Asymptotic Expansion Regularization for Inverse Source Problems in Two-Dimensional Singularly Perturbed Nonlinear Parabolic PDEs (CSIAM Transactions on Applied Mathematics, 2023)
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 thisapproach is the use of the asymptotic-expansion theory, which allows us to determinethe conditions for the existence and uniqueness of a solution to a given PDE witha sharp transition layer. As a by-product, we derive a simpler link equation betweenthe 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. Variousnumerical examples are provided to demonstrate the efficiency of our new approach.
Keywords: Inverse source problem; singular perturbed PDE; reaction-diffusion-advection equation; regularization; convergence.