相关论文: Kurdyka-Lojasiewicz-Simon inequality for gradient …

Before proving (unconditional) energy stability for gradient flows, most existing studies either require a strong Lipschitz condition regarding the non-linearity or certain $L^{\infty}$ bounds on the numerical solutions (the maximum…

In Athreya, L\"ohr, Winter (2016), an invariance principle is stated for a class of strong Markov processes on tree-like metric measure spaces. It is shown that if the underlying spaces converge Gromov vaguely, then the processes converge…

This paper addresses the gradient flow -- the continuous-time representation of the gradient method -- with the smooth approximation of a non-differentiable objective function and presents convergence analysis framework. Similar to the…

Stochastic Gradient Descent (SGD) has been the method of choice for learning large-scale non-convex models. While a general analysis of when SGD works has been elusive, there has been a lot of recent progress in understanding the…

The time decay of fully discrete finite-volume approximations of porous-medium and fast-diffusion equations with Neumann or periodic boundary conditions is proved in the entropy sense. The algebraic or exponential decay rates are computed…

Stochastic gradient optimization methods are broadly used to minimize non-convex smooth objective functions, for instance when training deep neural networks. However, theoretical guarantees on the asymptotic behaviour of these methods…

We consider multi-gradient fluids endowed with a volumetric internal energy which is a function of mass density, volumetric entropy and their successive gradients. We obtained the thermodynamic forms of equation of motions and equation of…

We construct a new finite difference method for the flow of ideal viscous isentropic gas in one spatial dimension. For the continuity equation, the method is a standard upwind discretization. For the momentum equation, the method is an…

The Gromov-Wasserstein (GW) distance quantifies discrepancy between metric measure spaces and provides a natural framework for aligning heterogeneous datasets. Alas, as exact computation of GW alignment is NP hard, entropic regularization…

We study the time evolution of velocity and pressure gradients in isotropic turbulence, by quantifying their decorrelation time scales as one follows fluid particles in the flow. The Lagrangian analysis uses data in a public database…

The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard…

This work is the third part of a program initiated in arXiv:2111.13258, arXiv:2302.06571 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in…

We derive a universal thermodynamic bound constraining directional transport in both discrete and continuous nonequilibrium systems. For continuous-time Markov jump processes and overdamped diffusions governed by Fokker--Planck equations,…

We study the high-frequency limit of non-autonomous gradient flows in metric spaces of energy functionals comprising an explicitly time-dependent perturbation term which might oscillate in a rapid way, but fulfills a certain Lipschitz…

We consider singularly perturbed gradient flows in Hilbert spaces, driven by a time-dependent, nonconvex, and nonsmooth energy, and address the convergence of their solutions to curves of critical points of the driving energy functional.…

This work is the second part of a program initiated in arXiv:2111.13258 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in metric spaces.…

We investigate the Moreau-Yosida regularization and the associated proximal map in the context of discrete gradient flow for the 2-Wasserstein metric. Our main results are a stepwise contraction property for the proximal map and an "above…

We prove the convergence of a Wasserstein gradient flow of a free energy in inhomogeneous media. Both the energy and media can depend on the spatial variable in a fast oscillatory manner. In particular, we show that the gradient-flow…

In this article we study the anisotropic curve shortening flow for a planar network of three curves with fixed endpoints and which meet in a triple junction. We show that the anisotropic curvature energy fulfills a Lojasiewicz-Simon…

This paper is devoted to logarithmic Hardy-Littlewood-Sobolev inequalities in the two-dimensional Euclidean space, in presence of an external potential with logarithmic growth. The coupling with the potential introduces a new parameter,…