Efficient arithmetic operations for rank-structured matrices based on hierarchical low-rank updates View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2013-12

AUTHORS

Steffen Börm, Knut Reimer

ABSTRACT

Many matrices appearing in numerical methods for partial differential equations and integral equations are rank-structured, i.e., they contain submatrices that can be approximated by matrices of low rank. A relatively general class of rank-structured matrices are H2-matrices: they can reach the optimal order of complexity, but are still general enough for a large number of practical applications. We consider algorithms for performing algebraic operations with H2-matrices, i.e., for approximating the matrix product, inverse or factorizations in almost linear complexity. The new approach is based on local low-rank updates that can be performed in linear complexity. These updates can be combined with a recursive procedure to approximate the product of two H2-matrices, and these products can be used to approximate the matrix inverse and the LR or Cholesky factorization. Numerical experiments indicate that the new algorithm leads to preconditioners that require O(n) units of storage, can be evaluated in O(n) operations, and take O(nlogn) operations to set up. More... »

PAGES

247-258

References to SciGraph publications

  • 2006-02. -Matrix Arithmetics in Linear Complexity in COMPUTING
  • 2000. On H2-Matrices in LECTURES ON APPLIED MATHEMATICS
  • 2002-09. Data-sparse Approximation by Adaptive ℋ2-Matrices in COMPUTING
  • 2009-03. A Fast Direct Solver for a Class of Elliptic Partial Differential Equations in JOURNAL OF SCIENTIFIC COMPUTING
  • 2005-02. Approximation of Integral Operators by Variable-Order Interpolation in NUMERISCHE MATHEMATIK
  • 2010-04. Approximation of solution operators of elliptic partial differential equations by - and -matrices in NUMERISCHE MATHEMATIK
  • 2005-12. A fast adaptive solver for hierarchically semiseparable representations in CALCOLO
  • 2000-02. A Sparse ℋ-Matrix Arithmetic. in COMPUTING
  • 2004-06. The Eigenvalue Problem for the 2D Laplacian in ℋ-Matrix Arithmetic and Application to the Heat and Wave Equation in COMPUTING
  • 2003-08. Construction and Arithmetics of H-Matrices in COMPUTING
  • 2009. Hierarchische Matrizen, Algorithmen und Analysis in NONE
  • 1999-04. A Sparse Matrix Arithmetic Based on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Cal H$\end{document}-Matrices. Part I: Introduction to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\Cal H}$\end{document}-Matrices in COMPUTING
  • 2003-07. Existence of ℋ-matrix approximants to the inverse FE-matrix of elliptic operators with L∞-coefficients in NUMERISCHE MATHEMATIK
  • 2009-06. Domain decomposition based -LU preconditioning in NUMERISCHE MATHEMATIK
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00791-015-0233-3

