Numerical identification of a nonlinear diffusion law via regularization in Hilbert scales

TL;DR

This paper develops a regularization method in Hilbert scales for stable numerical reconstruction of a nonlinear diffusion coefficient from boundary measurements in elliptic PDEs, with proven convergence and demonstrated numerical effectiveness.

Contribution

It introduces a novel regularization approach in Hilbert scales for nonlinear diffusion law identification, with explicit operator analysis and convergence guarantees.

Findings

Proven convergence and rates for the regularized solutions.

Numerical tests confirm theoretical predictions.

Applicable under mild assumptions on the true parameter.

Abstract

We consider the reconstruction of a diffusion coefficient in a quasilinear elliptic problem from a single measurement of overspecified Neumann and Dirichlet data. The uniqueness for this parameter identification problem has been established by Cannon and we therefore focus on the stable solution in the presence of data noise. For this, we utilize a reformulation of the inverse problem as a linear ill-posed operator equation with perturbed data and operators. We are able to explicitly characterize the mapping properties of the corresponding operators which allow us to apply regularization in Hilbert scales. We can then prove convergence and convergence rates of the regularized reconstructions under very mild assumptions on the exact parameter. These are, in fact, already needed for the analysis of the forward problem and no additional source conditions are required. Numerical tests are presented to illustrate the theoretical statements.