Related papers: Avoiding breakdown in incomplete factorizations in…
Incomplete factorizations have long been popular general-purpose algebraic preconditioners for solving large sparse linear systems of equations. Guaranteeing the factorization is breakdown free while computing a high quality preconditioner…
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…
The solution of a sparse system of linear equations is ubiquitous in scientific applications. Iterative methods, such as the Preconditioned Conjugate Gradient method (PCG), are normally chosen over direct methods due to memory and…
Incomplete LU factorizations of sparse matrices are widely used as preconditioners in Krylov subspace methods to speed up solving linear systems. Unfortunately, computing the preconditioner itself can be time-consuming and sensitive to…
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…
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…
Preconditioning for overdetermined least-squares problems has received comparatively little attention, and designing methods that are both effective and memory-efficient remains challenging. We propose a class of ILU-based preconditioners…
LU and Cholesky matrix factorization algorithms are core subroutines used to solve systems of linear equations (SLEs) encountered while solving an optimization problem. Standard factorization algorithms are highly efficient but remain…
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.…
The convergence of the conjugate gradient method for solving large-scale and sparse linear equation systems depends on the spectral properties of the system matrix, which can be improved by preconditioning. In this paper, we develop a…
We introduce a parallel algorithm to construct a preconditioner for solving a large, sparse linear system where the coefficient matrix is a Laplacian matrix (a.k.a., graph Laplacian). Such a linear system arises from applications such as…
Many application problems that lead to solving linear systems make use of preconditioned Krylov subspace solvers to compute their solution. Among the most popular preconditioning approaches are incomplete factorization methods either as…
A new hybrid algorithm for LDU-factorization for large sparse matrix combining iterative solver, which can keep the same accuracy as the classical factorization, is proposed. The last Schur complement will be generated by iterative solver…
Prior to computing the Cholesky factorization of a sparse, symmetric positive definite matrix, a reordering of the rows and columns is computed so as to reduce both the number of fill elements in Cholesky factor and the number of arithmetic…
A linearly implicit conservative difference scheme is applied to discretize the attractive coupled nonlinear Schr\"odinger equations with fractional Laplacian. Complex symmetric linear systems can be obtained, and the system matrices are…
Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in machine learning, statistics, bioinformatics, computer vision, as well as signal and image processing. In theory, this problem can…
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…
With the emergence of mixed precision capabilities in hardware, iterative refinement schemes for solving linear systems $Ax=b$ have recently been revisited and reanalyzed in the context of three or more precisions. These new analyses show…
Low-precision computing is essential for efficiently utilizing memory bandwidth and computing cores. While many mixed-precision algorithms have been developed for iterative sparse linear solvers, effectively leveraging half-precision (fp16)…
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 =…