A 1-harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in â„ÂN, is formulated by the use of subdifferentials of a singular energy—the total variation. An abstract convergence result is established to show that solutions of approximate problem converge to a solution of the limit problem. As an application of our convergence result, a local-in-time solution of 1-harmonic map flow equation is constructed as a limit of the solutions of p-harmonic (p>1) map flow equation, when the initial data is smooth with small total variation under periodic boundary condition
We consider a constrained gradient system of total variation flow. Our system is often used in color image processing to remove a noise from picture. In this paper, using abstract convergence theory of convex functions, we show the global existence of solutions to our problem with piecewise constant initial data.
In this paper we study an optimal control problem for a singular diffusion equation associated with total variation energy. The singular diffusion equation is derived as an Allen-Cahn type equation, and then the observing optimal control problem corresponds to a temperature control problem in the solid-liquid phase transition. We show the existence of an optimal control for our singular diffusion equation by applying the abstract theory. Next we consider our optimal control problem from the view-point of numerical analysis. In fact we consider the approximating problem of our equation, and we show the relationship between the original control problem and its approximating one. Moreover we show the necessary condition of an approximating optimal pair, and give a numerical experiment of our approximating control problem.
Abstract. We consider a gradient flow system of total variation with constraint. Our system is often used in the color image processing to remove a noise from picture. In particular, we want to preserve the sharp edges of picture and color chromaticity. Therefore, the values of solutions to our model is constrained in some fixed compact Riemannian manifold. By this reason, it is very difficult to analyze such a problem, mathematically. The main object of this paper is to show the global solvability of a constrained singular diffusion equation associated with total variation. In fact, by using abstract convergence theory of convex functions, we shall prove the existence of solutions to our models with piecewise constant initial and boundary data.
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