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Register a new userOn the total variation Wasserstein gradient flow and the TV-JKO scheme
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We study the JKO scheme for the total variation, characterize the optimizers, prove some of their qualitative properties (in particular a form of maximum principle and in some cases, a minimum principle as well). Finally, we establish a convergence result as the time step goes to zero to a solution of a fourth-order nonlinear evolution equation, under the additional assumption that the density remains bounded away from zero. This lower bound is shown in dimension one and in the radially symmetric case.
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In this paper we study the Total Variation Flow (TVF) in metric random walk spaces, which unifies into a broad framework the TVF on locally finite weighted connected graphs, the TVF determined by finite Markov chains and some nonlocal evolution problems. Once the existence and uniqueness of solutions of the TVF has been proved, we study the asymptotic behaviour of those solutions and, with that aim in view, we establish some inequalities of Poincar{e} type. In particular, for finite weighted connected graphs, we show that the solutions reach the average of the initial data in finite time. Furthermore, we introduce the concepts of perimeter and mean curvature for subsets of a metric random walk space and we study the relation between isoperimetric inequalities and Sobolev inequalities. Moreover, we introduce the concepts of Cheeger and calibrable sets in metric random walk spaces and characterize calibrability by using the $1$-Laplacian operator. Finally, we study the eigenvalue problem whereby we give a method to solve the optimal Cheeger cut problem.
We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $(mathcal{X}, d, mu)$ satisfying a doubling condition and supporting a Poincare inequality. For such parabolic minimizers that coincide with a time-independent Cauchy-Dirichlet datum $u_0$ on the parabolic boundary of a space-time-cylinder $Omega times (0, T)$ with $Omega subset mathcal{X}$ an open set and $T > 0$, we prove existence in the weak parabolic function space $L^1_w(0, T; mathrm{BV}(Omega))$. In this paper, we generalize results from a previous work by Bogelein, Duzaar and Marcellini by introducing a more abstract notion for $mathrm{BV}$-valued parabolic function spaces. We argue completely on a variational level.
This work considers a chemotaxis system for multi-species that includes birth or death rate terms, which implies no mass preservation of the populations. We aim to show the convergence to a $L^{infty} - $weak solutions, that is local in time, of the JKO - scheme arising from the Optimal Transport Theory, in the spirit of [35,14]. Currently, $L^{infty}$ solutions have shown to be important in order to get uniqueness. Since death rate case does not ensure global solutions, for arbitrary initial data, in this framework, it could be interest to analyze the Blowing-up phenomenon of this system. Therefore, in the last section, we get sufficient conditions that implies blowing-up phenomenon in finite time and we draw several stages where this occurs. This last part can be seen as a partial generalization of the blowing-up results in [16].
We study a non-local version of the Cahn-Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. In difference to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes --- but not necessarily the fluxes themselves --- annihilate each other. Our main result is a rigorous proof of existence of weak solutions. The starting point is the formal representation of the dynamics as a constrained gradient flow in the Wasserstein metric. We then show that time-discrete approximations by means of the incremental minimizing movement scheme converge to a weak solution in the limit. Further, we compare the non-local model to the classical Cahn-Hilliard model in numerical experiments. Our results illustrate the significant speed-up in the decay of the free energy due to the higher degree of freedom for the velocity fields.
In this paper, we focus on non-asymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (non-constant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step (usually called Unajusted Langevin Algorithm in the Monte-Carlo literature) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under uniform ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for Total Variation and L 1-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using Stochastic Analysis adapted to decreasing step setting.
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