{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:23Z","timestamp":1787336903462,"version":"build-2736575974"},"reference-count":25,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","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 finite difference multigrid technique that efficiently solves optimal control problems associated with the steady state solid fuel ignition model. This is a nonlinear indefinite problem which gives rise to singular optimal control systems. Regarding the direct problem, necessary and sufficient conditions for existence of solutions are given and second-order convergence of numerical solution errors with respect to the mesh size is proven. A robust nonlinear multigrid method is implemented to validate this convergence estimate.<\/jats:p>\n                  <jats:p>Two optimal control strategies for solid fuel ignition phenomena are considered and compared. The corresponding optimality systems are solved to second-order accuracy by a multigrid method whose convergence properties are independent of the values of the weights in the cost functionals and of the number of grid points.<\/jats:p>","DOI":"10.1137\/s1064827599360194","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"263-284","source":"Crossref","is-referenced-by-count":21,"title":["The Numerical Solution of the Steady State Solid Fuel Ignition Model and Its Optimal Control"],"prefix":"10.1137","volume":"22","author":[{"given":"Alfio","family":"Borz\u00ec","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Karl","family":"Kunisch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0902035"},{"key":"R2","unstructured":"E. Arian and S. Ta\u2019asan,\n                      Smoothers for optimization problems\n                      , in Seventh Copper Mountain Conference on Multigrid Methods, Vol. CP3339, N. D. Melson, T. A. Manteuffel, S. F. McCormick, and C. C. Douglas, eds., NASA Conference Publication, NASA, Hampton, VA, 1995, pp. 15\u201330."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"Eyal Arian, Shlomo Ta\u2019asan, Shape optimization in one shot, Progr. Systems Control Theory, Vol. 19, Birkh\u00e4user Boston, Boston, MA, 1995, 23\u2013401352249","DOI":"10.1007\/978-1-4612-0839-6_2"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4546-9"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/0703001"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1977-0431719-X"},{"key":"R7","unstructured":"A. Brandt,\n                      Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics\n                      , GMD\u2010Studien 85, St. Augustin, Germany, 1984."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(86)90095-0"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0903012"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1969-12175-0"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0906055"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"W. Hackbusch,\n                      Multi\u2010Grid Methods and Applications\n                      , Springer\u2010Verlag, New York, 1985.","DOI":"10.1007\/978-3-662-02427-0"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF00934478"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"K. Ito and K. Kunisch,\n                      Optimal Control of the Solid Fuel Model with H1\u2010Cost\n                      , preprint, 1998.","DOI":"10.1002\/047134608X.W1029"},{"key":"R15","unstructured":"A. Kauffmann,\n                      Optimal Control of the Solid Fuel Ignition Model\n                      , Ph.D. thesis, Technical University of Berlin, Berlin 1998."},{"key":"R16","volume-title":"Monotone iterative techniques for nonlinear differential equations","author":"Ladde G.","year":"1985"},{"key":"R17","unstructured":"J.\u2010L. Lions,\n                      Control of Distributed Singular Systems\n                      , Gauthier\u2010Villars, Paris, 1985."},{"key":"R18","doi-asserted-by":"crossref","unstructured":"Th. Meis, H. Lehmann, H. Michael, Application of the multigrid method to a nonlinear indefinite problem, Lecture Notes in Math., Vol. 960, Springer, Berlin, 1982, 545\u201355784c:65144","DOI":"10.1007\/BFb0069943"},{"key":"R19","volume-title":"The finite difference method in partial differential equations","author":"Mitchell Andrew","year":"1980"},{"key":"R20","unstructured":"J. M. Ortega,\n                      Numerical Analysis\n                      , Academic Press, New York, 1972."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389659"},{"key":"R22","first-page":"117","volume":"14","author":"Kondratiev V.","year":"1997","journal-title":"Asymptot. Anal."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.1972.21.21079"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"K. St\u00fcben and U. Trottenberg,\n                      Multigrid methods: Fundamental algorithms, model problem analysis and applications\n                      , in Multigrid Methods, W. 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Trottenberg, eds., Lecture Notes in Math. 960, Springer\u2010Verlag, New York, 1982.","DOI":"10.1007\/BFb0069928"},{"key":"R25","volume-title":"Matrix iterative analysis","author":"Varga Richard","year":"1962"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827599360194","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:56Z","timestamp":1787333756000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827599360194"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,1]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["10.1137\/S1064827599360194"],"URL":"https:\/\/doi.org\/10.1137\/s1064827599360194","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,1]]}}}