Related papers: Shape flows for spectral optimization problems
We explicitly construct parameter transformations between gradient flows in metric spaces, called curves of maximal slope, having different exponents when the associated function satisfies a suitable convexity condition. These…
We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open…
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…
The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy's Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification…
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…
Geometric flows related to shape optimization problems of Bernoulli type are investigated. The evolution law is the sum of a curvature term and a nonlocal term of Hele-Shaw type. We introduce generalized set solutions, the definition of…
We study evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on…
In this paper we study a gradient flow approach to the problem of quantization of measures in one dimension. By embedding our problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles…
Accelerated gradient descent iterations are widely used in optimization. It is known that, in the continuous-time limit, these iterations converge to a second-order differential equation which we refer to as the accelerated gradient flow.…
This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space $(X,\mathsf{d})$ that can be characterized by Evolution Variational Inequalities. We present new results concerning the…
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…
For addressing optimisation tasks on finite dimensional quantum systems, we give a comprehensive account of the foundations of gradient flows on Riemannian manifolds including new developments: we extend former results from Lie groups such…
We derive the system of differential equations for the gradient flow characterizing the training process of linear in-context learning in full generality. Next, we explore the geometric structure of the gradient flows in two instances,…
In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions…
When considering a general system of equations describing the space-time evolution (flow) of one or several variables, the problem of the optimization over a finite period of time of a measure of the state variable at the final time is a…
Sampling a probability distribution with an unknown normalization constant is a fundamental problem in computational science and engineering. This task may be cast as an optimization problem over all probability measures, and an initial…
Uncertainty propagation and filtering can be interpreted as gradient flows with respect to suitable metrics in the infinite dimensional manifold of probability density functions. Such a viewpoint has been put forth in recent literature, and…
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…
Total variation gradient flows are important in several applied fields, including image analysis and materials science. In this paper, we review a few basic topics including definition of a solution, explicit examples and the notion of…
The evaluation and consideration of the mean flow in wave evolution equations are necessary for the accurate prediction of fluid particle trajectories under wave groups, with relevant implications in several domains, from the transport of…