{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:33:57Z","timestamp":1787337237590,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2001,1]]},"abstract":"<jats:p>A propagating interface can develop corners and discontinuities as it advances. Level set algorithms have been extensively applied for the problems in which the solution has advancing fronts. One of the most popular level set algorithms is the so-called {fast marching method} (FMM), which requires total $\\cal O(N\\log_2N)$ operations, where N is the number of grid points. The article is concerned with the development of an $\\cal O(N)$ level set algorithm called the group marching method (GMM). The new method is based on the narrow band approach as in the FMM. However, it is incorporating a correction-by-iteration strategy to advance a group of grid points at a time, rather than sorting the solution in the narrow band to march forward a single grid point. After selecting a group of grid points appropriately, the GMM advances the group in two iterations for the cost of slightly larger than one iteration. Numerical results are presented to show the efficiency of the method, applied to the eikonal equation in two and three dimensions.<\/jats:p>","DOI":"10.1137\/s1064827500367130","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"2178-2193","source":"Crossref","is-referenced-by-count":74,"title":["An $\\cal O(N)$ Level Set Method for Eikonal Equations"],"prefix":"10.1137","volume":"22","author":[{"given":"Seongjai","family":"Kim","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","first-page":"2","volume":"58","author":"Cerveny V.","year":"1985","journal-title":"J. Geophys.","ISSN":"https:\/\/id.crossref.org\/issn\/0340-062X","issn-type":"print"},{"key":"R2","unstructured":"V. Cerveny, I. A. Molotkov, and I. Psencik,\n                      Ray Methods in Seismology\n                      , University of Karlova Press, Prague, 1977."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1983-0690039-8"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"Michael Crandall, Panagiotis Souganidis, Developments in the theory of nonlinear first\u2010order partial differential equations, North\u2010Holland Math. Stud., Vol. 92, North\u2010Holland, Amsterdam, 1984, 131\u201314286j:35033","DOI":"10.1016\/S0304-0208(08)73688-0"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"J. Dellinger,\n                      Anisotropic finite\u2010difference traveltimes\n                      , in Expanded Abstracts, Society of Exploration Geophysicists, Tulsa, OK, 1991, pp. 1530\u20131533.","DOI":"10.1190\/1.1888998"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"J. Dellinger and W. W. Symes,\n                      Anisotropic finite\u2010difference traveltimes using a Hamilton\u2010Jacobi solver\n                      , in Expanded Abstracts, Society of Exploration Geophysicists, Tulsa, OK, 1997, pp. 1786\u20131789.","DOI":"10.1190\/1.1885780"},{"key":"R7","unstructured":"F. Friedlander,\n                      Sound Pulses\n                      , Cambridge University Press, Cambridge, UK, 1958."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1190\/1.1443639"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"S. Kim,\n                      ENO\u2010DNO\u2010PS: A stable, second\u2010order accuracy eikonal solver\n                      , in Expanded Abstracts, Society of Exploration Geophysicists, Tulsa, OK, 1999, pp. 1747\u20131750.","DOI":"10.1190\/1.1820874"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"S. Kim,\n                      On eikonal solvers for anisotropic traveltimes\n                      , in Expanded Abstracts, Society of Exploration Geophysicists, Tulsa, OK, 1999, pp. 1875\u20131878.","DOI":"10.1190\/1.1820911"},{"key":"R11","unstructured":"S. Kim,\n                      An \ud835\udd46(N) Second\u2010Order Level Set Method\n                      , in preparation, 2000."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1190\/1.1444693"},{"key":"R13","unstructured":"S. Kim, W. Symes, and M. A. El\u2010Mageed,\n                      Superconvergent difference formulas for traveltimes and amplitudes\n                      , in Mathematical and Numerical Aspects of Wave Propagation, J. A. DeSanto, ed., SIAM, Philadelphia, 1998, pp. 591\u2013593."},{"key":"R14","doi-asserted-by":"crossref","unstructured":"P. Lax,\n                      Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves\n                      , SIAM, Philadelphia, 1973.","DOI":"10.1137\/1.9781611970562"},{"key":"R15","first-page":"570","volume":"60","author":"Papakov G.","year":"1996","journal-title":"Prikl. Mat. Mekh."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.93.18.9389"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90002-2"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/0728049"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"A. Popovici and J. A. 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