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

TL;DR

This paper introduces a new convergent numerical method for reconstructing the initial condition of nonlinear parabolic equations from boundary measurements, avoiding the need for an initial guess and ensuring global convergence.

Contribution

The paper develops a novel iterative boundary value problem approach with proven global convergence for nonlinear parabolic inverse problems.

Findings

Method successfully reconstructs initial conditions from boundary data.

Global convergence is rigorously established using Carleman estimates.

Numerical examples demonstrate effectiveness and accuracy.

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.