Least squares solvers for ill-posed PDEs that are conditionally stable

TL;DR

This paper develops least squares methods for ill-posed PDEs with conditional stability, providing optimal error bounds without data consistency assumptions, by leveraging dual norms and inf-sup stability.

Contribution

It introduces a framework for designing least squares solvers tailored to conditionally stable ill-posed PDEs, with a focus on dual norm handling and stability verification.

Findings

Established a general error bound aligned with conditional stability

Demonstrated the effectiveness of the approach through numerical experiments

Provided a systematic way to verify stability via Fortin projectors

Abstract

This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least squares functional are determined by the ingredients of the conditional stability assumption. We are then able to establish a general error bound that, in view of the conditional stability assumption, is qualitatively the best possible, without assuming consistent data. The price for these advantages is to handle dual norms which reduces to verifying suitable inf-sup stability. This, in turn, is done by constructing appropriate Fortin projectors for all sample scenarios. The theoretical findings are illustrated by numerical experiments.