A Carleman-Picard approach for reconstructing zero-order coefficients in parabolic equations with limited data

TL;DR

This paper introduces a globally convergent computational method for reconstructing zero-order coefficients in parabolic equations from limited boundary data, combining polynomial-exponential basis, Picard iteration, and Carleman estimates.

Contribution

It presents a novel reduced dimensional method that guarantees convergence without requiring a good initial guess, using Carleman estimates for the inverse problem.

Findings

Proves convergence of the iterative scheme without initial guess

Demonstrates the method's effectiveness through numerical examples

Provides a rigorous mathematical foundation using Carleman estimates

Abstract

We propose a globally convergent computational technique for the nonlinear inverse problem of reconstructing the zero-order coefficient in a parabolic equation using partial boundary data. This technique is called the "reduced dimensional method". Initially, we use the polynomial-exponential basis to approximate the inverse problem as a system of 1D nonlinear equations. We then employ a Picard iteration based on the quasi-reversibility method and a Carleman weight function. We will rigorously prove that the sequence derived from this iteration converges to the accurate solution for that 1D system without requesting a good initial guess of the true solution. The key tool for the proof is a Carleman estimate. We will also show some numerical examples.