TY - JOUR
T1 - Asymptotic Expansion Regularization for Inverse Source Problems in Two-Dimensional Singularly Perturbed Nonlinear Parabolic PDEs
AU - Chaikovskii , Dmitrii
AU - Liubavin , Aleksei
AU - Zhang , Ye
JO - CSIAM Transactions on Applied Mathematics
VL - 4
SP - 721
EP - 757
PY - 2023
DA - 2023/10
SN - 4
DO - http://doi.org/10.4208/csiam-am.SO-2022-0017
UR - https://global-sci.org/intro/article_detail/csiam-am/22076.html
KW - Inverse source problem, singular perturbed PDE, reaction-diffusion-advection equation, regularization, convergence.
AB - <p style="text-align: justify;">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.</p>