The Carleman-based contraction principle to reconstruct the potential of nonlinear hyperbolic equations

TL;DR

This paper introduces a new numerical approach combining Carleman estimates and the contraction principle to efficiently solve the inverse problem of identifying potentials in nonlinear hyperbolic equations from boundary data.

Contribution

It develops a convergent numerical method that constructs a sequence of linear problems to approximate the potential, with proven convergence analysis.

Findings

Method successfully reconstructs potentials from boundary data.

Numerical examples demonstrate the efficiency and accuracy of the approach.

Convergence is established using Carleman estimates and the contraction principle.

Abstract

We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear Cauchy problems whose corresponding solutions converge to a function that can be used to efficiently compute an approximate solution to the inverse problem of interest. The convergence analysis is established by combining the contraction principle and Carleman estimates. We numerically solve the linear Cauchy problems using a quasi-reversibility method. Numerical examples are presented to illustrate the efficiency of the method.