{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T08:03:55Z","timestamp":1787385835927,"version":"build-2736575974"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2000,1]]},"abstract":"<jats:p>We present a variant of the AINV factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix. The new preconditioner is breakdown-free and, when used in conjunction with the conjugate gradient method, results in a reliable solver for highly ill-conditioned linear systems. We also investigate an alternative approach to a stable approximate inverse algorithm, based on the idea of diagonally compensated reduction of matrix entries. The results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented.<\/jats:p>","DOI":"10.1137\/s1064827599356900","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1318-1332","source":"Crossref","is-referenced-by-count":129,"title":["Robust Approximate Inverse Preconditioning for the Conjugate Gradient Method"],"prefix":"10.1137","volume":"22","author":[{"given":"Michele","family":"Benzi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jane K.","family":"Cullum","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Miroslav","family":"Tuma","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,2,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620200511"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"O. Axelsson,\n                      Iterative Solution Methods\n                      , Cambridge University Press, Cambridge, 1994.","DOI":"10.1017\/CBO9780511624100"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1680010207"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1177\/109434209901300202"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-0887(1998100)14:10<897::AID-CNM196>3.0.CO;2-L"},{"key":"R6","unstructured":"M. Benzi, J. Mar\u00edn, and M. T\u016fma,\n                      A two\u2010level parallel preconditioner based on sparse approximate inverses\n                      , in Iterative Methods in Scientific Computation IV, D. R. Kincaid and A. C. Elster, eds., IMACS Series in Computational and Applied Mathematics, 5, IMACS, New Brunswick, NJ, 1999, pp. 167\u2013178."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827594271421"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595294691"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02512364"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(98)00118-4"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827598339372"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1506(200004\/05)7:3<99::AID-NLA188>3.3.CO;2-X"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"A. Berman and R. J. Plemmons,\n                      Nonnegative Matrices in the Mathematical Sciences\n                      , Academic Press, New York, 1979.","DOI":"10.1016\/B978-0-12-092250-5.50009-6"},{"key":"R14","unstructured":"R. Bridson,\n                      Multi\u2010Resolution Approximate Inverses\n                      , M.Sc. thesis, Computer Science Department, Waterloo University, Waterloo, Ontario, Canada, 1999."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827598335842"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/S106482759833913X"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(97)00171-4"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827594270415"},{"key":"R19","unstructured":"T. Davis,\n                      University of Florida Sparse Matrix Collection\n                      , http:\/\/www.cise.ufl.edu\/\u02dcdavis\/sparse\/ (1999)."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1986-0842130-7"},{"key":"R21","unstructured":"M. R. Field,\n                      An Efficient Parallel Preconditioner for the Conjugate Gradient Algorithm\n                      , Hitachi Dublin Laboratory Technical Report HDL\u2010TR\u201097\u2010175, Dublin, Ireland, 1997."},{"key":"R22","unstructured":"M. R. Field,\n                      Improving the Performance of Factorised Sparse Approximate Inverse Preconditioner\n                      , Hitachi Dublin Laboratory Technical Report HDL\u2010TR\u201098\u2010199, Dublin, Ireland, 1998."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827594276552"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595288425"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0207(19970615)40:11<2109::AID-NME163>3.0.CO;2-1"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(98)00117-2"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1680010208"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1506(199811\/12)5:6<483::AID-NLA156>3.3.CO;2-Z"},{"key":"R29","unstructured":"S. A. Kharchenko, L. Yu. Kolotilina, A. A. Nikishin, and A. Yu. Yeremin,\n                      A Reliable AINV\u2010type Preconditioning Method for Constructing Sparse Approximate Inverse Preconditioners in Factored Form\n                      , Moscow University, Moscow, Russia, 1999, preprint."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1506(199910\/11)6:7<515::AID-NLA176>3.0.CO;2-0"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/0614004"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1145\/214392.214398"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1980-0559197-0"},{"key":"R34","first-page":"148","volume":"31","author":"Meijerink J. A.","year":"1977","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"R35","unstructured":"J. E. Morel, M. L. Hall, and M. J. Shashkov,\n                      A Local Support\u2010Operators Diffusion Discretization Scheme for Hexahedral Meshes\n                      , Report LA\u2010UR\u201099\u20104358, Los Alamos National Laboratory, Los Alamos, NM, 1999."},{"key":"R36","unstructured":"National Institute of Standards,\n                      Matrix Market\n                      , http:\/\/math.nist.gov\/MatrixMarket (1999)."},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620381007"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(91)90383-8"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827599356900","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:34:20Z","timestamp":1787333660000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827599356900"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,1]]},"references-count":38,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["10.1137\/S1064827599356900"],"URL":"https:\/\/doi.org\/10.1137\/s1064827599356900","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,1]]}}}