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A domain decomposition method for stochastic evolution equations (2401.07291v1)

Published 14 Jan 2024 in math.NA, cs.NA, and math.PR

Abstract: In recent years, SPDEs have become a well-studied field in mathematics. With their increase in popularity, it becomes important to efficiently approximate their solutions. Thus, our goal is a contribution towards the development of efficient and practical time-stepping methods for SPDEs. Operator splitting schemes are a powerful tool for deterministic and stochastic differential equations. An example is given by domain decomposition schemes, where we split the domain into sub-domains. Instead of solving one expensive problem on the entire domain, we deal with cheaper problems on the sub-domains. This is particularly useful in modern computer architectures, as the sub-problems may often be solved in parallel. While splitting methods have already been used to study domain decomposition methods for deterministic PDEs, this is a new approach for SPDEs. We provide an abstract convergence analysis of a splitting scheme for stochastic evolution equations and state a domain decomposition scheme as an application of the setting. The theoretical results are verified through numerical experiments.

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References (39)
  1. M. Ableidinger and E. Buckwar. Splitting integrators for the stochastic Landau–Lifshitz equation. SIAM J. Sci. Comput., 38:A1788–A1806, 01 2016.
  2. Sobolev Spaces. Elsevier/Academic Press, Amsterdam, second edition, 2003.
  3. A. Alamo and J. M. Sanz-Serna. A technique for studying strong and weak local errors of splitting stochastic integrators. SIAM J. Numer. Anal., 54(6):3239–3257, 2016.
  4. A semi-discrete scheme for the stochastic Landau-Lifshitz equation. Stoch. Partial Differ. Equ. Anal. Comput., 2(3):281–315, 2014.
  5. Modified Douglas splitting methods for reaction-diffusion equations. BIT, 57(2):261–285, 2017.
  6. Stochastic ferromagnetism, volume 58 of De Gruyter Studies in Mathematics. De Gruyter, Berlin, 2014. Analysis and numerics.
  7. C.-E. Bréhier and D. Cohen. Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations. Appl. Numer. Math., 186:57–83, 2023.
  8. Domain decomposition strategies for the stochastic heat equation. Int. J. Comput. Math., 89(18):2517–2542, 2012.
  9. D. S. Clark. Short proof of a discrete Gronwall inequality. Discrete Appl. Math., 16(3):279–281, 1987.
  10. Implicit space-time domain decomposition methods for stochastic parabolic partial differential equations. SIAM J. Sci. Comput., 36(1):C1–C24, 2014.
  11. Effects of noise on models of spiny dendrites. J. Comput. Neurosci., 34(2):245–257, 2013.
  12. M. Eisenmann. Methods for the Temporal Approximation of Nonlinear, Nonautonomous Evolution Equations. PhD thesis, TU Berlin, 2019.
  13. M. Eisenmann and E. Hansen. Convergence analysis of domain decomposition based time integrators for degenerate parabolic equations. Numer. Math., 140(4):913–938, 2018.
  14. M. Eisenmann and E. Hansen. A variational approach to the sum splitting scheme. IMA J. Numer. Anal., 42(1):923–950, 2022.
  15. M. Eisenmann and T. Stillfjord. A randomized operator splitting scheme inspired by stochastic optimization methods. ArXiv Preprint, arXiv:2210.05375, 2022.
  16. L. C. Evans. Partial Differential Equations. American Mathematical Society, Providence, RI, 1998.
  17. Exponential time integrators for stochastic partial differential equations in 3d reservoir simulation. Comput. Geosci., 16(2):323–334, 2012.
  18. J. Geiser and C. Kravvaritis. A domain decomposition method based on the iterative operator splitting method. Appl. Numer. Math., 59(3-4):608–623, 2009.
  19. Splitting methods in communication and imaging, science, and engineering. Sci. Comput. Cham: Springer, 2016.
  20. I. Gyöngy. Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I. Potential Anal., 9(1):1–25, 1998.
  21. E. Hansen and E. Henningsson. Additive domain decomposition operator splittings—convergence analyses in a dissipative framework. IMA J. Numer. Anal., 37(3):1496–1519, 2017.
  22. E. Hausenblas. Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math., 147(2):485–516, 2002.
  23. L. Ji. An overlapping domain decomposition splitting algorithm for stochastic nonlinear Schrö dinger equation. ArXiv Preprint, arXiv:2309.03393, 2023.
  24. Parallel domain decomposition methods for stochastic elliptic equations. SIAM J. Sci. Comput., 29(5):2096–2114, 2007.
  25. R. Kruse. Strong and weak approximation of semilinear stochastic evolution equations, volume 2093 of Lecture Notes in Mathematics. Springer, Cham, 2014.
  26. R. Kruse and S. Larsson. Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise. Electron. J. Probab., 17:no. 65, 19, 2012.
  27. R. Kruse and R. Weiske. The BDF2-Maruyama scheme for stochastic evolution equations with monotone drift. ArXiv Preprint, arXiv:2105.08767, 2021.
  28. An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. J. R. Stat. Soc. Ser. B Stat. Methodol., 73(4):423–498, 2011. With discussion and a reply by the authors.
  29. W. Liu and M. Röckner. Stochastic partial differential equations: an introduction. Universitext. Springer, Cham, 2015.
  30. An introduction to computational stochastic PDEs. Cambridge Texts in Applied Mathematics. Cambridge University Press, New York, 2014.
  31. T. P. Mathew. Domain decomposition methods for the numerical solution of partial differential equations, volume 61 of Lect. Notes Comput. Sci. Eng. Berlin: Springer, 2008.
  32. R. Mclachlan and G. Quispel. Splitting methods. Acta Numer., 11:341–434, 01 2002.
  33. T. Roubíček. Nonlinear partial differential equations with applications, volume 153 of International Series of Numerical Mathematics. Birkhäuser/Springer Basel AG, Basel, second edition, 2013.
  34. M. Sauer and W. Stannat. Analysis and approximation of stochastic nerve axon equations. Math. Comput., 85(301):2457–2481, 2016.
  35. T. Shardlow. Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optim., 20(1-2):121–145, 1999.
  36. T. Shardlow. Splitting for dissipative particle dynamics. SIAM J. Sci. Comput., 24(4):1267–1282, 2003.
  37. P. N. Vabishchevich. Difference schemes with domain decomposition for solving non-stationary problems. U.S.S.R. Comput. Math. Math. Phys., 29(6):155–160, 1989.
  38. M. C. Veraar. Stochastic integration in banach spaces and applications to parabolic evolution equations. PhD Thesis, Delft University, 2006.
  39. Domain decomposition methods for linear and semilinear elliptic stochastic partial differential equations. Applied Mathematics and Computation, 195(2):630–640, 2008.

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