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

Abstract

<p>In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our method consists of offline and online stages. In the offline stage, we explore the low-dimensional structures in the solution space of parabolic <a href="https://www.sciencedirect.com/topics/computer-science/partial-differential-equation" title="Learn more about partial differential equations from ScienceDirect's AI-generated Topic Pages">partial differential equations</a> (PDEs) in the forward problems with a given class of source functions and construct a small number of <a href="https://www.sciencedirect.com/topics/computer-science/proper-orthogonal-decomposition" title="Learn more about proper orthogonal decomposition from ScienceDirect's AI-generated Topic Pages">proper orthogonal decomposition</a> (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problems extremely fast in the online stage. Thus, we develop a fast algorithm to solve the <a href="https://www.sciencedirect.com/topics/computer-science/optimization-problem" title="Learn more about optimization problem from ScienceDirect's AI-generated Topic Pages">optimization problem</a> in parabolic inverse source problems, which is referred to as the POD method. Moreover, we design an <em>a posteriori</em> algorithm to find the optimal <a href="https://www.sciencedirect.com/topics/computer-science/regularization-parameter" title="Learn more about regularization parameter from ScienceDirect's AI-generated Topic Pages">regularization parameter</a> in the <a href="https://www.sciencedirect.com/topics/computer-science/optimization-problem" title="Learn more about optimization problem from ScienceDirect's AI-generated Topic Pages">optimization problem</a> using the proposed POD method without knowing the noise level. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD method in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD method for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that the POD method provides considerable computational savings over the <a href="https://www.sciencedirect.com/topics/computer-science/finite-element-method" title="Learn more about finite element method from ScienceDirect's AI-generated Topic Pages">finite element method</a> while maintaining the same accuracy.</p>