Carleman estimates and the contraction principle for an inverse source problem for nonlinear hyperbolic equation

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.

References in corpus (12)

• Thermoacoustic tomography with variable sound speed
• Thermoacoustic tomography arising in brain imaging
• Time reversal in thermoacoustic tomography - an error estimate
• The d-bar approach to approximate inverse scattering at fixed energy in three dimensions
• Inversion of circular means and the wave equation on convex planar domains
• Electrical Impedance Tomography with Restricted Dirichlet-to-Neumann Map Data
• A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data
• Photoacoustic Imaging Taking into Account Attenuation
• An inverse source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method
• The gradient descent method for the convexification to solve boundary value problems of quasi-linear PDEs and a coefficient inverse problem
• Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data
• Convexification for a 3D inverse scattering problem with the moving point source

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• The Carleman contraction mapping method for quasilinear elliptic equations with over-determined boundary data
• Determining initial conditions for nonlinear hyperbolic equations with time dimensional reduction and the Carleman contraction