Topic
Gauss–Seidel method
About: Gauss–Seidel method is a research topic. Over the lifetime, 1182 publications have been published within this topic receiving 24073 citations. The topic is also known as: Gauss-Seidel method & Liebmann method.
Papers published on a yearly basis
Papers
More filters
TL;DR: This work proves an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance, that guarantees the convergence of bounded sequences under the assumption that the function f satisfies the Kurdyka–Łojasiewicz inequality.
Abstract: In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–Łojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.
1,282 citations
TL;DR: A unified theory for a diverse group of iterative algorithms, such as Jacobi and Gauss–Seidel iterations, diagonal preconditioning, domain decomposition methods, multigrid methods,Multilevel nodal basis preconditionsers and hierarchical basis methods, is presented by using the notions of space decomposition and subspace correction.
Abstract: The main purpose of this paper is to give a systematic introduction to a number of iterative methods for symmetric positive definite problems. Based on results and ideas from various existing works...
1,176 citations
TL;DR: On developpe un schema de relaxation multigrille, application a des ecoulements transsoniques d'application a des Ecoulements Transsoniques.
Abstract: On developpe un schema de relaxation multigrille. Application a des ecoulements transsoniques
1,131 citations
TL;DR: Monotonicity and stability properties of the fast sweeping algorithm are proven and it is shown that 2 n Gauss-Seidel iterations is enough for the distance function in n dimensions.
Abstract: In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding Eikonal equation in a certain direction simultaneously. The method has an optimal complexity of O(N) for N grid points and is extremely simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven. Convergence and error estimates of the algorithm for computing the distance function is studied in detail. It is shown that 2 n Gauss-Seidel iterations is enough for the distance function in n dimensions. An estimation of the number of iterations for general Eikonal equations is also studied. Numerical examples are used to verify the analysis.
998 citations
TL;DR: It is shown that the block Gauss-Seidel method is globally convergent for m=2 and that for m>2 convergence can be established both when the objective function f is componentwise strictly quasiconvex with respect to m-2 components and when f is pseudoconvex.
717 citations