Asymptotic expansion regularization for inverse source problems in two-dimensional singularly perturbed nonlinear parabolic PDEs

TL;DR

This paper introduces an asymptotic expansion-regularization method for solving inverse source problems in complex two-dimensional nonlinear singularly perturbed PDEs, improving accuracy and efficiency even with noisy data.

Contribution

The paper presents a novel AER method utilizing asymptotic-expansion theory for inverse source problems in nonlinear singularly perturbed PDEs, with proven accuracy and efficiency.

Findings

The AER method effectively solves inverse source problems.

The approach maintains accuracy with noisy data.

Numerical examples demonstrate high efficiency.

Abstract

In this paper, we develop an asymptotic expansion-regularization (AER) method for inverse source problems in two-dimensional nonlinear and nonstationary singularly perturbed partial differential equations (PDEs). The key idea of this approach is the use of the asymptotic-expansion theory, which allows us to determine the conditions for the existence and uniqueness of a solution to a given PDE with a sharp transition layer. As a by-product, we derive a simpler link equation between the source function and first-order asymptotic approximation of the measurable quantities, and based on that equation we propose an efficient inversion algorithm, AER, for inverse source problems. We prove that this simplification will not decrease the accuracy of the inversion result, especially for inverse problems with noisy data. Various numerical examples are provided to demonstrate the efficiency of our new approach.