Solving forward and inverse problems in a non-linear 3D PDE via an asymptotic expansion based approach

TL;DR

This paper introduces an asymptotic expansion method for efficiently solving forward and inverse nonlinear 3D PDE problems, ensuring accuracy near transition layers and robustness to noise.

Contribution

It develops a novel asymptotic expansion approach for 3D nonlinear PDEs, including a simplified inverse source model and a regularization algorithm for noisy data.

Findings

Proved existence and uniqueness of solutions with sharp transition layers.

Derived a simplified inverse source model close to the original.

Demonstrated the effectiveness of the numerical approach on a model problem.

Abstract

This paper concerns the use of asymptotic expansions for the efficient solving of forward and inverse problems involving a nonlinear singularly perturbed time-dependent reaction--diffusion--advection equation. By using an asymptotic expansion with the local coordinates in the transition-layer region, we prove the existence and uniqueness of a smooth solution with a sharp transition layer for a three-dimensional partial differential equation. Moreover, with the help of asymptotic expansion, a simplified model is derived for the corresponding inverse source problem, which is close to the original inverse problem over the entire region except for a narrow transition layer. We show that such simplification does not reduce the accuracy of the inversion results when the measurement data contain noise. Based on this simpler inversion model, an asymptotic-expansion regularization algorithm is proposed for efficiently solving the inverse source problem in the three-dimensional case. A model problem shows the feasibility of the proposed numerical approach.