Related papers: Computing rough solutions of the stochastic nonlin…
In this work (Part I), we study three time-discretization procedures of the Dynamical Low-Rank Approximation (DLRA) of high-dimensional stochastic differential equations (SDEs). Specifically, we consider the Dynamically Orthogonal (DO)…
This article proposes a novel approach for determining exact solutions to nonlinear ordinary differential equations. The recommended iterative method provides the solution via a rapidly converging series that readily approaches a closed…
In this paper, we consider the problem of minimizing the average of a large number of nonsmooth and convex functions. Such problems often arise in typical machine learning problems as empirical risk minimization, but are computationally…
An ill-posed Cauchy problem for the wave equation is considered: the solution is to be determined by the Cauchy data on some part of the time-space boundary. By means of Fourier method we obtain a regularization algorithm for this problem,…
In this paper, based on real-time nonlinear receding horizon control methodology, a novel approach is developed for parameter estimation of time invariant and time varying nonlinear dynamical systems in chaotic environments. Here, the…
The aim of this paper is to develop a general method for constructing approximation schemes for viscosity solutions of fully nonlinear pathwise stochastic partial differential equations, and for proving their convergence. Our results apply…
Recently, several studies consider the stochastic optimization problem but in a heavy-tailed noise regime, i.e., the difference between the stochastic gradient and the true gradient is assumed to have a finite $p$-th moment (say being upper…
Large-scale simulations of the wave equation in electromagnetism, seismology, and acoustics, can be solved efficiently by finite difference methods. The accuracy of these numerical solutions usually depends on the minimization of…
In this paper, a stochastic algorithm for the efficient simulation and optimal control of networked wave equations based on the random batch method is proposed and analyzed. The random approximation is constructed by dividing the time…
We propose smoothed primal-dual algorithms for solving stochastic and smooth nonconvex optimization problems with linear inequality constraints. Our algorithms are single-loop and only require a single stochastic gradient based on one…
The wavenumber integration model is considered to be the most accurate algorithm for arbitrary horizontally stratified media in computational ocean acoustics. In contrast to the normal mode approach, it considers not only the discrete…
We introduce a new framework for analyzing (Quasi-}Newton type methods applied to non-smooth optimization problems. The source of randomness comes from the evaluation of the (approximation) of the Hessian. We derive, using a variant of…
Stochastic partial differential equations have been used in a variety of contexts to model the evolution of uncertain dynamical systems. In recent years, their applications to geophysical fluid dynamics has increased massively. For a…
This study employs spectral methods to capture the behaviour of wave equation with dispersive-nonlinearity. We describe the evolution of hump initial data and track the conservation of the mass and energy functionals. The…
In partial differential equations-based (PDE-based) inverse problems with many measurements, many large-scale discretized PDEs must be solved for each evaluation of the misfit or objective function. In the nonlinear case, evaluating the…
In this paper, we present an energy-preserving exponentially integrable numerical method for stochastic wave equation with cubic nonlinearity and additive noise. We first apply the spectral Galerkin method to discretizing the original…
We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path…
We formally prove correct a C program that implements a numerical scheme for the resolution of the one-dimensional acoustic wave equation. Such an implementation introduces errors at several levels: the numerical scheme introduces method…
Non-smooth optimization is a core ingredient of many imaging or machine learning pipelines. Non-smoothness encodes structural constraints on the solutions, such as sparsity, group sparsity, low-rank and sharp edges. It is also the basis for…
Stochastic spectral methods have achieved great success in the uncertainty quantification of many engineering problems, including electronic and photonic integrated circuits influenced by fabrication process variations. Existing techniques…