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STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR stochastic partial differential equations WITH MULTIPLICATIVE NOISE

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Journal of Computational Mathematics 2021年第4期39卷 574-598页

作者： Xiaobing Feng Yukun Li Yi Zhang Department of Mathematics The University of TennesseeKnoxvilleTN 37996USA Department of Mathematics University of Central FloridaOrlandoFL 32816USA Department of Mathematics and Statistics The University of North Carolina at GreensboroGreensboroNC 27402USA

This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations(SPDEs)with multiplicative *** nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitz *** assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations(SODEs)under“minimum assumptions”were *** a result,the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear *** are several difficulties which need to be overcome for this ***,obviously the spatial discretization,which does not appear in the SODE case,adds an extra layer of *** turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness *** this paper we use a finite element interpolation technique to discretize the nonlinear drift ***,in order to prove the strong convergence of the proposed fully discrete finite element method,stability estimates for higher order moments of the H1-seminorm of the numerical solution must be established,which are difficult and delicate.A judicious combination of the properties of the drift and diffusion terms and some nontrivial techniques is used in this paper to achieve the ***,stability estimates for the second and higher order moments of the L^(2)-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally *** is done by utilizing the interpolation theory and the higher moment estimates for the H1-seminorm of the numerical *** overcoming these difficulties,it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of ***

关键词： stochastic partial differential equations One-sided Lipschitz Strong convergence

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On a semilinear stochastic partial differential equation with double-parameter fractional noises

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Science China Mathematics 2014年第4期57卷 855-872页

作者： LIU JunFeng YAN LiTan Department of Mathematics Nanjing Audit University Department of Mathematics Donghua University

We study the existence,uniqueness and Hlder regularity of the solution to a stochastic semilinear equation arising from 1-dimensional integro-differential scalar conservation *** equation is driven by double-parameter fractional *** addition,the existence and moment estimate are also obtained for the density of the law of such a solution.

关键词： stochastic partial differential equations double-parameter fractional noises H61der regularity density of the law Malliavin calculus

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QUALIFICATION OF UNCERTAINTY FOR SIMULATING SOLUTE TRANSPORT IN THE HETEROGENEOUS MEDIA WITH SPARSE GRID COLLOCATION METHOD

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Journal of Hydrodynamics 2009年第6期21卷 779-789页

作者： SHI Liang-sheng YANG Jin-zhong State Key Laboratory of Water Resources and Hydropower Engineering Science Wuhan University Wuhan 430072 China The Sonny Astani Department of Civil and Environmental Engineering University of Southern California Los Angeles CA 90089 USA

The sparse grid collocation method is discussed to qualify the uncertainty of solute transport. The Karhunen-Loeve （KL） expansion is employed to decompose the log transformed hydraulic conductivity. The head, velocity and concentration fields are represented by the Lagrange polynomial expansion. A sparse grid collocation method is then used to reduce the original stochastic partial differential equations to a set of deterministic equations which is collocated at selected interpolation （collocation） points. The collocation points are constructed by the Smolyak algorithm. The accuracy, efficiency and convergence property of sparse grid collocation method are investigated by numerical experiments. The analysis shows that stochastic collocation strategy helps to decouple stochastic computations, and all the numerical computation is possible to be implemented by existing deterministic finite element codes. The proposed method provides an efficient way to evaluate the uncertainty of the solute transport in the heterogeneous media.

关键词： stochastic partial differential equations collocation method solute transport uncertainty

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RANDOM ATTRACTORS FOR A stochastic HYDRODYNAMICAL EQUATION IN HEISENBERG PARAMAGNET

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Acta Mathematica Scientia 2011年第2期31卷 529-540页

作者：郭柏灵郭春晓蒲学科 Institute of Applied Physics and Computational Mathematics Department of Mathematics China University of Mining and Technology Beijing College of Mathematics and Physics Chongqing University

This article studies the asymptotic behaviors of the solution for a stochastic hydrodynamical equation in Heisenberg paramagnet in a two-dimensional periodic domain. We obtain the existence of random attractors in H1.

