On $\ell^1$-regularization under continuity of the forward operator in weaker topologies

TL;DR

This paper investigates the stability and convergence of $ ext{ell}^1$-regularization for solving linear operator equations with noisy data, especially when the solution's sparsity slightly fails and the forward operator's continuity is considered in weaker topologies.

Contribution

It extends the theory of $ ext{ell}^1$-regularization to weak*-to-weak* continuous operators, including non-reflexive cases, and discusses convergence rates under variational source conditions.

Findings

Convergence rates depend on the degree of sparsity failure.

Linear convergence occurs in proper sparsity and well-posed cases.

The theory is verified with an example operator.

Abstract

Our focus is on the stable approximate solution of linear operator equations based on noisy data by using $\ell^1$-regularization as a sparsity-enforcing version of Tikhonov regularization. We summarize recent results on situations where the sparsity of the solution slightly fails. In particular, we show how the recently established theory for weak*-to-weak continuous linear forward operators can be extended to the case of weak*-to-weak* continuity. This might be of interest when the image space is non-reflexive. We discuss existence, stability and convergence of regularized solutions. For injective operators, we will formulate convergence rates by exploiting variational source conditions. The typical rate function obtained under an ill-posed operator is strictly concave and the degree of failure of the solution sparsity has an impact on its behavior. Linear convergence rates just occur in the two borderline cases of proper sparsity, where the solutions belong to $\ell^0$, and of well-posedness. For an exemplary operator, we demonstrate that the technical properties used in our theory can be verified in practice. In the last section, we briefly mention the difficult case of oversmoothing regularization where $x^\dag$ does not belong to $\ell^1$.