  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.
