Domain decomposition solution of elliptic boundary-value problems via Monte Carlo and quasi-Monte Carlo methods

SIAM Journal on Scientific Computing

Volumn 27, Issue 2, 2006, Pages 440-457

  Acebrón, Juan A ^a,b   Busico, Maria Pia ^c   Lanucara, Piero ^c   Spigler, Renato ^d  

^a UNIVERSITY OF ALCALÁ   (Spain)

^b UNIVERSITY OF PADOVA   (Italy)

^c CASPUR   (Italy)

^d ROMA TRE UNIVERSITY   (Italy)

Author keywords

Domain decomposition; Monte Carlo methods; Parallel computing; Quasi Monte Carlo methods

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; INTERPOLATION; MONTE CARLO METHODS; PARALLEL PROCESSING SYSTEMS; PROBABILISTIC LOGICS;

DOMAIN DECOMPOSITION; GRID COMPUTING; MULTIPROCESSOR ARCHITECTURES; QUASI-MONTE CARLO METHODS;

BOUNDARY VALUE PROBLEMS;

EID: 33644936721     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/030600692     Document Type: Article

References (38)

1. L. ARNOLD, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1974.
2. P. BALDI, Exact asymptotics for the probability of exit from a domain and applications to simulation, Ann. Probab., 23 (1995), pp. 1644-1670.
3. S. C. BRENNER, Lower bounds for two-level additive Schwarz preconditioners with small overlap, SIAM J. Sci. Comput., 21 (2000), pp. 1657-1669.
4. F. M. BUCHMANN AND W. PETERSEN, Solving Dirichlet problems numerically using the Feynman-Kac representation, BIT, 43 (2003), pp. 519-540.
5. R. E. CAFLISCH, Monte Carlo and quasi-Monte Carlo methods, in Acta Numerica, Acta Numer. 7, Cambridge University Press, Cambridge, UK, 1998, pp. 1-49.
6. R. E. CAFLISCH, W. MOROKOFF, AND A. B. OWEN, Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension, J. Comput. Finance, 1 (1997), pp. 27-46.
7. T. F. CHAN AND T. P. MATHEW, Domain decomposition algorithms, in Acta Numerica, Acta Numer., Cambridge University Press, Cambridge, UK, 1994, pp. 61-143.
8. M. DRYJA AND O. B. WIDLUND, Domain decomposition algorithms with small overlap, SIAM J. Sci. Comput., 15 (1994), pp. 604-620.
9. M. DRYJA, B. F. SMITH, AND O. B. WIDLUND, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), pp. 1662-1694.
10. M. FREIDLIN, Functional Integration and Partial Differential Equations, Ann. Math. Stud. 109, Princeton University Press, Princeton, NJ, 1985.
11. E. HAUSENBLAS, A Monte-Carlo method with inherent parallelism for numerical solving partial differential equations with boundary conditions, in Parallel Computation, Lecture Notes in Comput. Sci. 1557, Springer-Verlag, Berlin, 1999, pp. 117-126.
12. D. J. HIGHAM, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), pp. 525-546.
13. N. HOFMANN AND P. MATHÉ, On quasi-Monte Carlo simulation of stochastic differential equations, Math. Comp., 66 (1997), pp. 573-589.
14. K. M. JANSONS AND G. D. LYTHE, Efficient numerical solution of stochastic differential equations using exponential timestepping, J. Statist. Phys., 100 (2000), pp. 1097-1109.
15. K. M. JANSONS AND G. D. LYTHE, Exponential timestepping with boundary test for stochastic differential equations, SIAM J. Sci. Comput., 24 (2003) pp. 1809-1822.
16. M. H. KALOS AND P. A. WITHLOCK, Monte Carlo Methods, Vol. I: Basics, Wiley, New York, 1986.
17. I. KARATZAS AND S. E. SHREVE, Brownian Motion and Stochastic Calculus, 2nd ed., Springer-Verlag, Berlin, 1991.
