The Grothendieck duality and sparse minimizing in spaces of Sobolev solutions to elliptic systems

TL;DR

This paper explores the use of Banach spaces of Sobolev solutions to elliptic operators for inverse problems, establishing sparse minimizers and a representer theorem using Grothendieck duality, with implications for neural network regularization.

Contribution

It introduces a novel approach leveraging Grothendieck duality in Sobolev spaces to prove the existence of sparse minimizers and a representer theorem for elliptic PDE-based inverse problems.

Findings

Existence of sparse minimizers in Sobolev solution spaces.

A representer theorem utilizing fundamental solutions as kernels.

Discussion of the infinite data limit case.

Abstract

We present an instructive example of using Banach spaces of solutions to (linear, generally, non-scalar) elliptic operator $A$ to investigate variational inverse problems related to neural networks and/or to regularization of solutions to boundary value problems. More precisely, inspired by kernel's method for optimization problems in locally convex spaces, we prove the existence of the so-called sparse minimizers for the related variational problem and produce a representer theorem where a suitable fundamental solution of the operator $A$ is used as a reproducing kernel. The Grothendieck type duality for the Sobolev spaces of solutions to elliptic operator $A$ plays an essential role in the considerations. The case where the number of data passes to infinity is also discussed. Some typical situations related to the standard elliptic operators, the corresponding function spaces and fundamental solutions are considered.