Related papers: Low rank approximation method for perturbed linear…
We consider the problem of approximating a given matrix by a low-rank matrix so as to minimize the entrywise $\ell_p$-approximation error, for any $p \geq 1$; the case $p = 2$ is the classical SVD problem. We obtain the first provably good…
Differentiable systems in this paper means systems of equations that are described by differentiable real functions in real matrix variables. This paper proposes algorithms for finding minimal rank solutions to such systems over (arbitrary…
Rank deficient Hankel matrices are at the core of several applications. However, in practice, the coefficients of these matrices are noisy due to e.g. measurements errors and computational errors, so generically the involved matrices are…
In this study, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the…
Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In…
In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a…
Tensor methods are among the most prominent tools for the numerical solution of high-dimensional problems where functions of multiple variables have to be approximated. These methods exploit the tensor structure of function spaces and apply…
We investigate an optimization problem governed by an elliptic partial differential equation with uncertain parameters. We introduce a robust optimization framework that accounts for uncertain model parameters. The resulting non-linear…
In this paper, we consider optimal low-rank regularized inverse matrix approximations and their applications to inverse problems. We give an explicit solution to a generalized rank-constrained regularized inverse approximation problem,…
This work proposes a systematic model reduction approach based on rank adaptive tensor recovery for partial differential equation (PDE) models with high-dimensional random parameters. Since the standard outputs of interest of these models…
Low-rank approximation is a technique to approximate a tensor or a matrix with a reduced rank to reduce the memory required and computational cost for simulation. Its broad applications include dimension reduction, signal processing,…
A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…
The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic…
For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives.…
We consider adaptive approximations of the parameter-to-solution map for elliptic operator equations depending on a large or infinite number of parameters, comparing approximation strategies of different degrees of nonlinearity: sparse…
Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this…
In this paper, we extend the dynamical low-rank approximation method to the space of finite signed measures. Under this framework, we derive stochastic low-rank dynamics for stochastic differential equations (SDEs) coming from classical…
We describe novel subgradient methods for a broad class of matrix optimization problems involving nuclear norm regularization. Unlike existing approaches, our method executes very cheap iterations by combining low-rank stochastic…
Rank minimization (RM) is a wildly investigated task of finding solutions by exploiting low-rank structure of parameter matrices. Recently, solving RM problem by leveraging non-convex relaxations has received significant attention. It has…
In this paper, we set the mathematical foundations of the Dynamical Low-Rank Approximation (DLRA) method for stochastic differential equations (SDEs). DLRA aims at approximating the solution as a linear combination of a small number of…