    DOI

    http://dx.doi.org/10.1007/s00791-015-0233-3

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1053685599


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Kiel University", 
              "id": "https://www.grid.ac/institutes/grid.9764.c", 
              "name": [
                "Department of Computer Science, University of Kiel, Kiel, Germany"
              ], 
              "type": "Organization"
            }, 
            "familyName": "B\u00f6rm", 
            "givenName": "Steffen", 
            "id": "sg:person.016063530723.23", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016063530723.23"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Kiel University", 
              "id": "https://www.grid.ac/institutes/grid.9764.c", 
              "name": [
                "Department of Computer Science, University of Kiel, Kiel, Germany"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Reimer", 
            "givenName": "Knut", 
            "id": "sg:person.014406133140.72", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014406133140.72"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/s006070050015", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1001171166", 
              "https://doi.org/10.1007/s006070050015"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1002/nla.691", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1015225989"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1002/nla.691", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1015225989"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00211-009-0278-7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1016593139", 
              "https://doi.org/10.1007/s00211-009-0278-7"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00211-009-0278-7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1016593139", 
              "https://doi.org/10.1007/s00211-009-0278-7"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00211-004-0564-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1016928699", 
              "https://doi.org/10.1007/s00211-004-0564-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00211-004-0564-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1016928699", 
              "https://doi.org/10.1007/s00211-004-0564-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-642-59709-1_2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1017730404", 
              "https://doi.org/10.1007/978-3-642-59709-1_2"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-642-00222-9", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1024806832", 
              "https://doi.org/10.1007/978-3-642-00222-9"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-642-00222-9", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1024806832", 
              "https://doi.org/10.1007/978-3-642-00222-9"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10092-005-0103-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1027299812", 
              "https://doi.org/10.1007/s10092-005-0103-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00211-009-0218-6", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1028190341", 
              "https://doi.org/10.1007/s00211-009-0218-6"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00211-009-0218-6", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1028190341", 
              "https://doi.org/10.1007/s00211-009-0218-6"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10915-008-9240-6", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1031697362", 
              "https://doi.org/10.1007/s10915-008-9240-6"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00607-002-1450-4", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1032782830", 
              "https://doi.org/10.1007/s00607-002-1450-4"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/j.laa.2006.10.021", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1035754670"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/j.camwa.2005.03.011", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1041938180"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00607-005-0146-y", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1042939469", 
              "https://doi.org/10.1007/s00607-005-0146-y"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00607-005-0146-y", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1042939469", 
              "https://doi.org/10.1007/s00607-005-0146-y"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00607-003-0019-1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1048274165", 
              "https://doi.org/10.1007/s00607-003-0019-1"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00211-002-0445-6", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1050237178", 
              "https://doi.org/10.1007/s00211-002-0445-6"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00607-003-0061-z", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1053674363", 
              "https://doi.org/10.1007/s00607-003-0061-z"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/060651173", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062849059"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/060669747", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062849692"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/09074543x", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062855879"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/pl00021408", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1085169737", 
              "https://doi.org/10.1007/pl00021408"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.4171/091", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1099348362"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2013-12", 
        "datePublishedReg": "2013-12-01", 
        "description": "Many matrices appearing in numerical methods for partial differential equations and integral equations are rank-structured, i.e., they contain submatrices that can be approximated by matrices of low rank. A relatively general class of rank-structured matrices are H2-matrices: they can reach the optimal order of complexity, but are still general enough for a large number of practical applications. We consider algorithms for performing algebraic operations with H2-matrices, i.e., for approximating the matrix product, inverse or factorizations in almost linear complexity. The new approach is based on local low-rank updates that can be performed in linear complexity. These updates can be combined with a recursive procedure to approximate the product of two H2-matrices, and these products can be used to approximate the matrix inverse and the LR or Cholesky factorization. Numerical experiments indicate that the new algorithm leads to preconditioners that require O(n) units of storage, can be evaluated in O(n) operations, and take O(nlogn) operations to set up.", 
        "genre": "research_article", 
        "id": "sg:pub.10.1007/s00791-015-0233-3", 
        "inLanguage": [
          "en"
        ], 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1134510", 
            "issn": [
              "1432-9360", 
              "1433-0369"
            ], 
            "name": "Computing and Visualization in Science", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "6", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "16"
          }
        ], 
        "name": "Efficient arithmetic operations for rank-structured matrices based on hierarchical low-rank updates", 
        "pagination": "247-258", 
        "productId": [
          {
            "name": "readcube_id", 
            "type": "PropertyValue", 
            "value": [
              "ffb665b438892498538abe512e57b655fbf348b6830f1607e9f7721799598b21"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s00791-015-0233-3"
            ]
          }, 
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1053685599"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s00791-015-0233-3", 
          "https://app.dimensions.ai/details/publication/pub.1053685599"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2019-04-10T17:33", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000001_0000000264/records_8672_00000516.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "http://link.springer.com/10.1007%2Fs00791-015-0233-3"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00791-015-0233-3'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00791-015-0233-3'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00791-015-0233-3'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00791-015-0233-3'


     

    This table displays all metadata directly associated to this object as RDF triples.

