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We consider a gradient flow of the total variation in a negative Sobolevspace $H^{-s}$ $(0\leq s \leq 1)$ 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 variationalproblem which gives an implicit-time discrete scheme for the flow. By a dualitybased method, we give a simple numerical scheme to calculate this minimizingproblem numerically and discuss convergence of a forward-backward splittingscheme. Several numerical experiments are given.

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