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.

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Cited by in corpus (4)

• A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications
• Carleman contraction mapping for a 1D inverse scattering problem with experimental time-dependent data
• 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