A fast solver for a class of linear systems

Abstract

The solution of linear systems is a problem of fundamental theoretical importance but also one with a myriad of applications in numerical mathematics, engineering and science. Linear systems that are generated by real-world applications frequently fall into special classes. Recent research led to a fast algorithm for solving symmetric diagonally dominant (SDD) linear systems. We give an overview of this solver and survey the underlying notions and tools from algebra, probability and graph algorithms. We also discuss some of the many and diverse applications of SDD solvers.

Efficient and Practical Tree Preconditioning for Solving Laplacian Systems

“…A proof using a lower bound for all eigenvalues and an upper bound on the number of eigenvalues above a certain threshold yields that PCG computes an −approximation in O (stretch T (G)) 1/3 log(1/ ) iterations (see Lemma 17.2 in [24]). In the past several years, this basic framework for solving linear systems with Laplacian matrices has been extended significantly, with two major research directions: finding trees that can optimize the stretch with respect to arbitrary large graphs [1], and changing this basic framework to use a more sophisticated hierarchical graph approximation scheme in which preconditioners themselves can be solved via iterative (and possibly recursive) schemes [16]. Unfortunately, both of these directions lead to highly complex algorithms and their practical performance has been evaluated only very recently [13].…”

“…The red edges in Fig. 3 define a spanning tree T l having a low stretch factor, which can be evaluated to be O(n log n) using an inductive argument (we refer to [16] for more details). These bounds reflect the numerical evaluation of the condition numbers for both trees T h and T l (plotted as dashed…”

“…These linear systems arise in a variety of applications, which are related to solving the Poisson equation on discretized domains, including physical (e.g. fluid) simulation, complex system analysis, geometry processing and computer graphics [16,17], among many others. One class of techniques, which is especially useful in solving large systems of equations with Laplacians matrices of certain (sparse) graphs is the conjugate gradient method.…”

Simple Parallel and Distributed Algorithms for Spectral Graph Sparsification

“…They were introduced by Spielman and Teng [24] as a basic component of the first nearly-linear time solvers for linear systems on symmetric diagonally dominant (SDD) matrices 1 . Such linear system solvers are a key algorithmic primitive with numerous applications [17,25].…”

“…They were introduced by Spielman and Teng [24] as a basic component of the first nearly-linear time solvers for linear systems on symmetric diagonally dominant (SDD) matrices 1 . Such linear system solvers are a key algorithmic primitive with numerous applications [17,25].The Spielman and Teng sparsification algorithm produces sparsifiers with O(n log c n/ǫ 2 ) edges for some fairly large constant c, where n is the number of vertices in the graph. At a high level their algorithm is based on graph decompositions into edge-disjoint sets that get sparsified independently via uniform sampling.…”