Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction
This work provides theoretical stability guarantees and a numerical algorithm for an inverse coefficient problem in hyperbolic PDEs, which is relevant for applications like medical imaging and geophysics.
The authors establish local Hölder stability for determining the spatial component of a time-dependent coefficient in a general hyperbolic equation using Carleman estimates, and propose an iterative numerical reconstruction method based on Tikhonov regularization. Numerical examples in 1D demonstrate the method's performance.
In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators and an auxiliary Carleman estimate, we establish local Hölder stability with both partial boundary and interior measurements under certain geometrical conditions. For numerical reconstruction, we minimize a Tikhonov functional which penalizes the gradient of the unknown function. Based on the resulting variational equation, we design an iteration method which is updated by solving a Poisson equation at each step. One-dimensional prototype examples illustrate the numerical performance of the proposed iteration.