Related papers: Convergence of a particle method for diffusive gra…
A nonlinear diffusion equation, interpreted as a Wasserstein gradient flow, is numerically solved in one space dimension using a higher-order minimizing movement scheme based on the BDF (backward differentiation formula) discretization. In…
The paper contains the proof that the diffusion ensemble of point wise particles with the intensity depending on the grain of spatial resolution serves as the satisfactory approximation of one quantum particle dynamics.
Existing analyses of optimization in deep learning are either continuous, focusing on (variants of) gradient flow, or discrete, directly treating (variants of) gradient descent. Gradient flow is amenable to theoretical analysis, but is…
As a counterpoint to classical stochastic particle methods for diffusion, we develop a deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic particle methods are incompatible with diffusive partial…
To increase the reliability of simulations by particle methods for incompressible viscous flow problems, convergence studies and improvements of accuracy are considered for a fully explicit particle method for incompressible Navier--Stokes…
Many tasks in machine learning and signal processing can be solved by minimizing a convex function of a measure. This includes sparse spikes deconvolution or training a neural network with a single hidden layer. For these problems, we study…
In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality…
In this paper we establish a rigorous gradient flow structure for one-dimensional Kimura equations with respect to some Wasserstein-Shahshahani optimal transport geometry. This is achieved by first conditioning the underlying stochastic…
We present and study a Particle method for the stationary solutions of a class of transport equations. This method is inspired by non-stationary Particle methods, the time variable being replaced by one spatial variable. Particles…
A popular method to perform adversarial attacks on neuronal networks is the so-called fast gradient sign method and its iterative variant. In this paper, we interpret this method as an explicit Euler discretization of a differential…
We consider a fully discrete and explicit scheme for the mean curvature flow of boundaries, based on an elementary diffusion step and a precise redistancing operation. We give an elementary convergence proof for the scheme under the…
When applying the finite-differences method to numerically solve the one-dimensional diffusion equation, one must choose discretization steps $\Delta x$, $\Delta t$ in space and time, respectively. By applying large-deviation theory on the…
This article overviews how gradient flows, and discretizations thereof, are useful to design and analyze optimization and sampling algorithms. The interplay between optimization, sampling, and gradient flows is an active research area; our…
This paper is devoted to the investigation of gradient flows in asymmetric metric spaces (for example, irreversible Finsler manifolds and Minkowski normed spaces) by means of discrete approximation. We study basic properties of curves and…
As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method…
We present a novel approximate inference method for diffusion processes, based on the Wasserstein gradient flow formulation of the diffusion. In this formulation, the time-dependent density of the diffusion is derived as the limit of…
By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the…
Pinched flow fractionation is shown to be an efficient and selective way to quickly separate particles by size in a very polydisperse semi-concentrated suspension. In an effort to optimize the method, we discuss the quantitative influence…
Elastic flow for closed curves can involve significant deformations. Mesh-based approximation schemes require tangentially redistributing vertices for long-time computations. We present and analyze a method that uses the Dirichlet energy…
We provide the proof of convergence of the directional diffusion splitting scheme for two-dimensional parabolic and elliptic advection-diffusion-reaction problems with certain restrictions on problem data