Communications in Mathematical Sciences

Volume 4 (2006)

Number 4

Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems

Pages: 707 – 729



Dror Givon

Ioannis G. Kevrekidis

Raz Kupferman


We study the convergence of the slow (or "essential") components of singularly perturbed stochastic differential systems to solutions of lower dimensional stochastic systems (the "effective", or "coarse" dynamics). We prove strong, mean-square convergence in systems where both fast and slow components are driven by noise, with full coupling between fast and slow components. We analyze a class of "projective integration" methods, which consist of a hybridization between a standard solver for the slow components, and short runs for the fast dynamics, which are used to estimate the effect that the fast components have on the slow ones. We obtain explicit bounds for the discrepancy between the results of the projective integration method and the slow components of the original system.


Dimension reduction; stochastic differential equations; scale separation; singular perturbations; projective integration

2010 Mathematics Subject Classification

Primary 60H10. Secondary 60F15, 65C30.

Published 1 January 2006