{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article2372","name":"Combinatorial problems in solving linear systems","abstract":"Numerical linear algebra and combinatorial optimization are vast\r\nsubjects; as is their interaction. In virtually all cases there should\r\nbe a notion of sparsity for a combinatorial problem to arise. Sparse\r\nmatrices, therefore, form the basis of the interaction of these two\r\nseemingly disparate subjects. As the core of many of today's numerical linear\r\nalgebra computations consists of sparse linear system solutions, we\r\nwill cover combinatorial problems, notions, and algorithms relating to\r\nthose computations.\r\n\r\nThis talk is thus concerned with direct and iterative methods for sparse \r\nlinear systems and their intercation with combinatorial optimization.\r\nOn the direct methods side, we discuss matrix ordering; bipartite matching \r\nand matrix scaling for better pivoting; task assignment and scheduling \r\nfor parallel multifrontal solvers. On the iterative method side, we discuss\r\npreconditioning techniques including incomplete factor preconditioners\r\n(notion of level of fill-in), support graph preconditioners (graph\r\nembedding concepts), and algebraic multigrids (independent sets in\r\nundirected graphs).\r\n\r\nIn a separate part of the talk, we discuss methods that aim to exploit\r\nsparsity during linear system solution. These methods include block\r\ndiagonalization of the matrix; efficient triangular system solutions\r\nfor right-hand side vectors of single nonzero entries. Towards the\r\nend, we mention, quite briefly as they are topics of other invited\r\ntalks, some other areas whose interactions with combinatorial\r\noptimization are of great benefit to numerical linear algebra. These\r\ninclude graph and hypergraph partitioning for load balancing problems,\r\nand colouring problems in numerical optimization. On closing, we\r\ncompile and list a set of open problems.","keywords":["Combinatorial scientific computing","graph theory","combinatorial optimization","sparse matrices","linear system solution"],"author":[{"@type":"Person","name":"Duff, Iain S.","givenName":"Iain S.","familyName":"Duff"},{"@type":"Person","name":"Ucar, Bora","givenName":"Bora","familyName":"Ucar"}],"position":8,"pageStart":1,"pageEnd":37,"dateCreated":"2009-07-24","datePublished":"2009-07-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Duff, Iain S.","givenName":"Iain S.","familyName":"Duff"},{"@type":"Person","name":"Ucar, Bora","givenName":"Bora","familyName":"Ucar"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09061.8","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume735","volumeNumber":9061,"name":"Dagstuhl Seminar Proceedings, Volume 9061","dateCreated":"2009-07-24","datePublished":"2009-07-24","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article2372","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume735"}}}