Efficient low rank approximations for parabolic control problems with unknown heat source

TL;DR

This paper introduces a low-rank approximation method using the Arnoldi algorithm to efficiently solve inverse heat source problems in parabolic equations, reducing computational costs while maintaining accuracy across multiple dimensions.

Contribution

The paper presents a novel application of the Arnoldi algorithm for low-rank approximation to efficiently solve inverse parabolic control problems with unknown heat sources, bypassing traditional computationally expensive methods.

Findings

Effective reduction of computational complexity in high dimensions.

Maintains accuracy comparable to classical methods.

Validated with numerical experiments in 1D, 2D, and 3D.

Abstract

An inverse problem of finding an unknown heat source for a class of linear parabolic equations is considered. Such problems can typically be converted to a direct problem with non-local conditions in time instead of an initial value problem. Standard ways of solving these non-local problems include direct temporal and spatial discretization as well as the shooting method, which may be computationally expensive in higher dimensions. In the present article, we present approaches based on low-rank approximation via Arnoldi algorithm to bypass the computational limitations of the mentioned classical methods. Regardless of the dimension of the problem, we prove that the Arnoldi approach can be effectively used to turn the inverse problem into a simple initial value problem at the cost of only computing one-dimensional matrix functions while still retaining the same accuracy as the classical approaches. Numerical results in dimensions d=1,2,3 are provided to validate the theoretical findings and to demonstrate the efficiency of the method for growing dimensions.