Some applications of weighted norm inequalities to the error analysis of PDE constrained optimization problems

TL;DR

This paper demonstrates how weighted norm inequalities and Muckenhoupt weights can be effectively used to analyze and discretize PDE-constrained optimization problems, simplifying the process and enabling broader generalizations.

Contribution

It introduces a Hilbert space-based approach using weighted inequalities for PDE-constrained optimization, applicable to problems with nonuniform ellipticity, pointwise observations, and singular sources.

Findings

Developed error estimates for numerical schemes in 2D and 3D

Simplified analysis using weighted Sobolev spaces

Extended applicability to problems with singular sources

Abstract

The purpose of this work is to illustrate how the theory of Muckenhoupt weights, Muckenhoupt weighted Sobolev spaces and the corresponding weighted norm inequalities can be used in the analysis and discretization of PDE constrained optimization problems. We consider: a linear quadratic constrained optimization problem where the state solves a nonuniformly elliptic equation; a problem where the cost involves pointwise observations of the state and one where the state has singular sources, e.g. point masses. For all three examples we propose and analyze numerical schemes and provide error estimates in two and three dimensions. While some of these problems might have been considered before in the literature, our approach allows for a simpler, Hilbert space-based, analysis and discretization and further generalizations.