TY - JOUR
T1 - Waveform Relaxation Methods for Stochastic Differential Equations
AU - H. Schurz & K. R. Schneider
JO - International Journal of Numerical Analysis and Modeling
VL - 2
SP - 232
EP - 254
PY - 2006
DA - 2006/03
SN - 3
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/898.html
KW - waveform relaxation methods, stochastic differential equations, stochastic-numerical methods, iteration methods, large scale systems.
AB - <p style="text-align: justify;">$L^p$-convergence of waveform 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 $\varepsilon$ of $\mathcal{F}_t$-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.</p>