Model reduction for stochastic systems with nonlinear drift

TL;DR

This paper develops a novel model reduction technique for high-dimensional stochastic differential equations with nonlinear drift, using new Gramian concepts to identify dominant subspaces and achieve accurate reduced models.

Contribution

It introduces new Gramian-based methods for model reduction of nonlinear stochastic systems, with theoretical analysis and application to reaction diffusion equations.

Findings

Effective dimension reduction for nonlinear stochastic systems.

High accuracy of reduced models demonstrated in reaction diffusion examples.

Error bounds and stability analysis provided for the proposed methods.

Abstract

In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such nonlinearities in high dimensional settings occur, e.g., when stochastic reaction diffusion equations are discretized in space. We provide a brief discussion around existence, uniqueness and stability of solutions. (Almost) stability then is the basis for new concepts of Gramians that we introduce and study in this work. With the help of these Gramians, dominant subspace are identified leading to a balancing related highly accurate reduced order SDE. We provide an algebraic error criterion and an error analysis of the propose model reduction schemes. The paper is concluded by applying our method to spatially discretized reaction diffusion equations.