Singular solutions, graded meshes, and adaptivity for total-variation regularized minimization problems

TL;DR

This paper investigates finite element methods for total-variation regularized problems, showing that adaptive, locally refined meshes can achieve near-linear convergence even with discontinuities, under certain dual solution conditions.

Contribution

It introduces adaptive mesh refinement strategies that improve convergence rates for total-variation minimization, relaxing the need for Lipschitz continuous dual solutions.

Findings

Near-linear convergence on refined meshes

Validity conditions for dual solutions discussed

Adaptive methods outperform uniform meshes

Abstract

Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems require the existence of a Lipschitz continuous dual solution. We discuss the validity of this condition and devise numerical methods using locally refined meshes that lead to improved convergence rates despite the occurrence of discontinuities. It turns out that nearly linear convergence is possible on suitably constructed meshes.