Related papers: Mixed precision sketching for least-squares proble…
We provide an exact analysis of a class of randomized algorithms for solving overdetermined least-squares problems. We consider first-order methods, where the gradients are pre-conditioned by an approximation of the Hessian, based on a…
Kernel Ridge Regression (KRR) is a simple yet powerful technique for non-parametric regression whose computation amounts to solving a linear system. This system is usually dense and highly ill-conditioned. In addition, the dimensions of the…
We present a simple way to discretize and precondition mixed variational formulations. Our theory connects with, and takes advantage of, the classical theory of symmetric saddle point problems and the theory of preconditioning symmetric…
Quadratic optimization problems (QPs) are ubiquitous, and solution algorithms have matured to a reliable technology. However, the precision of solutions is usually limited due to the underlying floating-point operations. This may cause…
We consider statistical and algorithmic aspects of solving large-scale least-squares (LS) problems using randomized sketching algorithms. Prior results show that, from an \emph{algorithmic perspective}, when using sketching matrices…
We propose new approximate alternating projection methods, based on randomized sketching, for the low-rank nonnegative matrix approximation problem: find a low-rank approximation of a nonnegative matrix that is nonnegative, but whose…
Power iteration can improve the accuracy of randomized SVD, but requires multiple data passes, making it impractical in streaming or memory-constrained settings. We introduce a lightweight yet effective sketch-power iteration, allowing…
In this paper, we propose a robust solver for the finite element discrete problem of the stationary incompressible magnetohydrodynamic (MHD) equations in three dimensions. By the mixed finite element method, both the velocity and the…
This research investigates using a mixed-precision iterative refinement method using posit numbers instead of the standard IEEE floating-point format. The method is applied to solve a general linear system represented by the equation $Ax =…
We establish a new iterative method for solving a class of large and sparse linear systems of equations with three-by-three block coefficient matrices having saddle point structure. Convergence properties of the proposed method are studied…
Modern computer architectures support low-precision arithmetic, which present opportunities for the adoption of mixed-precision algorithms to achieve high computational throughput and reduce energy consumption. As a growing number of…
In this paper, we consider an efficient iterative approach to the solution of the discrete Helmholtz equation with Dirichlet, Neumann and Sommerfeld-like boundary conditions based on a compact sixth order approximation scheme and…
Hierarchical Matrix (H-matrix) is an approximation technique which splits a target dense matrix into multiple submatrices, and where a selected portion of submatrices are low-rank approximated. The technique substantially reduces both time…
Techniques that rigorously bound the overall rounding error exhibited by a numerical program are of significant interest for communities developing numerical software. However, there are few available tools today that can be used to…
This paper presents a scalable physics-based block preconditioner for mixed-dimensional models in beam-solid interaction and their application in engineering. In particular, it studies the linear systems arising from a regularized…
Embedded computer vision applications increasingly require the speed and power benefits of single-precision (32 bit) floating point. However, applications which make use of Levenberg-like optimization can lose significant accuracy when…
A methodology for using random sketching in the context of model order reduction for high-dimensional parameter-dependent systems of equations was introduced in [Balabanov and Nouy 2019, Part I]. Following this framework, we here construct…
We propose iterative projection methods for solving square or rectangular consistent linear systems Ax = b. Existing projection methods use sketching matrices (possibly randomized) to generate a sequence of small projected subproblems, but…
We present a new approach for computing compact sketches that can be used to approximate the inner product between pairs of high-dimensional vectors. Based on the Weighted MinHash algorithm, our approach admits strong accuracy guarantees…
This paper presents a new approach to solve linear and nonlinear model predictive control (MPC) problems that requires small memory footprint and throughput and is particularly suitable when the model and/or controller parameters change at…