A fast block low-rank dense solver with applications to finite-element matrices
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TLDR
A fast solver for the dense "frontal" matrices that arise from the multifrontal sparse elimination process of 3D elliptic PDEs, using the HODLR direct solver as a preconditioner to the GMRES iterative scheme to reach machine accuracy much faster than a conventional LU solver.About:
This article is published in Journal of Computational Physics.The article was published on 2016-01-01 and is currently open access. It has received 112 citations till now. The article focuses on the topics: Solver & Matrix (mathematics).read more
Citations
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Fast Direct Methods for Gaussian Processes
TL;DR: In this paper, the authors show that for the most commonly used covariance functions, the matrix $C$ can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an $\mathcal {O} (n\,\log^2, n)$ algorithm for inversion.
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Performance and Scalability of the Block Low-Rank Multifrontal Factorization on Multicore Architectures
TL;DR: This article identifies the challenges posed by the use of BLR approximations in multifrontal solvers and put forward several algorithmic variants of the BLR factorization that overcome these challenges by improving its efficiency and scalability.
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Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations
Kenneth L. Ho,Lexing Ying +1 more
TL;DR: This paper introduces the hierarchical interpolative factorization for integral equations (HIF‐IE) associated with elliptic problems in two and three dimensions, and conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity.
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An Efficient Multicore Implementation of a Novel HSS-Structured Multifrontal Solver Using Randomized Sampling
TL;DR: A sparse linear system solver that is based on a multifrontal variant of Gaussian elimination and exploits low-rank approximation of the resulting dense frontal matrices, resulting in speedups up to sevenfold for problems in the test suite.
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A Distributed-Memory Package for Dense Hierarchically Semi-Separable Matrix Computations Using Randomization
TL;DR: A distributed-memory library for computations with dense structured matrices using Hierarchically Semi-Separable (HSS) representations and the compression algorithm that computes the HSS form of an input dense matrix relies on randomized sampling with a novel adaptive sampling mechanism.
References
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GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
Youcef Saad,Martin H. Schultz +1 more
TL;DR: An iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace.
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Methods of Conjugate Gradients for Solving Linear Systems
TL;DR: An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns and it is shown that this method is a special case of a very general method which also includes Gaussian elimination.
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The university of Florida sparse matrix collection
Timothy A. Davis,Yifan Hu +1 more
TL;DR: The University of Florida Sparse Matrix Collection, a large and actively growing set of sparse matrices that arise in real applications, is described and a new multilevel coarsening scheme is proposed to facilitate this task.
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Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
TL;DR: This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation, and presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions.
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New development in freefem
TL;DR: First the freefem++ software deals with mesh adaptation for problems in two and three dimension, second, it solves numerically a problem with phase change and natural convection, and finally to show the possibilities for HPC the software solves a Laplace equation by a Schwarz domain decomposition problem on parallel computer.