A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data

Abstract

We propose a new numerical method for the solution of the problem of the reconstruction of the initial condition of a quasilinear parabolic equation from the measurements of both Dirichlet and Neumann data on the boundary of a bounded domain. Although this problem is highly nonlinear, we do not require an initial guess of the true solution. The key in our method is the derivation of a boundary value problem for a system of coupled quasilinear elliptic equations whose solution is the vector function of the spatially dependent Fourier coefficients of the solution to the governing parabolic equation. We solve this problem by an iterative method. The global convergence of the system is rigorously established using a Carleman estimate. Numerical examples are presented.

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• 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
• Identification of the Source for Full Parabolic Equations
• The Carleman convexification method for Hamilton-Jacobi equations on the whole space
• The Carleman contraction mapping method for quasilinear elliptic equations with over-determined boundary data