The multi-index Monte Carlo method for semilinear stochastic partial differential equations

TL;DR

This paper introduces a multi-index Monte Carlo method (MIMC) for efficiently approximating statistics of solutions to semilinear parabolic SPDEs, demonstrating superior performance over existing methods through theoretical analysis and numerical experiments.

Contribution

The paper develops a novel MIMC approach that leverages exponential integrator solutions and proves its convergence and efficiency for low-regularity SPDEs, outperforming multilevel Monte Carlo methods.

Findings

MIMC achieves substantial computational savings compared to multilevel Monte Carlo.

Theoretical bounds on MIMC's cost as accuracy improves are established.

Numerical experiments confirm MIMC's superior performance on test problems.

Abstract

Stochastic partial differential equations (SPDEs) are often difficult to solve numerically due to their low regularity and high dimensionality. These challenges limit the practical use of computer-aided studies and pose significant barriers to statistical analysis of SPDEs. In this work, we introduce a highly efficient multi-index Monte Carlo method (MIMC) designed to approximate statistics of mild solutions to semilinear parabolic SPDEs. Key to our approach is the proof of a multiplicative convergence property for coupled solutions generated by an exponential integrator numerical solver, which we incorporate with MIMC. We further describe theoretically how the asymptotic computational cost of MIMC can be bounded in terms of the input accuracy tolerance, as the tolerance goes to zero. Notably, our methodology illustrates that for an SPDE with low regularity, MIMC offers substantial performance improvements over other viable methods. Numerical experiments comparing the performance of MIMC with the multilevel Monte Carlo method on relevant test problems validate our theoretical findings. These results also demonstrate that MIMC significantly outperforms state-of-the-art multilevel Monte Carlo, thereby underscoring its potential as a robust and tractable tool for solving semilinear parabolic SPDEs.