Combined Error Estimates for Local Fluctuations of SPDEs

TL;DR

This paper develops a comprehensive error estimate for numerical approximations of local fluctuations in parabolic SPDEs, accounting for multiple error sources in a unified framework, enhancing understanding of small noise effects.

Contribution

Introduces CERES, a new combined error estimate for local fluctuation analysis of SPDEs, integrating five main error sources into a single, rigorous bound.

Findings

CERES effectively bounds all major errors in SPDE local fluctuation approximation.

The method combines advanced Galerkin, covariance, and Lyapunov techniques.

Results facilitate more accurate numerical simulations of SPDEs under small noise.

Abstract

In this work, we study the numerical approximation of local fluctuations of certain classes of parabolic stochastic partial differential equations (SPDEs). Our focus is on effects for small spatially-correlated noise on a time scale before large deviation effects have occurred. In particular, we are interested in the local directions of the noise described by a covariance operator. We introduce a new strategy and prove a Combined ERror EStimate (CERES) for the five main errors: the spatial discretization error, the local linearization error, the noise truncation error, the local relaxation error to steady state, and the approximation error via an iterative low-rank matrix algorithm. In summary, we obtain one CERES describing, apart from modelling of the original equations and standard round-off, all sources of error for a local fluctuation analysis of an SPDE in one estimate. To prove our results, we rely on a combination of methods from optimal Galerkin approximation of SPDEs, covariance moment estimates, analytical techniques for Lyapunov equations, iterative numerical schemes for low-rank solution of Lyapunov equations, and working with related spectral norms for different classes of operators.