{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T07:53:17Z","timestamp":1789717997334,"version":"4.0.1"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM\/ASA J. Uncertainty Quantification"],"published-print":{"date-parts":[[2014,1]]},"abstract":"<jats:p>An asymptotic analysis of the tail probabilities for the dynamics of a soliton wave $U(x,t)$ under a stochastic time-dependent force is developed. The dynamics of the soliton wave $U(x,t)$ is described by the Korteweg--de Vries (KdV) equation with homogeneous Dirichlet boundary conditions under a stochastic time-dependent force, which is modeled as a time-dependent Gaussian noise with amplitude $\\epsilon$. The tail probability considered is $w(b):=P(\\sup_{t\\in[0,T]}U(x,t)&gt;b)$, as $b\\to\\infty$, for some constant $T&gt;0$ and a fixed $x$, which can be interpreted as tail probability of the amplitude of a water wave on the shallow surface of a fluid or long internal wave in a density-stratified ocean. Our goal is to characterize the asymptotic behaviors of $w(b)$ and evaluate the tail probability of the event that the soliton wave exceeds a certain threshold value under a random force term. Such rare-event calculation of $w(b)$ is especially useful for fast estimation of the risk of the potential damage that could be caused by the water wave in a density-stratified ocean modeled by the stochastic KdV equation. In this work, the asymptotic approximation of the probability that the soliton wave exceeds a high-level $b$ is derived. In addition, we develop a provably efficient rare-event simulation algorithm to compute $w(b)$ that runs in polynomial time of $\\log b$ with a prescribed relative accuracy.<\/jats:p>","DOI":"10.1137\/130944473","type":"journal-article","created":{"date-parts":[[2014,11,20]],"date-time":"2014-11-20T10:27:06Z","timestamp":1416479226000},"page":"698-716","source":"Crossref","is-referenced-by-count":5,"title":["Rare-Event Simulation for the Stochastic Korteweg--de Vries Equation"],"prefix":"10.1137","volume":"2","author":[{"given":"Gongjun","family":"Xu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guang","family":"Lin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jingchen","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2014,11,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"M. 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Glynn (2008),\n                      Strongly efficient algorithms for light-tailed random walks: An old folk song sung to a faster new tune\n                      , in Proceedings of the Eighth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC 2008), P. L'Ecuyer and A. Owen, eds., Springer, Berlin, pp. 227-248.","DOI":"10.1007\/978-3-642-04107-5_13"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1231340166"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"V. Bogachev (1998),\n                      Gaussian Measures\n                      , Math. Surveys Monogr. 62, American Mathematical Society, Providence, RI.","DOI":"10.1090\/surv\/062"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01425510"},{"key":"atypb10","unstructured":"F. Calogero and A. Degasperis (1982),\n                      Solitons and the Spectral Transform \\textupI\n                      , North-Holland, Amsterdam."},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"J.P. Chile\u0300s and P. Delfiner (1999),\n                      Geostatistics: Modeling Spatial Uncertainty\n                      , John Wiley & Sons, New York.","DOI":"10.1002\/9780470316993"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(99)00072-X"},{"key":"atypb13","unstructured":"R. Dodd, J. Eilbeck, J. Gibbon, and H. Morris (1982),\n                      Solitons and Nonlinear Wave Equations\n                      , Academic Press, London."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176997026"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1145\/1243991.1243995"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-12035"},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"L. Faddeev and L. Takhtajan (1987),\n                      Hamiltonian Methods in the Theory of Solitons\n                      , Springer-Verlag, Berlin.","DOI":"10.1007\/978-3-540-69969-9"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.19.1095"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1023\/A:1013549126956"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1080\/13616670109409780"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/23\/7\/014"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1086\/186160"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(93)91121-K"},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"V. Konotop and L. 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