2021/08/31 by Loc H. Nguyen, Michael V. Klibanov
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Applied mathematics #Contraction (grammar) #Contraction mapping #Contraction principle #Fixed point #Geometry #Hyperbolic partial differential equation #Internal medicine #Inverse #Inverse problem #Mathematical analysis #Mathematics #Microwave Imaging and Scattering Analysis #Nonlinear system #Numerical methods in inverse problems #Partial differential equation #cs.NA #math.AP #math.NA #msc:35R30 #msc:65M32
paper · pdf · doi:10.1088/1361-6420/ac4d09
arxiv created 2022/01/18 · openalex publication_date 2022/01/19 · openalex created_date 2022/01/26 · arxiv updated 2022/02/16 · openalex updated_date 2026/06/26
Abstract The main aim of this paper is to solve an inverse source problem for a general nonlinear hyperbolic equation. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the inverse problem. To find this fixed point, we define a recursive sequence with an arbitrary initial term by the same manner as in the classical proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution with the exponential rate. Therefore, our new method can be considered as an analog of the contraction principle. We rigorously study the stability of our method with respect to noise. Numerical examples are presented.