Oksanen Lauri, Yang Tianyu, Yang Yang
Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland.
Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, United States of America.
Inverse Probl. 2024 Dec 1;40(12):125031. doi: 10.1088/1361-6420/ad98bc. Epub 2024 Dec 13.
We develop a linearized boundary control method for the inverse boundary value problem of determining a density in the acoustic wave equation. The objective is to reconstruct an unknown perturbation in a known background density from the linearized Neumann-to-Dirichlet map. A key ingredient in the derivation is a linearized Blagoves̆c̆enskiĭ's identity with a free parameter. When the linearization is at a constant background density, we derive two reconstructive algorithms with stability estimates based on the boundary control method. When the linearization is at a non-constant background density, we establish an increasing stability estimate for the recovery of the density perturbation. The proposed reconstruction algorithms are implemented and validated with several numerical experiments to demonstrate the feasibility.
我们针对声波方程中确定密度的逆边值问题，开发了一种线性化边界控制方法。目标是根据线性化的诺伊曼到狄利克雷映射，在已知背景密度中重构未知扰动。推导过程中的一个关键要素是带有自由参数的线性化布拉戈韦先斯基恒等式。当在恒定背景密度下进行线性化时，我们基于边界控制方法推导了两种具有稳定性估计的重构算法。当在非恒定背景密度下进行线性化时，我们为密度扰动的恢复建立了递增稳定性估计。所提出的重构算法通过若干数值实验得以实现和验证，以证明其可行性。
