A duality based approach to the minimizing total variation flow in the space $H^{-s}$

TL;DR

This paper introduces a duality-based numerical method for minimizing total variation flows in negative Sobolev spaces, unifying classical and higher-order flows, with convergence analysis and numerical experiments.

Contribution

It develops a novel duality-based approach for discretizing and solving total variation flows in $H^{-s}$ spaces, including convergence and practical implementation.

Findings

The scheme effectively computes total variation flows in $H^{-s}$ spaces.

Numerical experiments demonstrate the method's accuracy and stability.

Convergence of the forward-backward splitting scheme is established.

Abstract

We consider a gradient flow of the total variation in a negative Sobolev space $H^{-s}$ $(0\leq s \leq 1)$ under the periodic boundary condition. If $s=0$, the flow is nothing but the classical total variation flow. If $s=1$, this is the fourth order total variation flow. We consider a convex variational problem which gives an implicit-time discrete scheme for the flow. By a duality based method, we give a simple numerical scheme to calculate this minimizing problem numerically and discuss convergence of a forward-backward splitting scheme. Several numerical experiments are given.