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Int. J. Numer. Anal. Mod., 3 (2006), pp. 232-254. |
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Waveform relaxation methods for stochastic differential equations Henri Schurz 1, Klaus R. Schneider 2 1 Department of Mathematics, Southern Illinois University, 1245 Lincoln Drive, Carbondale, IL 62901-4408, USA and Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA2 Department of Laserdynamics, Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, Berlin, D-10117, Germany Received by the editors October 29, 2004 Abstract L-p-convergence of wave form relaxation methods (WRMs) for numerical solving of systems of ordinary stochastic differential equations (SDEs) is studied. For this purpose, we convert the problem to an operator equation X = Pi X + G in a Banach space epsilon of F-t-adapted random elements describing the initial-or boundary value problem related to SDEs with weakly coupled, Lipschitz-continuous subsystems. The main convergence result of WRMs for SDEs depends on the spectral radius of a matrix associated to a decomposition of Pi. A generalization to one-sided Lipschitz continuous coefficients and a discussion on the example of singularly perturbed SDEs complete this paper. AMS subject classifications: 65C30, 65L20, 65D30, 34F05, 37H10, 60H10 Key words: waveform relaxation methods; stochastic differential equations; stochastic-numerical methods; iteration methods; large scale systems Email: hschurz@math.siu.edu (H. Schurz), schneide@wias-berlin.de (K. R. Schneider) |