{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:25:13Z","timestamp":1776831913208,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"229","license":[{"start":{"date-parts":[[2000,5,21]],"date-time":"2000-05-21T00:00:00Z","timestamp":958867200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter.<\/p>","DOI":"10.1090\/s0025-5718-99-01134-5","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"237-267","source":"Crossref","is-referenced-by-count":19,"title":["Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations"],"prefix":"10.1090","volume":"69","author":[{"given":"G.","family":"Milstein","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Tretyakov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,5,21]]},"reference":[{"issue":"217","key":"1","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1090\/S0025-5718-97-00776-X","article-title":"A stochastic particle method for the McKean-Vlasov and the Burgers equation","volume":"66","author":"Bossy, Mireille","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","series-title":"Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-0409-8","volume-title":"Mathematical modelling of heat and mass transfer processes","volume":"348","author":"Danilov, V. G.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0792337891"},{"issue":"6","key":"3","doi-asserted-by":"publisher","first-page":"1729","DOI":"10.1137\/0732078","article-title":"Adaptive finite element methods for parabolic problems. IV. Nonlinear problems","volume":"32","author":"Eriksson, Kenneth","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"4","series-title":"Springer Series in Computational Physics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58239-4","volume-title":"Computational techniques for fluid dynamics. 1","author":"Fletcher, C. A. J.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/3540530584","edition":"2"},{"key":"5","series-title":"Annals of Mathematics Studies","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1515\/9781400881598","volume-title":"Functional integration and partial differential equations","volume":"109","author":"Freidlin, Mark","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0691083541"},{"key":"6","series-title":"Lectures in Mathematics ETH Z\\\"{u}rich","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-9191-2","volume-title":"Markov processes and differential equations: asymptotic problems","author":"Freidlin, Mark","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/3764353929"},{"issue":"2","key":"7","first-page":"87","article-title":"Some problems in the theory of quasi-linear equations","volume":"14","author":"Gel\u2032fand, I. M.","year":"1959","journal-title":"Uspehi Mat. Nauk","ISSN":"https:\/\/id.crossref.org\/issn\/0042-1316","issn-type":"print"},{"key":"8","isbn-type":"print","first-page":"77","article-title":"Composite geometric continuous Lagrange-Hermite curves","author":"Geise, Gerhard","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/3055006798"},{"key":"9","first-page":"191","article-title":"Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of the time","volume":"51(93)","author":"Il\u2032in, A. M.","year":"1960","journal-title":"Mat. Sb. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"},{"key":"10","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","volume-title":"Solving ordinary differential equations. I","volume":"8","author":"Hairer, E.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540566708","edition":"2"},{"key":"11","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3968-0","volume-title":"Multiple scale and singular perturbation methods","volume":"114","author":"Kevorkian, J.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0387942025"},{"key":"12","series-title":"Applications of Mathematics (New York)","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12616-5","volume-title":"Numerical solution of stochastic differential equations","volume":"23","author":"Kloeden, Peter E.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/3540540628"},{"key":"13","doi-asserted-by":"publisher","first-page":"567","DOI":"10.2307\/1968941","article-title":"On an inequality of Marcel Riesz","volume":"40","author":"Young, L. C.","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"14","series-title":"Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-8455-5","volume-title":"Numerical integration of stochastic differential equations","volume":"313","author":"Milstein, G. N.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/079233213X"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1137\/1140061","article-title":"Solution of the first boundary value problem for equations of parabolic type by means of the integration of stochastic differential equations","volume":"40","author":"Mil\u2032shte\u012dn, G. N.","year":"1995","journal-title":"Teor. Veroyatnost. i Primenen.","ISSN":"https:\/\/id.crossref.org\/issn\/0040-361X","issn-type":"print"},{"issue":"1","key":"16","first-page":"210","article-title":"Application of the numerical integration of stochastic equations for the solution of boundary value problems with Neumann boundary conditions","volume":"41","author":"Mil\u2032shte\u012dn, G. N.","year":"1996","journal-title":"Teor. Veroyatnost. i Primenen.","ISSN":"https:\/\/id.crossref.org\/issn\/0040-361X","issn-type":"print"},{"issue":"1-2","key":"17","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1080\/17442509708834131","article-title":"Weak approximation of a diffusion process in a bounded domain","volume":"62","author":"Milstein, G. N.","year":"1997","journal-title":"Stochastics Stochastics Rep.","ISSN":"https:\/\/id.crossref.org\/issn\/1045-1129","issn-type":"print"},{"key":"18","unstructured":"G.N. Milstein.  The probability approach to numerical solution of nonlinear parabolic equations. Preprint No. 380, Weierstra\u00df-Institut f\u00fcr Angewandte Analysis und Stochastik, Berlin, 1998 (submitted)."},{"issue":"6","key":"19","doi-asserted-by":"publisher","first-page":"2142","DOI":"10.1137\/S0036142996278967","article-title":"Numerical methods in the weak sense for stochastic differential equations with small noise","volume":"34","author":"Milstein, G. N.","year":"1997","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"20","unstructured":"G.N. Milstein, M.V. Tretyakov. Numerical methods for nonlinear parabolic equations with small parameter based on probability approach. Preprint No. 396, Weierstra\u00df-Institut f\u00fcr Angewandte Analysis und Stochastik, Berlin, 1998."},{"key":"21","doi-asserted-by":"publisher","first-page":"712","DOI":"10.2307\/1968951","article-title":"Rings with minimal condition for left ideals","volume":"40","author":"Hopkins, Charles","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"1","key":"22","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/BF01438265","article-title":"Discretization and simulation of stochastic differential equations","volume":"3","author":"Pardoux, \u00c9.","year":"1985","journal-title":"Acta Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0167-8019","issn-type":"print"},{"key":"23","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-85268-1","volume-title":"Numerical approximation of partial differential equations","volume":"23","author":"Quarteroni, Alfio","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3540571116"},{"key":"24","series-title":"Interscience Tracts in Pure and Applied Mathematics, No. 4","volume-title":"Difference methods for initial-value problems","author":"Richtmyer, Robert D.","year":"1967","edition":"2"},{"key":"25","volume-title":"Computational fluid dynamics","author":"Roache, Patrick J.","year":"1976"},{"key":"26","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03206-0","volume-title":"Numerical methods for singularly perturbed differential equations","volume":"24","author":"Roos, H.-G.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/3540607188"},{"key":"27","volume-title":"{\\cyr Teoriya raznostnykh skhem}","author":"Samarski\u012d, A. A.","year":"1977"},{"key":"28","series-title":"Grundlehren der Mathematischen Wissenschaften","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-0152-3","volume-title":"Shock waves and reaction-diffusion equations","volume":"258","author":"Smoller, Joel","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0387907521"},{"key":"29","series-title":"Applied Mathematical Sciences","isbn-type":"print","volume-title":"Partial differential equations. III","volume":"117","author":"Taylor, Michael E.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0387946527"},{"key":"30","series-title":"Springer Series in Computational Physics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61271-8","volume-title":"Methods for the localization of singularities in numerical solutions of gas dynamics problems","author":"Vorozhtsov, E. V.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0387503633"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2000-69-229\/S0025-5718-99-01134-5\/S0025-5718-99-01134-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-229\/S0025-5718-99-01134-5\/S0025-5718-99-01134-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:12:01Z","timestamp":1776723121000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-229\/S0025-5718-99-01134-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,5,21]]},"references-count":30,"journal-issue":{"issue":"229","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["S0025-5718-99-01134-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01134-5","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1999,5,21]]}}}