Sparse finite element methods for operator equations with stochastic data

Applications of Mathematics

Volumn 51, Issue 2, 2006, Pages 145-180

  Von Petersdorff, Tobias ^a   Schwab, Christoph ^b  

^a Bldg 142   (United States)

^b ETH ZURICH   (Switzerland)

Author keywords

Monte Carlo method; random data; wavelet compression of operators; wavelet finite element method

Indexed keywords

EID: 84867925855     PISSN: 08627940     EISSN: 15729109     Source Type: Journal    
DOI: 10.1007/s10492-006-0010-1     Document Type: Article

References (29)

1. I. Babuška 1961 On randomized solution of Laplace's equation Čas. Pěst. Mat. 86 269 276
2. I. Babuška 1971 Error-bounds for finite element method Numer. Math. 16 322 333 44 #6166 10.1007/BF02165003
3. I. Babuška R. Tempone G. Zouraris 2004 Galerkin finite element approximations of stochastic elliptic partial differential equations SIAM J. Numer. Anal. 42 800 825 2005h:65012 10.1137/S0036142902418680
4. V. I. Bogachev: Gaussian Measures. AMS Mathematical Surveys and Monographs Vol. 62. AMS, Providence, 1998.
5. L. Breiman: Probability. Addison-Wesley, Reading, 1968.
6. W. A. Light, E.W. Cheney: Approximation Theory in Tensor Product Spaces. Lecture Notes in Mathematics Vol. 1169. Springer-Verlag, Berlin, 1985.
7. P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Elsevier Publ. North Holland, Amsterdam, 1978.
8. M. Dahmen H. Harbrecht R. Schneider 2006 Compression techniques for boundary integral equations- optimal complexity estimates SIAM J. Numer. Anal. 43 2251 2271 2206435 10.1137/S0036142903428852
9. W. Dahmen R. Stevenson 1999 Element-by-element construction of wavelets satisfying stability and moment conditions SIAM J. Numer. Anal. 37 319 352 2001c:65144 10.1137/S0036142997330949
10. S. C. Eisenstat H. C. Elman M. H. Schultz 1983 Variational iterative methods for nonsymmetric systems of linear equations SIAM J. Numer. Anal. 20 345 357 84h:65030 10.1137/0720023
11. M. Griebel P. Oswald T. Schiekofer 1999 Sparse grids for boundary integral equations Numer. Math. 83 279 312 2000h:65195 10.1007/s002110050450
12. S. Hildebrandt N. Wienholtz 1964 Constructive proofs of representation theorems in separable Hilbert space Commun. Pure Appl. Math. 17 369 373 29 #3881
13. G. C. Hsiao W. L. Wendland 1977 A finite element method for some integral equations of the first kind J. Math. Anal. Appl. 58 449 481 57 #1945 10.1016/0022-247X(77)90186-X
14. S. Janson: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge, 1997.
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.
17. P. Malliavin: Stochastic Analysis. Springer-Verlag, Berlin, 1997.
18. W. McLean: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge, 2000.
19. 3 RAIRO Anal. Numér. 7 105 129
20. 3 Eng. Anal. Bound. Elem. 27 469 490 10.1016/S0955-7997(02)00156-X
21. T. von Petersdorff C. Schwab 1996 Wavelet approximations for first kind boundary integral equations in polygons Numer. Math. 74 479 516 97m:65223 10.1007/s002110050226
22. T. von Petersdorff C. Schwab 2004 Numerical solution of parabolic equations in high dimensions M2AN, Math. Model. Numer. Anal. 38 93 127 2005d:65169 10.1051/m2an:2004005
23. R. Schneider: Multiskalen-und Wavelet-Matrixkompression. Advances in Numerical Mathematics. Teubner, Stuttgart, 1998.
24. C. Schwab R. A. Todor 2003 Sparse finite elements for elliptic problems with stochastic loading Numer. Math. 95 707 734 2004j:65192 10.1007/s00211-003- 0455-z
25. C. Schwab R. A. Todor 2003 Sparse finite elements for stochastic elliptic problems-higher order moments Computing 71 43 63 2004k:65226 10.1007/s00607-003-0024-4
26. S. A. Smolyak 1963 Quadrature and interpolation formulas for tensor products of certain classes of functions Sov. Math. Dokl. 4 240 243
27. V. N. Temlyakov: Approximation of Periodic Functions. Nova Science Publ., New York, 1994.
28. G.W. Wasilkowski H. Wozniakowski 1995 Explicit cost bounds of algorithms for multivariate tensor product problems J. Complexity 11 1 56 95k:65138 10.1006/jcom.1995.1001
29. N. Wiener 1938 The homogeneous Chaos Amer. J. Math. 60 987 936