Doubly Degenerate Diiusion Equations as Steepest Descent
نویسنده
چکیده
For p 2 (1; 1) and n > 0 we consider the scalar doubly degenerate diiusion equation @ t s ? div(jrs n j p?2 rs n) = 0 in (0; 1) (1) with no{{ux boundary conditions. We argue that this evolution problem can be understood as steepest descent of the convex functional sign(m ? 1) Z s m ; provided m := n + p ? 2 p ? 1 > 0 ; (2) w. r. t. the Wasserstein metric of order p on the space of probability densities. This appearently new viewpoint is diierent and in some respects (slow vs. fast diiusion and similarity solutions) more natural than the conventional interpretation of (1) as steepest descent of the functional Z s n+1 w. r. t. the H ?1;p {metric. Our interpretation is made rigorous by a convergence result for a time{discrete scheme which involves the functional (2) and the Wasserstein metric.
منابع مشابه
On Discontinuity-preserving Optic Flow
We investigate a modiication of Horn and Schunck's approach which leads to a better preservation of ow discontinuities. It replaces the quadratic smoothness term by a nonquadratic one. Energy minimization by steepest descent leads to a system of two nonlinear diiusion{reaction equations with a single common nonlinear diiusivity. This enables us to make use from techniques for nonlinear diiusion...
متن کاملResidual norm steepest descent based iterative algorithms for Sylvester tensor equations
Consider the following consistent Sylvester tensor equation[mathscr{X}times_1 A +mathscr{X}times_2 B+mathscr{X}times_3 C=mathscr{D},]where the matrices $A,B, C$ and the tensor $mathscr{D}$ are given and $mathscr{X}$ is the unknown tensor. The current paper concerns with examining a simple and neat framework for accelerating the speed of convergence of the gradient-based iterative algorithm and ...
متن کاملOn mean curvature diffusion in nonlinear image filtering
Mean curvature diiusion is shown to be a position vector diiusion, tending to scalar diiusion as a at image region is approached, and providing noise removal by steepest descent surface minimization. At edges, it switches to a nondiiusion state due to two factors: the Laplacian of position vanishes and the magnitude of the surface normal attains a local maximum.
متن کاملExistence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory
Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Abstract We obtain solutions of the nonlinear degenerate parabolic equation ∂ ρ ∂ t = div ρ ∇c ⋆ [ ∇ (F ′ (ρ) + V) ] as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires less uniform convexity assumption than that imposed by Alt and ...
متن کاملA new Levenberg-Marquardt approach based on Conjugate gradient structure for solving absolute value equations
In this paper, we present a new approach for solving absolute value equation (AVE) whichuse Levenberg-Marquardt method with conjugate subgradient structure. In conjugate subgradientmethods the new direction obtain by combining steepest descent direction and the previous di-rection which may not lead to good numerical results. Therefore, we replace the steepest descentdir...
متن کامل