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Notes on the discretization of TV-norm regularized inverse potential problems

L Baratchart, D P Hardin, C Villalobos-Guillén

TL;DR

This work develops a rigorous discretization framework for TV-regularized inverse problems where the forward map acts on 3D-valued measures supported on a fixed set. By introducing Generated-by-Singular-Measures (GSM) spaces and an R-topology that couples weak-* convergence with total-variation convergence, the authors obtain practical discretizations whose minimizers converge to the true regularized solution. They derive explicit critical-point conditions for minimizers, establish stability and error estimates under data perturbations and discretization, and prove convergence of recovered supports to the true support as discretization is refined. The resulting theory provides a solid foundation for numerically solving inverse magnetization and related problems with guarantees on convergence and support recovery. Overall, the approach offers a principled path from continuous BV-type regularization to finite-dimensional, provably convergent discretizations suitable for magnetic source localization and similar applications.

Abstract

We describe a method to discretize optimization problems arising in the regularization of linear inverse problem having compact forward operator defined on 3-D valed measures, compactly supported on a fixed set. The criterion is a quadratic residual attached to the data, with an additive penalization of the total variation of the measure.

Notes on the discretization of TV-norm regularized inverse potential problems

TL;DR

This work develops a rigorous discretization framework for TV-regularized inverse problems where the forward map acts on 3D-valued measures supported on a fixed set. By introducing Generated-by-Singular-Measures (GSM) spaces and an R-topology that couples weak-* convergence with total-variation convergence, the authors obtain practical discretizations whose minimizers converge to the true regularized solution. They derive explicit critical-point conditions for minimizers, establish stability and error estimates under data perturbations and discretization, and prove convergence of recovered supports to the true support as discretization is refined. The resulting theory provides a solid foundation for numerically solving inverse magnetization and related problems with guarantees on convergence and support recovery. Overall, the approach offers a principled path from continuous BV-type regularization to finite-dimensional, provably convergent discretizations suitable for magnetic source localization and similar applications.

Abstract

We describe a method to discretize optimization problems arising in the regularization of linear inverse problem having compact forward operator defined on 3-D valed measures, compactly supported on a fixed set. The criterion is a quadratic residual attached to the data, with an additive penalization of the total variation of the measure.
Paper Structure (6 sections, 15 theorems, 59 equations)

This paper contains 6 sections, 15 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. R-Topology and GSM spaces
  4. Critical point equations of $\mathcal{F}_{\lambda, f}$
  5. Some inequalities
  6. Convergence of the support

Key Result

Lemma 3.1

The sequence ${\boldsymbol \mu}_n\in\mathcal{M}(S)^N$ converges in the R-topology to ${\boldsymbol \mu}\in\mathcal{M}(S)^N$ if and only if ${\boldsymbol \mu}_n$ converges weak-star to ${\boldsymbol \mu}$ and $|{\boldsymbol \mu}_n|$ converges weak-star to $|{\boldsymbol \mu}|$.

Theorems & Definitions (35)

  • Lemma 3.1
  • proof
  • Remark 1
  • Lemma 3.2
  • Remark 2
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • ...and 25 more