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

We consider a gradient ow of the total variation in a negative Sobolev space H s (0 s 1) under the periodic boundary condition. If s = 0, the ow is nothing but the classical total variation ow. If s = 1, this is the fourth order total variation ow. We consider a convex variational problem which gives an implicit-time discrete scheme for the ow. 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