Sparse recovery for inverse potential problems in divergence form

Laurent Baratchart,Juliette Leblond,Cristobal Villalobos-Guillen,Douglas Hardin,Edward Saff

doi:10.1088/1742-6596/1476/1/012009

Abstract

We discuss recent results from [10] on sparse recovery for inverse potential problem with source term in divergence form. The notion of sparsity which is set forth is measure- theoretic, namely pure 1-unrectifiability of the support. The theory applies when a superset of the support is known to be slender, meaning it has measure zero and all connected components of its complement has infinite measure in ℝ3. We also discuss open issues in the non-slender case.

Highlights

Inverse potential problems with source term in divergence form consist in recovering a R3-valued distribution μ, knowing the potential Φ of div μ which is the solution to the Poisson-Hodge equation ∆Φ = div μ having “least growth” at infinity
In the preprint [10], notions of sparsity have been introduced concerning μ, when the latter is a finite R3-valued measure. They justify the use of Tikhonov-like regularization schemes that minimize the residuals while penalizing the total variation norm, in order to asymptotically reconstruct a sparse measure μ when the regularization parameter goes to zero, under a specific assumption on S: it should be slender, meaning it has measure zero and each connected component of R3 \ S has infinite measure
As expected from the non-reflexive character of spaces of measures, consistency estimates hold in the sense of weak-∗ convergence of subsequences to solutions of minimum total variation, or convergence in the Bregman distance when the so-called source condition holds

Summary

Introduction

As expected from the non-reflexive character of spaces of measures, consistency estimates hold in the sense of weak-∗ convergence of subsequences to solutions of minimum total variation, or convergence in the Bregman distance when the so-called source condition holds (which is, by the way, not the case here) In principle, such methods yield algorithms to approximate a solution of minimum total variation to the initial equation by a sequence of discrete measures, but imply nothing about the nature of limit points as discrete measures are weak-∗ dense in the space of measures supported on an open subset of Rn. We note that an infinite-dimensional recovery result for sparse measures, in the sense of being a sum of Dirac masses, was established in [17] for 1-D deconvolution issues, where a train of spikes is to be recovered from filtered observation thereof. We should stress in our case that the convolution kernel is singular and the null-space of the forward operator has infinite dimension

The inverse problem

Slenderness and null-space of the forward operator