Related papers: Optimization, Isoperimetric Inequalities, and Samp…

In this paper we propose a multigrid optimization algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind. This approach is enabled by the fact that the solution of the variational…

Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the…

Poincar\'e inequality is a fundamental property that rises naturally in different branches of mathematics. The associated Poincar\'e constant plays a central role in many applications since it governs the convergence of various practical…

We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev inequality to the best constants for…

We prove that every probability measure $\mu$ satisfying the stationary Fokker-Planck-Kolmogorov equation obtained by a $\mu$-integrable perturbation $v$ of the drift term $-x$ of the Ornstein-Uhlenbeck operator is absolutely continuous…

We introduce a notion of "gradient at a given scale" of functions defined on a metric measure space. We then use it to define Sobolev inequalities at large scale and we prove their invariance under large-scale equivalence (maps that…

This paper is devoted to the construction and study of an equilibrium Glauber-type dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space $\Gamma$ of all locally…

Training sparse networks to converge to the same performance as dense neural architectures has proven to be elusive. Recent work suggests that initialization is the key. However, while this direction of research has had some success,…

In this paper, we consider functionals of the form $H_\alpha(u)=F(u)+\alpha G(u)$ with $\alpha\in[0,+\infty)$, where $u$ varies in a set $U\neq\emptyset$ (without further structure). We first revisit a result stating that, excluding at most…

Gradient-flow (GF) viewpoints unify and illuminate optimization algorithms, yet most GF analyses focus on unconstrained settings. We develop a geometry-respecting framework for constrained problems by (i) reparameterizing feasible sets with…

We consider the problem of recovering a real-valued $n$-dimensional signal from $m$ phaseless, linear measurements and analyze the amplitude-based non-smooth least squares objective. We establish local convergence of subgradient descent…

Using tools from the theory of optimal transport, we establish several results concerning isometric actions of amenable topological groups with potentially unbounded orbits. Specifically, suppose $d$ is a compatible left-invariant metric on…

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $\Sigma$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current…

We introduce a novel primal-dual flow for affine constrained convex optimization problems. As a modification of the standard saddle-point system, our primal-dual flow is proved to possess the exponential decay property, in terms of a…

We develop a thermodynamic formalism for a strongly dissipative H\'enon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. For any $t\in\mathbb R$…

We study properties of an attractive-repulsive energy functional based on power-kernels, which can be used for halftoning of images. In the first part of this work, using a variational framework for probability measures, we examine…

We consider ultraweak variational formulations for (parametrized) linear first order transport equations in time and/or space. Computationally feasible pairs of optimally stable trial and test spaces are presented, starting with a suitable…

Let $\mu>0$ be a fixed constant, and we prove that minimizers to the following energy functional \begin{align*} E_f(u,\Omega):=\int_{\Omega}|\nabla u|^2+\mu P(\Omega) \end{align*}exist among pairs $(\Omega,u)$ such that $\Omega$ is an…

Motivated by recent work of Renegar, we present new computational methods and associated computational guarantees for solving convex optimization problems using first-order methods. Our problem of interest is the general convex optimization…

Sampling Gibbs measures at low temperatures is an important task but computationally challenging. Numerical evidence suggests that the infinite-swapping algorithm (isa) is a promising method. The isa can be seen as an improvement of the…