Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

Numerische Mathematik

Volumn 125, Issue 3, 2013, Pages 569-600

  Teckentrup, A L ^a   Scheichl, R ^a   Giles, M B ^b   Ullmann, E ^a  

^a University of Bath   (United Kingdom)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84885940668     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-013-0546-4     Document Type: Article

References (32)

1. Barrett, J. W., Elliott, C. E.: Total flux estimates for a finite-element approximation of elliptic equations. IMA J. Numer. Anal. 7, 129-148 (1987).
2. Barth, A., Schwab, Ch., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDE's with stochastic coefficients. Numer. Math. 119(1), 123-161 (2011).
3. Bourlard, M., Dauge, M., Lubama, M. S., Nicaise, S.: Coefficients of the singularities of elliptic boundary value problems in domains with conical points. III: Finite element methods on polygonal domains. SIAM J. Numer. Anal. 29(1), 136-155 (1992).
4. Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, 3rd edn, vol. 15. Springer, Berlin (2008).
5. Charrier, J.: Strong and weak error estimates for the solutions of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal 50(1), 216-246 (2012).
6. Charrier, J., Scheichl, R., Teckentrup, A. L.: Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal. 51(1), 322-352 (2013).
7. Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38(4), 1200-1216 (2001).
8. Cliffe, K. A., Giles, M. B., Scheichl, R., Teckentrup, A. L.: Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14(1), 3-15 (2011).
9. Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems. M2AN Math. Model. Numer. Anal. 33(3), 627-649 (1999).
10. Dereich, S., Heidenreich, F.: A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations. Stoch. Process. Appl. 121(7), 1565-1587 (2011).
11. Dietrich, C. R., Newsam, G. N.: Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18(4), 1088-1107 (1997).
12. Douglas, J., Dupont, T., Wheeler, M. F.: A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems. RAIRO Modèl. Math. Anal. Numèr. 2, 47-59 (1974).
13. Ghanem, R. G., Spanos, P. D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991).
14. Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin, Reprint of the 1998 edition (2001).
15. Giles, M. B.: Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Monte Carlo and Quasi-Monte Carlo methods 2006, pp. 343-358. Springer, Berlin (2007).
16. Giles, M. B.: Multilevel Monte Carlo path simulation. Oper. Res. 256, 981-986 (2008).
17. Giles, M. B., Reisinger, C.: Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance. SIFIN 1(3), 575-592 (2012).
18. Giles, M. B., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica, vol., pp. 145-236. Cambridge University Press, Cambridge (2002).
19. Gittelson, C. J., Könnö, J., Schwab, Ch., Stenberg, R.: The multilevel Monte Carlo finite element method for a stochastic Brinkman problem. SAM Research Report 2011-31, ETH Zurich (2011).
20. Graham, I. G., Kuo, F. Y., Nuyens, D., Scheichl, R., Sloan, I. H.: Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. J. Comput. Phys. 230(10), 3668-3694 (2011).
21. Graham, I. G., Scheichl, R., Ullmann, E.: Mixed Finite Element analysis of lognormal diffusion and multilevel Monte Carlo methods, In preparation (2013).
22. Graubner, S.: Multi-level Monte Carlo Methoden für stochastische partielle Differentialgleichungen. Diploma thesis, TU Darmstadt (2008). http://people. maths. ox. ac. uk/gilesm/files/Diplomarbeit. pdf.
23. Grisvard, P.: Elliptic Problems in Non-Smooth Domains. Pitman, Boston (1985).
24. Grisvard, P.: Singularities in boundary value problems. In: Researh Notes in Mathematics. Springer, Berlin (1992).
25. Hackbusch, W.: Elliptic differential equations. In: Springer Series in Computational Mathematics, vol. 18. Springer, Berlin (2010).
26. Heinrich, S.: Multilevel Monte Carlo methods. In: Lecture Notes in Computer Science, vol. 2179, pp. 3624-3651. Springer, Berin (2001).
27. Hoel, H., von Schwerin, E., Szepessy, A., Tempone, R.: Adaptive multilevel Monte Carlo simulation. In: Engquist, B., Runborg, O., Tsai, Y.-H. R. (eds.) Numerical Analysis of Multiscale Computations. Lecture Notes in Computational Science and Engineering, vol. 82, pp. 217-234. Springer, Berlin (2012).
28. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966).
29. Kloeden, P., Neuenkirch, A., Pavani, R.: Multilevel Monte Carlo for stochastic differential equations with additive fractional noise. Ann. App. Probab. 189, 255-276 (2011).
30. Nicaise, S., Sändig, A. M.: General interface problems - I. Math. Methods Appl. Sci. 17(6), 395-429 (1994).
31. Petzoldt, M.: Regularity and error estimators for elliptic problems with discontinuous coefficients. PhD thesis, WIAS Berlin (2001). http://webdoc. sub. gwdg. de/ebook/diss/2003/fu-berlin/2001/111/.
32. Pierce, N. A., Giles, M. B.: Adjoint and defect error bounding and correction for functional estimates. J. Comput. Phys. 200, 769-794 (2004).