Some applications of weighted norm inequalities to the error analysis of PDE constrained optimization problems
For researchers in PDE-constrained optimization, this provides a unified weighted-norm framework that simplifies analysis and discretization of problems with singularities or nonuniform ellipticity, though the approach is incremental.
This work applies Muckenhoupt weighted norm inequalities to analyze and discretize three PDE-constrained optimization problems, including nonuniformly elliptic equations, pointwise observations, and singular sources. Error estimates are provided in 2D and 3D, with the approach offering simpler Hilbert space-based analysis and discretization.
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.