[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-228128-en":3,"doc-seo-228128-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},228128,962088006270,"eBook King","https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Iterative Methods by Space Decomposition and Subspace Correction","The paper presents a systematic introduction to iterative methods for symmetric positive definite problems, unifying algorithms such as Jacobi and Gauss-Seidel iterations, diagonal preconditioning, domain decomposition, multigrid, multilevel nodal basis preconditioners, and hierarchical basis methods. Using space decomposition and subspace correction notions, it classifies schemes into parallel subspace correction (PSC) and successive subspace correction (SSC) methods. A general abstract convergence theory is established, and optimal convergence estimates follow by specifying the space decomposition and the associated subspace solvers.","# Iterative Methods by Space Decomposition and Subspace Correction\n\nAuthor(s):Jinchao Xu  \nSource:SIAM Review,Dec.,1992,Vol.34,No.4(Dec.,1992),pp.581-613  \nPublished by:Society for Industrial and Applied Mathematics  \nStable URL:http://www.jstor.com/stable/2132629  \nJSTOR is a not-for-profit service that helps scholars,researchers,and students discover,use,and build upon a widerange of content in^a trusted digital archive.We use information technology and tools to increase productivity andfacilitate new forms of scholarship.For more information about JSTOR,please contact support@jsfor.org.  \nYour use of the JSTOR archive indicates your acceptance of the Terms &Conditions of Use,available athttps://about.jstor.org/terms  \nSociety for Industrial and Applied Mathematics is collaborating with JSTOR to digitize,preserve and extend access to SIAM Review  \n# ITERATIVE METHODSBY SPACE DECOMPOSITION AND SUBSPACE CORRECTION*\n\nJINCHAO XUt  \nAbstract.The main purpose of this paper is to give a systematic introduction to a number of iterativemethods for symmetric positive definite problems.Based on results and ideas from various existing works oniterative methods,a unified theory for a diverse group of iterative algorithms,such as Jacobi and Gauss-Seideliterations,diagonal preconditioning,domain decomposition methods,multigrid methods,multilevel nodal ba-sis preconditioners and hierarchical basis methods,is presented.By using the notions of space decompositionand subspace correction,all these algorithms are classified into two groups,namely parallel subspace correction(PSC)and successive subspace correction(SSC)methods.These two types of algorithms are similar in natureto the familiar Jacobi and Gauss-Seidel methods,respectively.  \nA feature of this framework is that a quite general abstract convergence theory can be established.Inorder to apply the abstract theory to a particular problem,it is only necessary to specify a decompositionof the underlying space and the corresponding subspace solvers.For example,subspaces arising from thedomain decomposition method are associated with subdomains whereas with the multigrid method subspacesare provided by multiple “coarser”grids.By estimating only two parameters,optimal convergence estimationsfor a given algorithm can be obtained as a direct consequence of the abstract theory.  \nKey words.domain decomposition,Gauss-Seidel,finite elements,hierarchical basis,Jacobi,multigrid,  \nSchwarz,space decomposition,strengthened Cauchy-Schwarz inequalities,subspace correction  \nAMS(MOS)subject classifications.65M60,65N15,65N30  \n1.Introduction.In this paper,we shalldiscuss iterative algorithms to approximatethe solution of a linear equation  \n(1.1)  \nAu=f,  \nwhere A is a symmetric positive definite(SPD)operator on a finite-dimensional vectorspace V.There exist a large group of algorithms for solving the above problem.Classicexamples are Gauss-Seidel,Jacobi iterations,and diagonal preconditioning techniques;more contemporary algorithms include multigrid and domain decomposition methods.The goal of this paper is to present these algorithms in a unified framework.  \nThe central idea behind our framework is simple.A single step linear iterativemethod that uses an old approximation,uold,of the solution u of (1.1),to produce anew approximation,unew,usually consists of three steps:  \n(1)Form rod=f-Auold;  \n(2)Solve Ae=rold approximately:ê=Brold with B≈A-1;  \n(3)Update unew=uold+ê.  \nClearly the choice of B(an approximate inverse of A)is the core of this type of algorithm.The point of our theory is to chose B by solving appropriate subspace problems.Thesubspaces are provided by a decomposition of V:  \nHere V;are subspaces of V.Assume that Ai:Vi→Vi is a restriction operator of A onVi(see (3.2));then the above three steps can be carried out on each subspace V;with  \nJINCHAOXU  \nB=Ri≈for i=1,2,.…,J.We shall demonstrate how an iterative algorithm canbe obtained by repeatedly using the above three steps together with a d","cbCaip6kw9BBMsk3","https://ap.wps.com/l/cbCaip6kw9BBMsk3","pdf",2332485,1,34,"English","en",105,"# Introduction\n## Overview of the unified framework\n# Preliminaries\n## Iterative methods considered\n# Convergence theory and organization\n## Abstract convergence theory and outline","[{\"question\":\"What type of problems does the paper focus on?\",\"answer\":\"The paper focuses on symmetric positive definite (SPD) problems for a linear system Au=f.\"},{\"question\":\"How are iterative algorithms organized in the proposed framework?\",\"answer\":\"Algorithms are classified using space decomposition and subspace correction into two groups: parallel subspace correction (PSC) and successive subspace correction (SSC).\"},{\"question\":\"How is convergence analysis handled?\",\"answer\":\"A general abstract convergence theory is developed, and optimal convergence estimates are obtained by estimating two parameters after specifying the decomposition and subspace solvers.\"}]","Iterative Methods by Space Decomposition and Subspace Correction | 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type of problems does the paper focus on?","Question",{"text":76,"@type":77},"The paper focuses on symmetric positive definite (SPD) problems for a linear system Au=f.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How are iterative algorithms organized in the proposed framework?",{"text":81,"@type":77},"Algorithms are classified using space decomposition and subspace correction into two groups: parallel subspace correction (PSC) and successive subspace correction (SSC).",{"name":83,"@type":74,"acceptedAnswer":84},"How is convergence analysis handled?",{"text":85,"@type":77},"A general abstract convergence theory is developed, and optimal convergence estimates are obtained by estimating two parameters after specifying the decomposition and subspace 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