Fast Linear Solvers for Laplacian Systems
نویسنده
چکیده
Solving a system of linear equations is a fundamental problem that has deep implications in the computational sciences, engineering, and applied mathematics. The problem has a long history and, until recently, has not broken polynomial time bounds. In this report we present a survey of the algorithms that solve symmetric diagonally dominant linear systems in near-linear time. We also discuss a new linear solver which uses random walk approximations to achieve sublinear time, assuming a random walk step takes unit time. The recent success of these algorithms has inspired their advocacy as a new class of algorithmic primitives. We discuss the progress being made to this end and present a few ways these fast linear solvers may be applied to various graph problems.
منابع مشابه
Lx = 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 کاملLecture : Laplacian solvers ( 2 of 2 )
Last time, we talked about a very simple solver for Laplacian-based systems of linear equations, i.e., systems of linear equations of the form Ax = b, where the constraint matrix A is the Laplacian of a graph. This is not fully-general—Laplacians are SPSD matrices of a particular form—but equations of this form arise in many applications, certain other SPSD problems such as those based on SDD m...
متن کاملAlgorithm Design Using Spectral Graph Theory
Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. Laplace’s equation and its discrete form, the Laplacian matrix, appear ubiquitously in mathematical physics. Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial optimization, computer vision, computer graphics, and machine le...
متن کاملLecture 1: Low-stretch trees
The main theme of the workshop is fast algorithms, particularly those that relate to fast solvers for linear systems involving the Laplacian of a graph. In my lectures, I will discuss three key technical ingredients that underlie those solvers. In this first lecture, I will discuss “low-stretch trees”. Given a graph, the goal is to find a spanning subtree such that, on average, distances in the...
متن کاملLecture 25 : Element - wise Sampling of Graphs and Linear Equation Solving , Cont
Last time, we talked about a very simple solver for Laplacian-based systems of linear equations, i.e., systems of linear equations of the form Ax = b, where the constraint matrix A is the Laplacian of a graph. This is not fully-general—Laplacians are SPSD matrices of a particular form—but equations of this form arise in many applications, certain other SPSD problems such as those based on SDD m...
متن کاملSparse matrix factorizations for fast linear solvers with application to Laplacian systems
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 iterat...
متن کامل