Sparse matrix factorizations for fast linear solvers with application to Laplacian systems
نویسندگان
چکیده
In solving a linear system with iterative methods, one is usually confronted with the dilemma of having to choose between cheap, inefficient iterates over sparse search directions (e.g., coordinate descent), or expensive iterates in well-chosen search directions (e.g., conjugate gradients). In this paper, we propose to interpolate between these two extremes, and show how to perform cheap iterations along non-sparse search directions, provided that these directions admit a new kind of sparse factorization. For example, if the search directions are the columns of a hierarchical matrix, then the cost of each iteration is typically logarithmic in the number of variables. Using some graphtheoretical results on low-stretch spanning trees, we deduce as a special case a nearly-linear time algorithm to approximate the minimal norm solution of a linear system Bx = b where B is the incidence matrix of a graph. We thereby can connect our results to recently proposed nearly-linear time solvers for Laplacian systems, which emerge here as a particular application of our sparse matrix factorization. Key word. matrix factorization, linear system, Laplacian matrix, iterative algorithms, sparsity, hierarchical matrices AMS subject classifications. 15A06, 15A23, 15A24
منابع مشابه
Sparsified Cholesky Solvers for SDD linear systems
We show that Laplacian and symmetric diagonally dominant (SDD) matrices can be well approximated by linear-sized sparse Cholesky factorizations. Specifically, n × n matrices of these types have constant-factor approximations of the form LL , where L is a lowertriangular matrix with O(n) non-zero entries. This factorization allows us to solve linear systems in such matrices in O(n) work and O(lo...
متن کاملCAS WAVELET METHOD FOR THE NUMERICAL SOLUTION OF BOUNDARY INTEGRAL EQUATIONS WITH LOGARITHMIC SINGULAR KERNELS
In this paper, we present a computational method for solving boundary integral equations with loga-rithmic singular kernels which occur as reformulations of a boundary value problem for the Laplacian equation. Themethod is based on the use of the Galerkin method with CAS wavelets constructed on the unit interval as basis.This approach utilizes the non-uniform Gauss-Legendre quadrature rule for ...
متن کاملSparse Direct Linear Solvers: An Introduction
The minisymposium on sparse direct solvers included 11 talks on the state of the art in this area. The talks covered a wide spectrum of research activities in this area. The papers in this part of the proceedings are expanded, revised, and corrected versions of some the papers that appeared in the CD-ROM proceedings that were distributed at the conference. Not all the talks in the minisymposium...
متن کاملLx = b Laplacian Solvers and Their Algorithmic Applications
The ability to solve a system of linear equations lies at the heart of areas such as optimization, scientific computing, and computer science, and has traditionally been a central topic of research in the area of numerical linear algebra. An important class of instances that arise in practice has the form Lx = b, where L is the Laplacian of an undirected graph. After decades of sustained resear...
متن کاملRandomized Sparse Direct Solvers
We propose randomized direct solvers for large sparse linear systems, which integrate randomization into rank structured multifrontal methods. The use of randomization highly simplifies various essential steps in structured solutions, where fast operations on skinny matrix-vector products replace traditional complex ones on dense or structured matrices. The new methods thus significantly enhanc...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- SIAM J. Matrix Analysis Applications
دوره 38 شماره
صفحات -
تاریخ انتشار 2017