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We introduce the sparsified Cholesky and sparsified multigrid algorithms for solving systems of linear equations. These algorithms accelerate Gaussian elimination by sparsifying the nonzero matrix entries created by the elimination process.…
This paper introduces a new class of algorithms for solving large-scale linear inverse problems based on new flexible and inexact Golub-Kahan factorizations. The proposed methods iteratively compute regularized solutions by approximating a…
Discretizations of infinite-dimensional variational inequalities lead to linear and nonlinear complementarity problems with many degrees of freedom. To solve these problems in a parallel computing environment, we propose two active-set…
The relaxed physical factorization (RPF) preconditioner is a recent algorithm allowing for the efficient and robust solution to the block linear systems arising from the three-field displacement-velocity-pressure formulation of coupled…
We present a modified version of the PRESB preconditioner for two-by-two block system of linear equations with the coefficient matrix $$\textbf{A}=\left(\begin{array}{cc} F & -G^* G & F \end{array}\right),$$ where $F\in\mathbb{C}^{n\times…
With lowrank approximation the storage requirements for dense data are reduced down to linear complexity and with the addition of hierarchy this also works for data without global lowrank properties. However, the lowrank factors itself are…
In this work, we study a variant of nonnegative matrix factorization where we wish to find a symmetric factorization of a given input matrix into a sparse, Boolean matrix. Formally speaking, given $\mathbf{M}\in\mathbb{Z}^{m\times m}$, we…
A new domain decomposition preconditioner is introduced for efficiently solving linear systems Ax = b with a symmetric positive definite matrix A. The particularity of the new preconditioner is that it is not necessary to have access to the…
The dominant cost in solving least-square problems using Newton's method is often that of factorizing the Hessian matrix over multiple values of the regularization parameter ($\lambda$). We propose an efficient way to interpolate the…
Ising machines are emerging as a new technology for solving various classes of computationally hard problems of practical importance, yet their limits on structured SAT workloads, representative of numerous real-world applications, remain…
Operators with fractional perturbations are crucial components for robust preconditioning of interface-coupled multiphysics systems. However, in case the perturbation is strong, standard approaches can fail to provide scalable approximation…
We propose a localized divide and conquer algorithm for inverse factorization $S^{-1} = ZZ^*$ of Hermitian positive definite matrices $S$ with localized structure, e.g. exponential decay with respect to some given distance function on the…
We propose a flexible and theoretically supported framework for scalable nonnegative matrix factorization. The goal is to find nonnegative low-rank components directly from compressed measurements, accessing the original data only once or…
Optimization problems with constraints in the form of a partial differential equation arise frequently in the process of engineering design. The discretization of PDE-constrained optimization problems results in large-scale linear systems…
Wiedemann's paper, introducing his algorithm for sparse and structured matrix computations over arbitrary fields, also presented a pair of matrix preconditioners for computations over small fields. The analysis of the second of these is…
Designing and implementing systems as an interconnection of smaller subsystems is a common practice for modularity and standardization of components and design algorithms. Although not typically cast in this framework, many of these…
In this experimental work, we present a general framework based on the Bregman log determinant divergence for preconditioning Hermitian positive definite linear systems. We explore this divergence as a measure of discrepancy between a…
We investigate the problem of factorizing a matrix into several sparse matrices and propose an algorithm for this under randomness and sparsity assumptions. This problem can be viewed as a simplification of the deep learning problem where…
A class of splitting alternating algorithms is proposed for finding the sparse solution of linear systems with concatenated orthogonal matrices. Depending on the number of matrices concatenated, the proposed algorithms are classified into…
Despite hundreds of papers on preconditioned linear systems of equations, there remains a significant lack of comprehensive performance benchmarks comparing various preconditioners for solving symmetric positive definite (SPD) systems. In…