Waveform Relaxation Methods for Stochastic Differential Equations

The solution of complex and large scale systems plays a crucial role in recent scientific computations. In particular, large scale stochastic dynamical systems represent very complex systems incorporating the random appearances of physical processes in nature. The development of efficient numerical methods to study such large scale systems, which can be characterized as weakly coupled subsystems with quite different behavior, is an important challenge. Under some conditions, block-iterative methods are very efficient. One of these methods to solve large scale systems is given by the waveform relaxation method. This method was first proposed by Lelarasmee, Ruehli and Sangiovanni–Vincentelli [27] for the time-domain analysis of large scale integrated circuits. For the waveform algorithm concerning deterministic processes and related aspects, many research papers can be found, e.g. Bremer and Schneider [4], Bremer [5], Burrage [6], in’t Hout [12], Jackiewicz and Kwapisz [16], Jansen et al. [17], Jansen and Vandewalle [18], Leimkuhler [25, 26], Miekkala and Nevanlinna [30], Nevanlinna and Odeh [32], Sand and Burrage [36], Schneider [37, 38, 39], Ta’asan and Zhang [44], Zennaro [48], Zubik–Koval and Vandewalle [50], among many others. In what follows we present a theoretical foundation for the construction and convergence of waveform iterations applied to systems of ordinary stochastic differential equations (SDEs) which are decomposable into weakly coupled subsystems. The attention is restricted to Itô-interpreted SDEs and L-solutions (i.e. strong solutions in the Banach space of Lp(Ω,F , IP)-integrable random processes). For