Related papers: A Hybrid Factorization Algorithm for Sparse Matrix…
Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability; but, need to be used with an appropriate preconditioner (e.g.,…
LU-factorization of matrices is one of the fundamental algorithms of linear algebra. The widespread use of supercomputers with distributed memory requires a review of traditional algorithms, which were based on the common memory of a…
In this paper we introduce an algebraic recursive multilevel incomplete factorization preconditioner, based on a distributed Schur complement formulation, for solving general linear systems. The novelty of the proposed method is to combine…
While existing algorithms may be used to solve a linear system over a general field in matrix-multiplication time, the complexity of constructing a symmetric triangular factorization (LDL) has received relatively little formal study. The…
Low-rank matrix approximation is a fundamental tool in data analysis for processing large datasets, reducing noise, and finding important signals. In this work, we present a novel truncated LU factorization called Spectrum-Revealing LU…
The matrix LU factorization algorithm is a fundamental algorithm in linear algebra. We propose a generalization of the LU and LEU algorithms to accommodate the case of a commutative domain and its field of quotients. This algorithm…
The hierarchical matrix framework partitions matrices into subblocks that are either small or of low numerical rank, enabling linear storage complexity and efficient matrix-vector multiplication. This work focuses on the $H^2$-matrix format…
We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or…
Square matrices appear in many machine learning problems and models. Optimization over a large square matrix is expensive in memory and in time. Therefore an economic approximation is needed. Conventional approximation approaches factorize…
We propose a penalized likelihood framework for estimating multiple precision matrices from different classes. Most existing methods either incorporate no information on relationships between the precision matrices, or require this…
The emergence of low precision floating-point arithmetic in computer hardware has led to a resurgence of interest in the use of mixed precision numerical linear algebra. For linear systems of equations, there has been renewed enthusiasm for…
In this paper, a method via sparse-sparse iteration for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix is proposed. The resulting factorized sparse approximate inverse is used as a…
Matrix factorization (MF) is a widely used collaborative filtering (CF) algorithm for recommendation systems (RSs), due to its high prediction accuracy, great flexibility and high efficiency in big data processing. However, with the…
Sparse matrix factorization is a popular tool to obtain interpretable data decompositions, which are also effective to perform data completion or denoising. Its applicability to large datasets has been addressed with online and randomized…
This study presents a novel mixed-precision iterative refinement algorithm, GADI-IR, within the general alternating-direction implicit (GADI) framework, designed for efficiently solving large-scale sparse linear systems. By employing…
We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in.…
Solving sparse linear systems from discretized PDEs is challenging. Direct solvers have in many cases quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and…
The Schur decomposition of a square matrix $A$ is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following…
A distributed optimal control problem with the constraint of a linear elliptic partial differential equation is considered. A necessary optimality condition for this problem forms a saddle point system, the efficient and accurate solution…
Our interest lies in the robust and efficient solution of large sparse linear least-squares problems. In recent years, hardware developments have led to a surge in interest in exploiting mixed precision arithmetic within numerical linear…