Optimal explicit stabilized integrator of weak order one for stiff and ergodic stochastic differential equations

Abdulle, Assyr; Almuslimani, Ibrahim; Vilmart, Gilles

doi:10.1137/17M1145859

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Title		Optimal explicit stabilized integrator of weak order one for stiff and ergodic stochastic differential equations
Authors		Abdulle, Assyr Almuslimani, Ibrahim Vilmart, Gilles
Published in		SIAM/ASA Journal on Uncertainty Quantification. 2018, vol. 6, no. 2, p. 937-964
Abstract		A new explicit stabilized scheme of weak order one for stiff and ergodic stochastic differential equations (SDEs) is introduced. In the absence of noise, the new method coincides with the classical deterministic stabilized scheme (or Chebyshev method) for diffusion dominated advection-diffusion problems and it inherits its optimal stability domain size that grows quadratically with the number of internal stages of the method. For mean-square stable stiff stochastic problems, the scheme has an optimal extended mean-square stability domain that grows at the same quadratic rate as the deterministic stability domain size in contrast to known existing methods for stiff SDEs [A. Abdulle and T. Li. Commun. Math. Sci., 6(4), 2008, A. Abdulle, G. Vilmart, and K. C. Zygalakis, SIAM J. Sci. Comput., 35(4), 2013]. Combined with postprocessing techniques, the new methods achieve a convergence rate of order two for sampling the invariant measure of a class of ergodic SDEs, achieving a stabilized version of the non-Markovian scheme introduced in [B. Leimkuhler, C. Matthews, and M. V. Tretyakov, Proc. R. Soc. A, 470, 2014].
Keywords		Explicit stochastic methods — Stabilized methods — Postprocessor — Invariant measure — Ergodicity — Orthogonal Runge–Kutta Chebyshev — SK-ROCK — PSK-ROCK
Identifiers		DOI: 10.1137/17M1145859
Full text		Article (Accepted version) (1.9 MB) - Free access
Structures		Faculté des sciences / Section de mathématiques
Research group		Analyse numérique
Citation (ISO format)		ABDULLE, Assyr, ALMUSLIMANI, Ibrahim, VILMART, Gilles. Optimal explicit stabilized integrator of weak order one for stiff and ergodic stochastic differential equations. In: SIAM/ASA Journal on Uncertainty Quantification, 2018, vol. 6, n° 2, p. 937-964. doi: 10.1137/17M1145859 https://archive-ouverte.unige.ch/unige:105971