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We consider a gradient flow of the total variation in a negative Sobolev space \(H^{-s}\,(0\le s \le 1)\) 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 solve this minimizing problem numerically and discuss convergence of a forward–backward splitting scheme. Results of several numerical experiments are given.

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