A Random Batch Method for the Efficient Simulation and Optimal Control of Networked 1-D Wave Equations

TL;DR

This paper introduces a stochastic random batch method for efficiently simulating and controlling networked 1-D wave equations, with proven convergence and demonstrated computational benefits.

Contribution

It presents a novel random batch algorithm for networked wave equations, with theoretical convergence proofs and practical efficiency improvements.

Findings

Solution converges to original system as subintervals shrink

Optimal controls for randomized system converge to original controls

Numerical examples show computational efficiency gains

Abstract

In this paper, a stochastic algorithm for the efficient simulation and optimal control of networked wave equations based on the random batch method is proposed and analyzed. The random approximation is constructed by dividing the time interval into subintervals and restricting the dynamics to a randomly chosen subnetwork during each of these subintervals. It is proven that the solution for this randomized system converges in expectation to the solution on the original network when the length of the subintervals approaches zero. Furthermore, the optimal controls for the randomized system converge (in $H^2$ and in expectation) to the optimal controls for the original system. The computational advantage of the proposed method is demonstrated in two numerical examples.