    145 TRIPLES      21 PREDICATES      48 URIs      19 LITERALS      7 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00791-015-0233-3 schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author N9185a138103241cbb41a8b5fba31001e
    4 schema:citation sg:pub.10.1007/978-3-642-00222-9
    5 sg:pub.10.1007/978-3-642-59709-1_2
    6 sg:pub.10.1007/pl00021408
    7 sg:pub.10.1007/s00211-002-0445-6
    8 sg:pub.10.1007/s00211-004-0564-3
    9 sg:pub.10.1007/s00211-009-0218-6
    10 sg:pub.10.1007/s00211-009-0278-7
    11 sg:pub.10.1007/s00607-002-1450-4
    12 sg:pub.10.1007/s00607-003-0019-1
    13 sg:pub.10.1007/s00607-003-0061-z
    14 sg:pub.10.1007/s00607-005-0146-y
    15 sg:pub.10.1007/s006070050015
    16 sg:pub.10.1007/s10092-005-0103-3
    17 sg:pub.10.1007/s10915-008-9240-6
    18 https://doi.org/10.1002/nla.691
    19 https://doi.org/10.1016/j.camwa.2005.03.011
    20 https://doi.org/10.1016/j.laa.2006.10.021
    21 https://doi.org/10.1137/060651173
    22 https://doi.org/10.1137/060669747
    23 https://doi.org/10.1137/09074543x
    24 https://doi.org/10.4171/091
    25 schema:datePublished 2013-12
    26 schema:datePublishedReg 2013-12-01
    27 schema:description Many matrices appearing in numerical methods for partial differential equations and integral equations are rank-structured, i.e., they contain submatrices that can be approximated by matrices of low rank. A relatively general class of rank-structured matrices are H2-matrices: they can reach the optimal order of complexity, but are still general enough for a large number of practical applications. We consider algorithms for performing algebraic operations with H2-matrices, i.e., for approximating the matrix product, inverse or factorizations in almost linear complexity. The new approach is based on local low-rank updates that can be performed in linear complexity. These updates can be combined with a recursive procedure to approximate the product of two H2-matrices, and these products can be used to approximate the matrix inverse and the LR or Cholesky factorization. Numerical experiments indicate that the new algorithm leads to preconditioners that require O(n) units of storage, can be evaluated in O(n) operations, and take O(nlogn) operations to set up.
    28 schema:genre research_article
    29 schema:inLanguage en
    30 schema:isAccessibleForFree true
    31 schema:isPartOf Nacfea0b76604434f81885c3425b7ec7c
    32 Nb06cab1361604295acd7929f68403adb
    33 sg:journal.1134510
    34 schema:name Efficient arithmetic operations for rank-structured matrices based on hierarchical low-rank updates
    35 schema:pagination 247-258
    36 schema:productId N6fb0cea65c1f42e086aecc122f8af480
    37 N9ed39383d4c34476a30bd2cdd8748e2d
    38 Nf54e1940c8784c2491240490969eb087
    39 schema:sameAs https://app.dimensions.ai/details/publication/pub.1053685599
    40 https://doi.org/10.1007/s00791-015-0233-3
    41 schema:sdDatePublished 2019-04-10T17:33
    42 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    43 schema:sdPublisher Nf5bd953449824d0088ebd60038db2951
    44 schema:url http://link.springer.com/10.1007%2Fs00791-015-0233-3
    45 sgo:license sg:explorer/license/
    46 sgo:sdDataset articles
    47 rdf:type schema:ScholarlyArticle
    48 N6fb0cea65c1f42e086aecc122f8af480 schema:name dimensions_id
    49 schema:value pub.1053685599
    50 rdf:type schema:PropertyValue
    51 N9185a138103241cbb41a8b5fba31001e rdf:first sg:person.016063530723.23
    52 rdf:rest Nadd98ccb1a2b465f9c115059c7cd36f6
    53 N9ed39383d4c34476a30bd2cdd8748e2d schema:name readcube_id
    54 schema:value ffb665b438892498538abe512e57b655fbf348b6830f1607e9f7721799598b21
    55 rdf:type schema:PropertyValue
    56 Nacfea0b76604434f81885c3425b7ec7c schema:volumeNumber 16
    57 rdf:type schema:PublicationVolume
    58 Nadd98ccb1a2b465f9c115059c7cd36f6 rdf:first sg:person.014406133140.72
    59 rdf:rest rdf:nil
    60 Nb06cab1361604295acd7929f68403adb schema:issueNumber 6
    61 rdf:type schema:PublicationIssue
    62 Nf54e1940c8784c2491240490969eb087 schema:name doi
    63 schema:value 10.1007/s00791-015-0233-3
    64 rdf:type schema:PropertyValue
    65 Nf5bd953449824d0088ebd60038db2951 schema:name Springer Nature - SN SciGraph project
    66 rdf:type schema:Organization
    67 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    68 schema:name Mathematical Sciences
    69 rdf:type schema:DefinedTerm
    70 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    71 schema:name Pure Mathematics
    72 rdf:type schema:DefinedTerm
    73 sg:journal.1134510 schema:issn 1432-9360
    74 1433-0369
