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

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.