Performance of a Parallel Time Integrator for Noisy Nonlinear System

Abstract

The paper demonstrates the performance of a parallel time integration algorithm for simulating the trajectories (sample path) of a noisy non-linear dynamical system described by Ito stochastic differential equation (SDE). In particular the numerical algorithm is an extension of so-called parareal algorithm for ordinary differential equations (ODEs). We adapt the parareal algorithm to Euler-Maruyama scheme to tackle the Ito SDE describing a Duffing system driven by random noise. Note that the presenceof Wiener process in Ito SDEs leads to difficulties in the straightforward extension of numerical techniques of ODEs. This is due to the fact that the Wiener process, although continuous, is not differentiable and possesses unbounded variation in any integration subinterval. In this paper we conduct a numerical investigation to simulate the sample path of a Duffing oscillator driven by combined deterministic and random inputs. It turns out that for low to medium strength of noise, the parallel integrator is capable of computing the sample path of the oscillator reasonably well.