Error estimates for total-variation regularized minimization problems with singular dual solutions

TL;DR

This paper proves that Lipschitz continuity of dual solutions is not necessary for error estimates in total-variation regularized minimization, and derives bounds based on Sobolev regularity using Lipschitz truncation.

Contribution

It removes the need for Lipschitz continuity assumptions and provides new error estimates based on Sobolev regularity for dual solutions.

Findings

Lipschitz continuity of dual solutions is not required for error estimates.

Error bounds can be derived based on Sobolev regularity.

Analytic proofs support the new error estimation approach.

Abstract

Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems using the Crouzeix--Raviart finite element require the existence of a Lipschitz continuous dual solution, which is not generally given. We provide analytic proofs showing that the Lipschitz continuity of a dual solution is not necessary, in general. Using the Lipschitz truncation technique, we, in addition, derive error estimates that depend directly on the Sobolev regularity of a given dual solution.