Lefu Cai | Searcharxiv

arXiv2026

A Non-Decoupled Time-Domain Direct Sampling Method for Inverse Elastic Medium Scattering

This work is concerned with an inverse medium problem for elastic waves, in which unknown inhomogeneities are reconstructed from time-resolved boundary measurements. We propose a novel time-domain direct sampling method for locating scatterers from a single incident source, without imposing specific assumptions on the temporal profile of the excitation. In particular, the imaging functional introduces a time-shifted correlation strategy that replaces the traditional $P$-$S$ wave decomposition with a travel-time alignment mechanism, thereby enabling direct imaging from the coupled elastic wave field. To analyze the proposed time-domain imaging functional, we employ Parseval's identity for the Fourier--Laplace transform and reformulate the functional in the frequency domain. By exploiting properties of modified Bessel functions, we characterize the asymptotic behavior of the imaging functional and show that it attains its maximum at the target location, which enables reliable identification of the scatterer. Rigorous theoretical justifications are provided to substantiate the effectiveness of the proposed method. Numerical experiments are also presented to demonstrate its performance and applicability.

math.NA↗

arXiv2025

A Computationally Efficient Finite Element Method for Shape Reconstruction of Inverse Conductivity Problems

The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE constraints have been widely used to deal with this problem. However, such approaches typically require repeated iterations and solving the forward problem at each iteration, which leads to a heavy computational cost. To address this issue, we first reformulate the inverse conductivity problem as a minimization problem involving a regularized residual functional. We then transform this minimization problem into a variational problem and establish the equivalence between them. This reformulation enables the employment of the finite element method to reconstruct the shape of the object from finitely many measurements. Notably, the proposed approach allows us to identify the object directly without requiring any iterative procedure. {\it A prior} error estimates are rigorously established to demonstrate the theoretical soundness of the finite element method. Based on these estimates, we provide a criterion for selecting the regularization parameter. Additionally, several numerical examples are presented to verify the feasibility of the proposed approach in shape reconstruction.

math.NA↗