A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

TL;DR

This paper introduces a data-driven model reduction approach using POD basis functions to efficiently solve parabolic inverse source problems, with proven convergence and error estimates, demonstrated through numerical experiments.

Contribution

The paper develops a POD-based model reduction method with convergence analysis for parabolic inverse source problems, enhancing computational efficiency.

Findings

Significant reduction in computational time compared to finite element methods.

Convergence and error estimates for the POD algorithm are established.

Numerical examples confirm the method's accuracy and efficiency.

Abstract

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems efficiently. Our method consists of offline and online stages. In the off-line stage, we explore the low-dimensional structures in the solution space of the parabolic partial differential equations (PDEs) in the forward problem with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problem extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in the parabolic inverse source problems, which is referred to as the POD algorithm in this paper. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD algorithm in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD algorithm for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the POD algorithm provides considerable computational savings over the finite element method.