Related papers: Metric-driven numerical methods
We consider the proximal gradient method on Riemannian manifolds for functions that are possibly not geodesically convex. Starting from the forward-backward-splitting, we define an intrinsic variant of the proximal gradient method that uses…
For many optimization problems it is possible to define a distance metric between problem variables that correlates with the likelihood and strength of interactions between the variables. For example, one may define a metric so that the…
We introduce the Optimizing a Discrete Loss (ODIL) framework for the numerical solution of Partial Differential Equations (PDE) using machine learning tools. The framework formulates numerical methods as a minimization of discrete residuals…
In this paper, we propose a random gradient-free method for optimization in infinite dimensional Hilbert spaces, applicable to functional optimization in diverse settings. Though such problems are often solved through finite-dimensional…
We present a new numerical technique to solve large-scale eigenvalue problems. It is based on the projection technique, used in strongly correlated quantum many-body systems, where first an effective approximate model of smaller complexity…
We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully…
The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in…
Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient…
We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been…
We consider stochastic differential equations in a Hilbert space, perturbed by the gradient of a convex potential. We investigate the problem of convergence of a sequence of such processes. We propose applications of this method to…
We propose a deep learning based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz method is naturally nonlinear, naturally…
This paper deals with nonsmooth convex optimization problems in Euclidean spaces. We identify special elements of the subdifferential of a convex function, called specular gradients. Based on this observation, we propose three numerical…
We study the variable metric forward-backward splitting algorithm for convex minimization problems without the standard assumption of the Lipschitz continuity of the gradient. In this setting, we prove that, by requiring only mild…
In practice, optimization tasks have some structure that allows developing new algorithms for every problem with faster convergence rates. Using the structure of optimization tasks, we can propose algorithms with more optimistic convergence…
In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering…
In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the…
Optimal exploitation of supercomputing resources for the evaluation of electrostatic forces remains a challenge in molecular dynamics simulations of very large systems. The most efficient methods are currently based on particle-mesh Ewald…
This paper investigates a category of constrained fractional optimization problems that emerge in various practical applications. The objective function for this category is characterized by the ratio of a numerator and denominator, both…
We propose a first-order method for solving inequality constrained optimization problems. The method is derived from our previous work [12], a modified search direction method (MSDM) that applies the singular-value decomposition of…
Problems of quadratic optimization in Hilbert space often arise when solving ill-posed problems for differential equations. In this case, the target value of the functional is known. In addition, the structure of the functional allows…