Asymptotic Expansion Regularization for Inverse Source Problems in Two-Dimensional Singularly Perturbed Nonlinear Parabolic PDEs
Year: 2023
Author: Dmitrii Chaikovskii, Aleksei Liubavin, Ye Zhang
CSIAM Transactions on Applied Mathematics, Vol. 4 (2023), Iss. 4 : pp. 721–757
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/csiam-am.SO-2022-0017
CSIAM Transactions on Applied Mathematics, Vol. 4 (2023), Iss. 4 : pp. 721–757
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 37
Keywords: Inverse source problem singular perturbed PDE reaction-diffusion-advection equation regularization convergence.