Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction

Abstract

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.

Local Hölder stabilities for inverse problems of first-order hyperbolic equations

“…Regarding local Hölder stabilities for inverse source and coefficient problems for second-order hyperbolic equations with time-dependent coefficients, readers are referred to Jiang, Liu, and Yamamoto [6], Yu, Liu, and Yamamoto [9], Bellassoued and Yamamoto [1], Klibanov and Li [7], and Takase [8].…”

Determination of source and initial values for acoustic equations with a time-fractional attenuation

“…[4,13]). Moreover, we point out that the case of t-dependent coefficients can be handled by adapting the strategy developed in [37] for a classical hyperbolic equation, to the framework of Theorems 1.2 and 1.3.…”

Inverse problems for first-order hyperbolic equations with time-dependent coefficients

“…For the second-order hyperbolic equations with time-dependent coefficients, Jiang, Liu, and Yamamoto [15], and Yu, Liu, and Yamamoto [31] proved the local Hölder stability for inverse source and coefficient problems in the Euclidean space assuming the Carleman estimates existed. Takase [28] proved also local Hölder stability and obtained some sufficient conditions for the Carleman estimate by using geometric analysis on Lorentzian manifolds.…”