Keywords:
wavelet compression of operators; random data; Monte-Carlo method; wavelet finite element method
wavelet compression of operators; random data; Monte-Carlo method; wavelet finite element method
Summary:
Let $A\: V\rightarrow V^{\prime }$ be a strongly elliptic operator on a $d$-dimensional manifold $D$ (polyhedra or boundaries of polyhedra are also allowed). An operator equation $Au=f$ with stochastic data $f$ is considered. The goal of the computation is the mean field and higher moments $\mathcal M^1 u\in V$, $\mathcal M^2u\in V\otimes V$, $\ldots $, $\mathcal M^k u \in V\otimes \cdots \otimes V$ of the solution. We discretize the mean field problem using a FEM with hierarchical basis and $N$ degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment $\mathcal M^k u$ for $k\ge 1$. The key tool in both algorithms is a “sparse tensor product” space for the approximation of $\mathcal M^k u$ with $O(N (\log N)^{k-1})$ degrees of freedom, instead of $N^k$ degrees of freedom for the full tensor product FEM space. A sparse Monte-Carlo FEM with $M$ samples (i.e., deterministic solver) is proved to yield approximations to ${\mathcal M}^k u$ with a work of $O(M N(\log N)^{k-1})$ operations. The solutions are shown to converge with the optimal rates with respect to the Finite Element degrees of freedom $N$ and the number $M$ of samples. The deterministic FEM is based on deterministic equations for ${\mathcal M}^k u$ in $D^k\subset \mathbb{R}^{kd}$. Their Galerkin approximation using sparse tensor products of the FE spaces in $D$ allows approximation of ${\mathcal M}^k u$ with $O(N(\log N)^{k-1})$ degrees of freedom converging at an optimal rate (up to logs). For nonlocal operators wavelet compression of the operators is used. The linear systems are solved iteratively with multilevel preconditioning. This yields an approximation for $\mathcal M^k u$ with at most $O(N (\log N)^{k+1})$ operations.
References:
[1] I. Babuška: On randomized solution of Laplace’s equation. Čas. Pěst. Mat. 86 (1961), 269–276.
[2] I. Babuška: Error-bounds for finite element method. Numer. Math. 16 (1971), 322–333. DOI 10.1007/BF02165003 | MR 0288971
[3] I. Babuška, R. Tempone, and G. Zouraris: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004), 800–825. DOI 10.1137/S0036142902418680 | MR 2084236
[4] V. I. Bogachev: Gaussian Measures. AMS Mathematical Surveys and Monographs Vol. 62. AMS, Providence, 1998. MR 1642391
[5] L. Breiman: Probability. Addison-Wesley, Reading, 1968. MR 0229267 | Zbl 0174.48801
[6] W. A. Light, E. W. Cheney: Approximation Theory in Tensor Product Spaces. Lecture Notes in Mathematics Vol. 1169. Springer-Verlag, Berlin, 1985. DOI 10.1007/BFb0075392 | MR 0817984
[7] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Elsevier Publ. North Holland, Amsterdam, 1978. MR 0520174 | Zbl 0383.65058
[8] M. Dahmen, H. Harbrecht, and R. Schneider: Compression techniques for boundary integral equations—optimal complexity estimates. SIAM J. Numer. Anal. 43 (2006), 2251–2271. DOI 10.1137/S0036142903428852 | MR 2206435
[9] W. Dahmen, R. Stevenson: Element-by-element construction of wavelets satisfying stability and moment conditions. SIAM J. Numer. Anal. 37 (1999), 319–352. DOI 10.1137/S0036142997330949 | MR 1742747
[10] S. C. Eisenstat, H. C. Elman, M. H. Schultz: Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20 (1983), 345–357. DOI 10.1137/0720023 | MR 0694523
[11] M. Griebel, P. Oswald, T. Schiekofer: Sparse grids for boundary integral equations. Numer. Math. 83 (1999), 279–312. DOI 10.1007/s002110050450 | MR 1712687
[12] S. Hildebrandt, N. Wienholtz: Constructive proofs of representation theorems in separable Hilbert space. Commun. Pure Appl. Math. 17 (1964), 369–373. DOI 10.1002/cpa.3160170309 | MR 0166608
[13] G. C. Hsiao, W. L. Wendland: A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58 (1977), 449–481. DOI 10.1016/0022-247X(77)90186-X | MR 0461963
[14] S. Janson: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge, 1997. MR 1474726 | Zbl 0887.60009
[15] S. Larsen: Numerical analysis of elliptic partial differential equations with stochastic input data. Doctoral Dissertation, Univ. of Maryland, 1985.
[16] M. Ledoux, M. Talagrand: Probability in Banach Spaces. Isoperimetry and Processes. Springer-Verlag, Berlin, 1991. MR 1102015
[17] P. Malliavin: Stochastic Analysis. Springer-Verlag, Berlin, 1997. MR 1450093 | Zbl 0878.60001
[18] W. McLean: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge, 2000. MR 1742312 | Zbl 0948.35001
[19] J. C. Nédélec, J. P. Planchard: Une méthode variationelle d’éléments finis pour la résolution numérique d’un problème extérieur dans $\mathbb{R}^3$. RAIRO Anal. Numér. 7 (1973), 105–129. MR 0424022
[20] G. Schmidlin, C. Lage, and C. Schwab: Rapid solution of first kind boundary integral equations in $\mathbb{R}^3$. Eng. Anal. Bound. Elem. 27 (2003), 469–490. DOI 10.1016/S0955-7997(02)00156-X
[21] T. von Petersdorff, C. Schwab: Wavelet approximations for first kind boundary integral equations in polygons. Numer. Math. 74 (1996), 479–516. DOI 10.1007/s002110050226 | MR 1414419
[22] T. von Petersdorff, C. Schwab: Numerical solution of parabolic equations in high dimensions. M2AN, Math. Model. Numer. Anal. 38 (2004), 93–127. DOI 10.1051/m2an:2004005 | MR 2073932
[23] R. Schneider: Multiskalen- und Wavelet-Matrixkompression. Advances in Numerical Mathematics. Teubner, Stuttgart, 1998. MR 1623209
[24] C. Schwab, R. A. Todor: Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95 (2003), 707–734. DOI 10.1007/s00211-003-0455-z | MR 2013125
[25] C. Schwab, R. A. Todor: Sparse finite elements for stochastic elliptic problems—higher order moments. Computing 71 (2003), 43–63. DOI 10.1007/s00607-003-0024-4 | MR 2009650
[26] S. A. Smolyak: Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov. Math. Dokl. 4 (1963), 240–243. Zbl 0202.39901
[27] V. N. Temlyakov: Approximation of Periodic Functions. Nova Science Publ., New York, 1994. MR 1373654
[28] G. W. Wasilkowski, H. Wozniakowski: Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complexity 11 (1995), 1–56. DOI 10.1006/jcom.1995.1001 | MR 1319049
[29] N. Wiener: The homogeneous Chaos. Amer. J. Math. 60 (1938), 987–936. DOI 10.2307/2371268 | MR 1507356 | Zbl 0019.35406
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