18. P. E. KLOEDEN AND E. PLATEN, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
19. C. LÉCOT, A quasi-Monte Carlo method for the Boltzmann equation, Math. Comp., 56 (1991), pp. 621-644.
20. C. LÉCOT AND F. EL KHETTABI, Quasi-Monte Carlo simulation of diffusion, J. Complexity, 15 (1999), pp. 342-359.
21. R. MANNELLA, Absorbing boundaries and optimal stopping in a stochastic differential equation, Phys. Lett. A, 254 (1999), pp. 257-262.
22. M. MASCAGNI, Parallel Wiener integral methods for elliptic BVPs: A tale of two architectures, in Applications on Advanced Architecture Computers, Software Environ. Tools, G. Astfalk, ed., SIAM, Philadelphia, 1996, pp. 27-33.
23. W. J. MOROKOFF, Generating quasi-random paths for stochastic processes, SIAM Rev., 40 (1998), pp. 765-788.
24. W. J. MOROKOFF AND R. E. CAFLISCH, A quasi-Monte Carlo approach to particle simulation of the heat equation, SIAM J. Numer. Anal., 30 (1993), pp. 1558-1573.
25. W. J. MOROKOFF AND R. E. CAFLISCH, Quasi-random sequences and their discrepancies, SIAM J. Sci. Comput., 15 (1994), pp. 1251-1279.
26. W. J. MOROKOFF AND R. E. CAFLISCH, Quasi-Monte Carlo integration, J. Comput. Phys., 122 (1995), pp. 218-230.
27. B. MOSKOWITZ AND R. E. CAFLISCH, Smoothness and dimension reduction in quasi-Monte Carlo methods, Math. Comput. Modelling, 23 (1996), pp. 37-54.
28. H. NIEDERREITER, Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Regional Conf. Ser. in Appl. Math. 63, SIAM, Philadelphia, PA, 1992.
29. E. PEIRANO AND D. TALAY, Domain decomposition by stochastic methods, in Proceedings of the 14th International Conference on Domain Decomposition Methods, I. Herrera, D. Keyes, O. Widlund, and R. Yates, eds., National Autonomous University of Mexico, Mexico City, 2003, pp. 131-147.
30. E. PLATEN, An introduction to numerical methods for stochastic differential equations, in Acta Numerica, Acta Numer. 8, Cambridge University Press, Cambridge, UK, 1999, pp. 197-246.
31. W. H. PRESS, S. A. TEUKOLSKY, W. T. VETTERLING, AND B. P. FLANNERY, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 1996.
32. A. QUARTERONI AND A. VALLI, Theory and application of Steklov-Poincaré operators for boundary-value problems, in Applied and Industrial Mathematics, Math. Appl. 56, R. Spigler, ed., Kluwer, Dordrecht, The Netherlands, 1991, pp. 179-203.
33. A. QUARTERONI AND A. VALLI, Domain Decomposition Methods for Partial Differential Equations, Numer. Math. Sci. Comput., Oxford Science Publications, Clarendon Press, Oxford, UK, 1999.
34. T. J. RIVLIN, An Introduction to the Approximation of Functions, Dover, New York, 1981.
35. T. J. RIVLIN, The Chebyshev Polynomials, Wiley, New York, 1974.
36. R. SPIGLER, A probabilistic approach to the solution of PDE problems via domain decomposition methods, in Proceedings of the Second International Conference on Industrial and Applied Mathematics, Washington, DC, 1991, p. 12 (Book of Abstracts).
37. W. STRITTMATTER, Numerical simulation of the mean first passage time, Technical report THEP 87/12, University of Freiburg, Freiburg, Germany, 1987.
38. D. TALAY, Probabilistic numerical methods for partial differential equations: Elements of analysis, in Probabilistic Models for Nonlinear Partial Differential Equations, Lecture Notes in Math. 1627, Springer-Verlag, Berlin, 1996, pp. 148-196.