Convergence analysis for an implementable scheme to solve the linear-quadratic stochastic optimal control problem with stochastic wave equation

TL;DR

This paper develops a scalable, implementable numerical scheme for solving a stochastic linear-quadratic control problem governed by a stochastic wave equation, with proven strong convergence rates and validated efficiency.

Contribution

It introduces a discretization method combining finite elements and midpoint rule, along with a gradient algorithm that avoids Monte Carlo sampling, for the stochastic wave equation control problem.

Findings

Strong convergence rates for the discretized control problem.

Efficient gradient algorithm with linear computational cost per iteration.

Numerical validation confirms the method's scalability and accuracy.

Abstract

We study an optimal control problem for the stochastic wave equation driven by affine multiplicative noise, formulated as a stochastic linear-quadratic (SLQ) problem. By applying a stochastic Pontryagin's maximum principle, we characterize the optimal state-control pair via a coupled forward-backward SPDE system. We propose an implementable discretization using conforming finite elements in space and an implicit midpoint rule in time. By a new technical approach we obtain strong convergence rates for the discrete state-control pair without relying on Malliavin calculus. For the practical computation we develop a gradient-descent algorithm based on artificial iterates that employs an exact computation for the arising conditional expectations, thereby eliminating costly Monte Carlo sampling. Consequently, each iteration has a computational cost that is proportional to the number of spatial degrees of freedom, producing a scalable method that preserves the established strong convergence rates. Numerical results validate its efficiency.