Related papers: An efficient solver based on low-rank approximatio…
This article considers stochastic algorithms for efficiently solving a class of large scale non-linear least squares (NLS) problems which frequently arise in applications. We propose eight variants of a practical randomized algorithm where…
A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic…
The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline…
We study a low-rank iterative solver for the unsteady Navier-Stokes equations for incompressible flows with a stochastic viscosity. The equations are discretized using the stochastic Galerkin method, and we consider an all-at-once…
In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. A dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional…
Many data-analysis problems involve large dense matrices that describe the covariance of stationary noise processes; the computational cost of inverting these matrices, or equivalently of solving linear systems that contain them, is often a…
Identification of nonlinear dynamical systems is crucial across various fields, facilitating tasks such as control, prediction, optimization, and fault detection. Many applications require methods capable of handling complex systems while…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
We present a Multi-Index Quasi-Monte Carlo method for the solution of elliptic partial differential equations with random coefficients. By combining the multi-index sampling idea with randomly shifted rank-1 lattice rules, the algorithm…
We formulate a recursive estimation problem for multiple dynamical systems coupled through a low dimensional stochastic input, and we propose an efficient sub-optimal solution. The suggested approach is an approximation of the Kalman filter…
Fractional Fokker-Planck equation plays an important role in describing anomalous dynamics. To the best of our knowledge, the existing discussions mainly focus on this kind of equation involving one diffusion operator. In this paper, we…
In this article we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient…
We investigate a numerical behaviour of robust deterministic optimal control problem subject to a convection diffusion equation containing uncertain inputs. Stochastic Galerkin approach, turning the original optimization problem containing…
In the present paper, we consider large-scale differential Lyapunov matrix equations having a low rank constant term. We present two new approaches for the numerical resolution of such differential matrix equations. The first approach is…
We describe an exact and highly efficient numerical algorithm for solving a special but important class of convection-diffusion equations. These equations occur in many problems in physics, chemistry, or biology, and they are usually hard…
The radiative transfer equation models various physical processes ranging from plasma simulations to radiation therapy. In practice, these phenomena are often subject to uncertainties. Modeling and propagating these uncertainties requires…
We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity…
We study the problem of estimating low-rank matrices from linear measurements (a.k.a., matrix sensing) through nonconvex optimization. We propose an efficient stochastic variance reduced gradient descent algorithm to solve a nonconvex…
We consider a Poisson equation in $\mathbb R^d$ for the elliptic operator corresponding to an ergodic diffusion process. Optimal regularity and smoothness with respect to the parameter are obtained under mild conditions on the coefficients.…
We consider a randomised implementation of the finite element method (FEM) for elliptic partial differential equations on high-dimensional models. This is motivated by applications where model predictions are essential for real-time process…