{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:28:15Z","timestamp":1787333295929,"version":"3.56.0"},"reference-count":33,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>Using Wiener chaos expansion (WCE), we develop numerical algorithms for solving second-order linear parabolic stochastic partial differential equations (SPDEs). We propose a deterministic WCE-based algorithm for computing moments of the SPDE solutions without any use of the Monte Carlo technique. We also compare the proposed deterministic algorithm with two other numerical methods based on the Monte Carlo technique and demonstrate that the new method is more efficient for highly accurate solutions. Numerical tests verify that the scheme is of mean-square order $O( \\frac{\\Delta^{N\/2}}{\\sqrt{(N+1)!}})$ for diffusion and for diffusion-reaction SPDEs with constant or variable coefficients, where $\\Delta$ is the time step, and N is the Wiener chaos order.<\/jats:p>","DOI":"10.1137\/110849572","type":"journal-article","created":{"date-parts":[[2012,3,27]],"date-time":"2012-03-27T19:00:19Z","timestamp":1332874819000},"page":"A914-A936","source":"Crossref","is-referenced-by-count":14,"title":["A Multistage Wiener Chaos Expansion Method for Stochastic Advection-Diffusion-Reaction Equations"],"prefix":"10.1137","volume":"34","author":[{"given":"Z.","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"B.","family":"Rozovskii","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M. V.","family":"Tretyakov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G. E.","family":"Karniadakis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,3,27]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1080\/17442509808834159"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01184157"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.2307\/1969178"},{"key":"R4","unstructured":"P. L. Chow, J.L. Jiang, and J.L. Menaldi,\n                      Pathwise convergence of approximate solutions to Zakai's equation in a bounded domain\n                      , in Stochastic Partial Differential Equations and Applications (Trento, 1990), Pitman Res. Notes Math. Ser. 268, Longman Scientific and Technical, Harlow, 1992, pp. 111\u2013123."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-006-0875-0"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2010-02395-6"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(99)00171-2"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1080\/17442508808833486"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/050623140"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972700015094"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"I. Gy\u00f6ngy,\n                      Approximations of stochastic partial differential equations\n                      , in Stochastic Partial Differential Equations and Applications (Trento, 2002), Lecture Notes in Pure and Appl. Math. 227, Dekker, New York, 2002, pp. 287\u2013307.","DOI":"10.1201\/9780203910177-16"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1048516528"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2006.01.008"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/s00032-009-0100-0"},{"key":"R15","unstructured":"A. Jentzen and M. R\u00f6ckner,\n                      A Break of the Complexity of the Numerical Approximation of Nonlinear SPDES with Multiplicative Noise\n                      , BiBoS Preprint 11\u201301\u2013372, University of Bielefeld, Bielefeld, Germany, 2011."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1063\/1.1692063"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"F. Le Gland,\n                      Splitting-up approximation for SPDEs and SDEs with application to nonlinear filtering\n                      , in Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991), Lecture Notes in Control and Inform. Sci. 176, Springer, Berlin, 1992, pp. 177\u2013187.","DOI":"10.1007\/BFb0007332"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.4213\/rm719"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012993248918"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"S. Lototsky and B. Rozovskii,\n                      Stochastic differential equations: A Wiener chaos approach\n                      , in From Stochastic Calculus to Mathematical Finance, Springer, Berlin, 2006, pp. 433\u2013506.","DOI":"10.1007\/978-3-540-30788-4_23"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1214\/009117905000000738"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"R. Mikulevicius and B. Rozovskii,\n                      Separation of observations and parameters in nonlinear filtering\n                      , in Proceedings of the 32nd IEEE Conference on in Decision and Control, Vol. 2, 1993, pp. 1564\u20131569.","DOI":"10.1109\/CDC.1993.325450"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141096299065"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901395588"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"G. N. Milstein and M. V. Tretyakov,\n                      Stochastic Numerics for Mathematical Physics\n                      , Sci. Comput., Springer-Verlag, Berlin, 2004.","DOI":"10.1007\/978-3-662-10063-9"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-09-02250-9"},{"key":"R27","doi-asserted-by":"crossref","unstructured":"J. Picard,\n                      Approximation of nonlinear filtering problems and order of convergence\n                      , in Filtering and Control of Random Processes (Paris, 1983), Lecture Notes in Control and Inform. Sci. 61, Springer, Berlin, 1984, pp. 219\u2013236.","DOI":"10.1007\/BFb0006572"},{"key":"R28","unstructured":"B. L. Rozovski\u0131\u02d8,\n                      Stochastic Evolution Systems: Linear Theory and Applications to Nonlinear Filtering\n                      , Math. Appl. (Soviet Ser.) 35, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990."},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1007\/BF03024948"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1080\/01630569908816884"},{"key":"R31","unstructured":"J. Xu and J. Li,\n                      Sparse Wiener chaos approximations of Zakai equation for nonlinear filtering\n                      , in Proceedings of the 21st Annual IEEE International Conference on Chinese Control and Decision Conference (CCDC'09), Piscataway, NJ, 2009, pp. 910\u2013913."},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1137\/040605278"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01150-3"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/110849572","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:48:48Z","timestamp":1787330928000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/110849572"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1137\/110849572"],"URL":"https:\/\/doi.org\/10.1137\/110849572","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,1]]}}}