{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T01:14:53Z","timestamp":1787361293454,"version":"build-2736575974"},"reference-count":17,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We introduce a novel multisection method for the solution of integral equations on unbounded domains. The method is applied to the rough surface scattering problem in three dimensions, in particular to a Brakhage\u2013Werner-type integral equation for acoustic scattering by an unbounded rough surface with Dirichlet boundary condition, where the fundamental solution is replaced by some appropriate half-space Green's function. The basic idea of the multisection method is to solve an integral equation $A\\varphi = f$ by approximately solving the equation $P_{\\varrho}A P_{\\tau}\\varphi = P_{\\varrho} f$ for some positive constants $\\varrho, \\tau$. Here $P_{\\varrho}$ is a projection operator that truncates a function to a ball with radius $\\varrho&gt;0$. For a very general class of operators A, for which the Brakhage\u2013Werner equation from acoustic scattering is a particular example, we will show existence of approximate solutions to the multisection equation and show that approximate solutions to the multisection equation approximate the true solution $\\varphi_0$ of the operator equation $A\\varphi = f$. Finally, we describe a numerical implementation of the multisection algorithm and provide numerical examples for the case of rough surface scattering in three dimensions.<\/jats:p>","DOI":"10.1137\/060673321","type":"journal-article","created":{"date-parts":[[2008,4,16]],"date-time":"2008-04-16T18:05:23Z","timestamp":1208369123000},"page":"1780-1798","source":"Crossref","is-referenced-by-count":7,"title":["Convergence and Numerics of a Multisection Method for Scattering by Three-Dimensional Rough Surfaces"],"prefix":"10.1137","volume":"46","author":[{"given":"Eric","family":"Heinemeyer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marko","family":"Lindner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roland","family":"Potthast","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,16]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/050635262"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2006.1752"},{"key":"R3","unstructured":"S. N. Chandler-Wilde and M. Lindner,\n                      Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method\n                      , J. Integral Equations Appl., to appear."},{"key":"R4","doi-asserted-by":"crossref","unstructured":"D. Colton and R. Kress,\n                      Inverse Acoustic and Electromagnetic Scattering Theory\n                      , 2nd ed., Appl. Math. Sci. 93, Springer-Verlag, Berlin, 1998.","DOI":"10.1007\/978-3-662-03537-5"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"DeSanto,\n                      Scattering by rough surfaces\n                      , in Scattering: Scattering and Inverse Scattering in Pure and Applied Science, R. Pike and P. Sabatier, eds., Academic Press, New York, 2002, pp. 15\u201336.","DOI":"10.1016\/B978-012613760-6\/50003-6"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"H. W. Engl, M. Hanke, and A. Neubauer,\n                      Regularization of Inverse Problems\n                      , Math. Appl. 375, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"R. Hagen, S. Roch, and B. Silbermann,\n                      C$^*$-Algebras and Numerical Analysis\n                      , Marcel Dekker, New York, 2001.","DOI":"10.1201\/9781482270679"},{"key":"R8","unstructured":"E. Heinemeyer,\n                      Integral Equation Methods for Rough Surface Scattering Problems in Three Dimensions\n                      , Ph.D. dissertation, University of G\u00f6ttingen, G\u00f6ttingen, Germany, 2008."},{"key":"R9","doi-asserted-by":"crossref","unstructured":"R. Kress,\n                      Linear Integral Equations\n                      , Appl. Math. Sci. 82, Springer-Verlag, Berlin, 1989.","DOI":"10.1007\/978-3-642-97146-4_7"},{"key":"R10","unstructured":"M. Lindner,\n                      Infinite Matrices and Their Finite Sections: An Introduction to the Limit Operator Method\n                      , Front. Math., Birkh\u00e4user-Verlag, Basel, Boston, Berlin, 2006."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1002\/mma.210"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"J. A. Ogilvy,\n                      Theory of Wave Scattering from Random Rough Surfaces\n                      , Adam Hilger, Bristol, UK, 1991.","DOI":"10.1121\/1.401410"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"V. S. Rabinovich, S. Roch, and B. Silbermann,\n                      Limit Operators and Their Applications in Operator Theory\n                      , Oper. Theory. Adv. Appl. 150, Birkh\u00e4user-Verlag, Basel, Boston, Berlin, 2004.","DOI":"10.1007\/978-3-0348-7911-8"},{"key":"R14","unstructured":"S. Roch,\n                      Finite sections of band-dominated operators\n                      , Mem. Amer. Math. Soc., to appear."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1080\/13616670109409779"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"A. G. Voronovich,\n                      Wave Scattering from Rough Surfaces\n                      , 2nd ed., Springer-Verlag, Berlin, 1998.","DOI":"10.1007\/978-3-642-59936-1"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1088\/0959-7174\/11\/1\/201"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060673321","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:59Z","timestamp":1787325239000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060673321"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":17,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060673321"],"URL":"https:\/\/doi.org\/10.1137\/060673321","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}