A duality based approach to the minimizing total variation flow in the space

We consider a gradient flow of the total variation in a negative Sobolev space under the periodic boundary condition. If , the flow is nothing but the classical total variation flow. If , 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 solve this minimizing problem numerically and discuss convergence of a forward-backward splitting scheme. Results of several numerical experiments are given.