Related papers: Existence results on Lagrange multiplier approach …
This article presents a novel resolution to the problem of spline interpolation versus least-squares fitting on smooth Riemannian manifolds utilizing the method of gradient flows of networks. This approach represents a contribution to both…
MLPGradientFlow is a software package to solve numerically the gradient flow differential equation $\dot \theta = -\nabla \mathcal L(\theta; \mathcal D)$, where $\theta$ are the parameters of a multi-layer perceptron, $\mathcal D$ is some…
We propose two closely--related Lagrangian numerical methods for the simulation of physical processes involving advection, reaction and diffusion. The methods are intended to be used in settings where the flow is nearly incompressible and…
The equality between dissipation and energy drop is a structural property of gradient-flow dynamics. The classical implicit Euler scheme fails to reproduce this equality at the discrete level. We discuss two modifications of the Euler…
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 exploits the theory of geometric gradient flows to introduce an alternative regularization of the thin-film equation. The solution properties of this regularization are investigated via a sequence of numerical simulations whose…
The scarcity of labeled data is a long-standing challenge for many machine learning tasks. We propose our gradient flow method to leverage the existing dataset (i.e., source) to generate new samples that are close to the dataset of interest…
This paper concerns an extension of discrete gradient methods to finite-dimensional Riemannian manifolds termed discrete Riemannian gradients, and their application to dissipative ordinary differential equations. This includes Riemannian…
Two primary scalar auxiliary variable (SAV) approaches are widely applied for simulating gradient flow systems, i.e., the nonlinear energy-based approach and the Lagrange multiplier approach. The former guarantees unconditional energy…
We consider a general formulation of gradient flow evolution for problems whose natural framework is the one of metric spaces. The applications we deal with are concerned with the evolution of {\it capacitary measures} with respect to the…
In this work, we investigate the use of data-driven equation discovery for dynamical systems to model and forecast continuous-time dynamics of unconstrained optimization problems. To avoid expensive evaluations of the objective function and…
Mathematical Programs with Vanishing Constraints (MPVCs) are a notoriously challenging class of problems owing to their lack of constraint qualification. Therefore, to tackle these problems, relaxation-based approaches are typically used.…
This paper proposes a theoretical framework for establishing the energy dissipation of general implicit-explicit linear multistep methods (IMEX-LMMs) for gradient flows, by constructing a dissipative modified energy consisting of the…
We present a distributed Lagrange multiplier formulation of the Finite Element Immersed Boundary Method to couple incompressible fluids with compressible solids. This is a generalization of the formulation presented in Heltai and Costanzo…
The paper studies numerical methods that preserve a Lyapunov function of a dynamical system, i.e. numerical approximations whose energy decreases, just like in the original differential equation. With this aim, a discrete gradient method is…
We show numerical results of triangular cavity flow problems solved by a Lagrange-Galerkin scheme free from numerical quadrature. The scheme has recently developed by us, where a locally linearized velocity and the backward Euler…
In this work, we investigate links between the formulation of the flow of marginals of reversible diffusion processes as gradient flows in the space of probability measures and path wise large deviation principles for sequences of such…
In this paper we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition we numerically investigate the residual diffusion phenomenon in chaotic advection. The residual…
Accurate and efficient prediction of multi-scale flows remains a formidable challenge. Constructing theoretical models and numerical methods often involves the design and optimization of parameters. While gradient descent methods have been…
In this paper we introduce the essential Lagrange multiplier and establish the solid mathematical foundation of constrained optimization in Hilbert spaces with sharp results on the mathematical foundation of quadratic-programming based…