Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction. (15th February 2018)

• Record Type: Journal Article
• Authors: Yu, Jie; Liu, Yikan; Yamamoto, Masahiro
• Is Part Of: Inverse problems. Volume 34:Number 4(2018:Apr.)
• Publication Date: 2018-02-15
• Subjects: hyperbolic equation -- coefficient inverse problem -- Carleman estimate -- local Hölder stability -- iteration method; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aaa4a0
• Languages: English
• ISSNs: 0266-5611

Abstract

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 either partial boundary or 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.