Related papers: Nonlocal Solutions to Dynamic Equilibrium Models: …

Many practical approximations in physics and engineering invoke a relatively long physical domain with a relatively thin cross-section. In this scenario we typically expect the system to have structures that vary slowly in the long…

This work presents a method for constructing online-efficient reduced models of large-scale systems governed by parametrized nonlinear scalar conservation laws. The solution manifolds induced by transport-dominated problems such as…

This work presents a unified framework that combines global approximations with locally built models to handle challenging nonconvex and nonsmooth composite optimization problems, including cases involving extended real-valued functions. We…

We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only…

One of the very few mathematically rigorous nonlinear model reduction methods is the restriction of a dynamical system to a low-dimensional, sufficiently smooth, attracting invariant manifold. Such manifolds are usually found using local…

This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our…

This paper presents smoothing schemes for obtaining approximate stationary points of unconstrained or linearly-constrained composite nonconvex-concave min-max (and hence nonsmooth) problems by applying well-known algorithms to composite…

In this paper, we propose a new method that combines the inexact Newton method with a procedure to obtain a feasible inexact projection for solving constrained smooth and nonsmooth equations. The local convergence theorems are established…

We introduce novel approximate systems for dispersive and diffusive-dispersive equations with nonlinear fluxes. For purely dispersive equations, we construct a first-order, strictly hyperbolic approximation. Local well-posedness of smooth…

A solution manifold is the collection of points in a $d$-dimensional space satisfying a system of $s$ equations with $s<d$. Solution manifolds occur in several statistical problems including hypothesis testing, curved-exponential families,…

Motivated by the mathematics literature on the algebraic properties of so-called polynomial vector flows, we propose a technique for approximating nonlinear differential equations by linear differential equations. Although the idea of…

We extend the class of SQP methods for equality constrained optimization to the setting of differentiable manifolds. The use of retractions and stratifications allows us to pull back the involved mappings to linear spaces. We study local…

Although stable solutions of dynamical systems are typically considered more important than unstable ones, unstable solutions have a critical role in the dynamical integrity of stable solutions. In fact, usually, basins of attraction…

We develop model-based methods for solving stochastic convex optimization problems, introducing the approximate-proximal point, or aProx, family, which includes stochastic subgradient, proximal point, and bundle methods. When the modeling…

We examine a wide class of stochastic approximation algorithms for solving (stochastic) nonlinear problems on Riemannian manifolds. Such algorithms arise naturally in the study of Riemannian optimization, game theory and optimal transport,…

Stochastic nonconvex optimization problems with nonlinear constraints have a broad range of applications in intelligent transportation, cyber-security, and smart grids. In this paper, first, we propose an inexact-proximal accelerated…

We develop a validated numerical procedure for continuation of local stable/unstable manifold patches attached to equilibrium solutions of ordinary differential equations. The procedure has two steps. First we compute an accurate high order…

Two algorithms are proposed, analyzed, and tested for solving continuous optimization problems with nonlinear equality constraints. Each is an extension of a stochastic momentum-based method from the unconstrained setting to the setting of…

While it is well known that nonlinear methods of approximation can often perform dramatically better than linear methods, there are still questions on how to measure the optimal performance possible for such methods. This paper studies…

Stochastic dynamical systems often contain nonlinearities which make it hard to compute probability density functions or statistical moments of these systems. For the moment computations, nonlinearities in the dynamics lead to unclosed…