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In this paper we present a computer-assisted procedure for proving the existence of transverse heteroclinic orbits connecting hyperbolic equilibria of polynomial vector fields. The idea is to compute high-order Taylor approximations of…
Many recent studies on first-order methods (FOMs) focus on \emph{composite non-convex non-smooth} optimization with linear and/or nonlinear function constraints. Upper (or worst-case) complexity bounds have been established for these…
Improved local numerical solution for the ADER-DG numerical method with a local DG predictor for solving the initial value problem for a first-order ODE system is proposed. The improved local numerical solution demonstrates convergence…
First order methods endowed with global convergence guarantees operate using global lower bounds on the objective. The tightening of the bounds has been shown to increase both the theoretical guarantees and the practical performance. In…
We provide an interior point method based on quasi-Newton iterations, which only requires first-order access to a strongly self-concordant barrier function. To achieve this, we extend the techniques of Dunagan-Harvey [STOC '07] to maintain…
A framework is developed for applying accelerated methods to general hyperbolic programming, including linear, second-order cone, and semidefinite programming as special cases. The approach replaces a hyperbolic program with a convex…
The discovery of conservation laws is a cornerstone of scientific progress. However, identifying these invariants from observational data remains a significant challenge. We propose a hybrid framework to automate the discovery of conserved…
We consider unconstrained minimization of smooth convex functions. We propose a novel variational perspective using forced Euler-Lagrange equation that allows for studying high-resolution ODEs. Through this, we obtain a faster convergence…
Simulations of the dynamics generated by partial differential equations (PDEs) provide approximate, numerical solutions to initial value problems. Such simulations are ubiquitous in scientific computing, but the correctness of the results…
A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1st. order ODE-solver and routines for, among other things: the…
This study addresses some algorithms for solving structured unconstrained convex optimiza- tion problems using first-order information where the underlying function includes high-dimensional data. The primary aim is to develop an…
In this paper, we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems. The algorithm is based on availability of families of stable finite element pairs and on the availability of…
We design and compute first-order implicit-in-time variational schemes with high-order spatial discretization for initial value gradient flows in generalized optimal transport metric spaces. We first review some examples of gradient flows…
This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide…
We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…
In this paper, we propose first-order feasible methods for difference-of-convex (DC) programs with smooth inequality and simple geometric constraints. Our strategy for maintaining feasibility of the iterates is based on a "retraction" idea…
In this paper, a new implicit-explicit local method with an arbitrary order is produced for stiff initial value problems. Here, a general method for one-step time integrations has been created, considering a direction free approach for…
In this paper, we study a class of stochastic and finite-sum convex optimization problems with deterministic constraints. Existing methods typically aim to find an $\epsilon$-$expectedly\ feasible\ stochastic\ optimal$ solution, in which…
The goal of this work is to fill a gap in [Yang, SIAM J. Matrix Anal. Appl, 41 (2020), 1797--1825]. In that work, an approximation procedure was proposed for orthogonal low-rank tensor approximation; however, the approximation lower bound…
We propose a numerical method to solve general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on non-uniform grids. The properties of the method are based on the introduction of an…