Convergence guarantees for coefficient reconstruction in PDEs from boundary measurements by variational and Newton type methods via range invariance

TL;DR

This paper establishes convergence guarantees for regularization methods in PDE coefficient reconstruction from boundary data by leveraging range invariance and penalization techniques, applicable to elliptic and time-dependent PDEs.

Contribution

It introduces a novel approach to achieve convergence in PDE coefficient identification by extending parameters and then restoring dependencies through penalization.

Findings

Convergence of variational and Newton methods is proven under range invariance conditions.

The framework applies to both reduced and all-at-once PDE settings.

Three examples demonstrate the effectiveness in elliptic and time-dependent PDEs.

Abstract

A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations -- such as coefficient identification in partial differential equiations (PDE)s from boundary observations -- can often be achieved by extending the seached for parameter in the sense of allowing it to depend on additional variables. This clearly counteracts unique identifiability of the parameter, though. The second key idea of this paper is now to restore the original restricted dependency of the parameter by penalization. This is shown to lead to convergence of variational (Tikhonov type) and iterative (Newton type) regularization methods. We concretize the abstract convergence analysis in a framework typical of parameter identification in PDEs in a reduced and an all-at-once setting. This is further illustrated by three examples of coefficient identification from boundary observations in elliptic and time-dependent PDEs.