    75 schema:name Computing and Visualization in Science
    76 rdf:type schema:Periodical
    77 sg:person.014406133140.72 schema:affiliation https://www.grid.ac/institutes/grid.9764.c
    78 schema:familyName Reimer
    79 schema:givenName Knut
    80 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014406133140.72
    81 rdf:type schema:Person
    82 sg:person.016063530723.23 schema:affiliation https://www.grid.ac/institutes/grid.9764.c
    83 schema:familyName Börm
    84 schema:givenName Steffen
    85 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016063530723.23
    86 rdf:type schema:Person
    87 sg:pub.10.1007/978-3-642-00222-9 schema:sameAs https://app.dimensions.ai/details/publication/pub.1024806832
    88 https://doi.org/10.1007/978-3-642-00222-9
    89 rdf:type schema:CreativeWork
    90 sg:pub.10.1007/978-3-642-59709-1_2 schema:sameAs https://app.dimensions.ai/details/publication/pub.1017730404
    91 https://doi.org/10.1007/978-3-642-59709-1_2
    92 rdf:type schema:CreativeWork
    93 sg:pub.10.1007/pl00021408 schema:sameAs https://app.dimensions.ai/details/publication/pub.1085169737
    94 https://doi.org/10.1007/pl00021408
    95 rdf:type schema:CreativeWork
    96 sg:pub.10.1007/s00211-002-0445-6 schema:sameAs https://app.dimensions.ai/details/publication/pub.1050237178
    97 https://doi.org/10.1007/s00211-002-0445-6
    98 rdf:type schema:CreativeWork
    99 sg:pub.10.1007/s00211-004-0564-3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1016928699
    100 https://doi.org/10.1007/s00211-004-0564-3
    101 rdf:type schema:CreativeWork
    102 sg:pub.10.1007/s00211-009-0218-6 schema:sameAs https://app.dimensions.ai/details/publication/pub.1028190341
    103 https://doi.org/10.1007/s00211-009-0218-6
    104 rdf:type schema:CreativeWork
    105 sg:pub.10.1007/s00211-009-0278-7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1016593139
    106 https://doi.org/10.1007/s00211-009-0278-7
    107 rdf:type schema:CreativeWork
    108 sg:pub.10.1007/s00607-002-1450-4 schema:sameAs https://app.dimensions.ai/details/publication/pub.1032782830
    109 https://doi.org/10.1007/s00607-002-1450-4
    110 rdf:type schema:CreativeWork
    111 sg:pub.10.1007/s00607-003-0019-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1048274165
    112 https://doi.org/10.1007/s00607-003-0019-1
    113 rdf:type schema:CreativeWork
    114 sg:pub.10.1007/s00607-003-0061-z schema:sameAs https://app.dimensions.ai/details/publication/pub.1053674363
    115 https://doi.org/10.1007/s00607-003-0061-z
    116 rdf:type schema:CreativeWork
    117 sg:pub.10.1007/s00607-005-0146-y schema:sameAs https://app.dimensions.ai/details/publication/pub.1042939469
    118 https://doi.org/10.1007/s00607-005-0146-y
    119 rdf:type schema:CreativeWork
    120 sg:pub.10.1007/s006070050015 schema:sameAs https://app.dimensions.ai/details/publication/pub.1001171166
    121 https://doi.org/10.1007/s006070050015
    122 rdf:type schema:CreativeWork
    123 sg:pub.10.1007/s10092-005-0103-3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1027299812
    124 https://doi.org/10.1007/s10092-005-0103-3
    125 rdf:type schema:CreativeWork
    126 sg:pub.10.1007/s10915-008-9240-6 schema:sameAs https://app.dimensions.ai/details/publication/pub.1031697362
    127 https://doi.org/10.1007/s10915-008-9240-6
    128 rdf:type schema:CreativeWork
    129 https://doi.org/10.1002/nla.691 schema:sameAs https://app.dimensions.ai/details/publication/pub.1015225989
    130 rdf:type schema:CreativeWork
    131 https://doi.org/10.1016/j.camwa.2005.03.011 schema:sameAs https://app.dimensions.ai/details/publication/pub.1041938180
    132 rdf:type schema:CreativeWork
    133 https://doi.org/10.1016/j.laa.2006.10.021 schema:sameAs https://app.dimensions.ai/details/publication/pub.1035754670
    134 rdf:type schema:CreativeWork
    135 https://doi.org/10.1137/060651173 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062849059
    136 rdf:type schema:CreativeWork
    137 https://doi.org/10.1137/060669747 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062849692
    138 rdf:type schema:CreativeWork
    139 https://doi.org/10.1137/09074543x schema:sameAs https://app.dimensions.ai/details/publication/pub.1062855879
    140 rdf:type schema:CreativeWork
    141 https://doi.org/10.4171/091 schema:sameAs https://app.dimensions.ai/details/publication/pub.1099348362
    142 rdf:type schema:CreativeWork
    143 https://www.grid.ac/institutes/grid.9764.c schema:alternateName Kiel University
    144 schema:name Department of Computer Science, University of Kiel, Kiel, Germany
    145 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...