关键词： stochastic partial differential equations Heisenberg paramagnet Randomattractor

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WEAK ERROR ESTIMATES FOR TRAJECTORIES OF SPDEs UNDER SPECTRAL GALERKIN DISCRETIZATION

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Journal of Computational Mathematics 2018年第2期36卷 159-182页

作者： Charles-Edouard Brehier Martin Hairer Andrew M. Stuart Univ Lyon CNRS Universite Claude Bernard Lyon 1 UMR5208 Institut Camille Jordan F-69622 Villeurbanne France Department of Mathematics Imperial College London London SW7 2AZ UK Computing and Mathematical Sciences California Institute of Technology 1200 E. California Blvd. MC 305-16 Pasadena CA 91125 USA

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation （analytic semigroup） and damped-driven wave equations （bounded semigroup） type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories： test functions are regular, but depend on the values of the process on the interval [0, T]. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function （It6 map） of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the It6 map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup： the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

关键词： stochastic partial differential equations Weak approximation Spectral Galerkin discretization

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Tanaka Formula and Local Time for a Class of Interacting Branching Measure-valued Diffusions

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Acta Mathematica Sinica,English Series 2024年第4期40卷 1059-1098页

作者： Donald A.DAWSON Jean VAILLANCOURT Hao WANG School of Mathematics and Statistics Carleton UniversityOttawaCanada Department of Decision Sciences HECMontrealCanada Department of Mathematics University of OregonEugeneUSA

We construct superprocesses with dependent spatial motion(SDSMs)in Euclidean spaces R^(d)with d≥1 and show that,even when they start at some unbounded initial positive Radon measure such as Lebesgue measure on R^(d),... 详细信息

关键词： Measure-valued diffusions stochastic partial differential equations superprocesses branching processes local time Tanaka formula

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Global Well-posedness of the stochastic Generalized Kuramoto-Sivashinsky Equation with Multiplicative Noise

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Acta Mathematicae Applicatae Sinica 2018年第3期34卷 566-584页

作者： Wei WU Shang-bin CUI Jin-qiao DUAN Science and Information College Qingdao Agricultural University Qingdao 266109 China Department of Mathematics Sun Yat-Sen University Guangzhou 510275 China Department of Applied Mathematics Illinois Institute of Technology Chicago IL 60616 USA Center for Mathematical Sciences Huazhong University of Science and Technology Wuhan 430074 China

Global well-posedness of the initial-boundary value problem for the stochastic generalized KuramotoSivashinsky equation in a bounded domain D with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data u0 ∈L2（D×Ω）, this problem has a unique global solution u in the space L2（Ω, C（[0, T ], L2（D））） for any T 〉 0, and the solution map u0 →u is Lipschitz continuous.

关键词： Kuramoto-Sivashinsky equation stochastic partial differential equations multiplicative noise well-posedness

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Smooth Solutions for a stochastic Hydrodynamical Equation in Heisenberg Paramagnet

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Acta Mathematica Sinica,English Series 2011年第9期27卷 1855-1868页

作者： Xue Ke PU Bo Ling GUO Yong Qian HAN College of Mathematics and Physics Chongqing University Chongqing 400044 P. R. China Institute of Applied Physics and Computational Mathematics P. O. Box 8009 Beijing 100088 P. R. China

In this article, we consider a stochastic hydrodynamical equation in Heisenberg paramagnet driven by additive noise. We prove the existence and uniqueness of smooth solutions to this equation with difference method.

关键词： stochastic partial differential equations Heisenberg paramagnet smooth solution difference method

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Mean-Square Approximation of Navier-Stokes equations with Additive Noise in Vorticity-Velocity Formulation

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Numerical Mathematics(Theory,Methods and Applications) 2021年第1期14卷 1-30页

作者： G.N.Milstein M.V.Tretyakov Ural Federal University EkaterinburgRussia School of Mathematical Sciences University of NottinghamNottinghamUK

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions and additive noise in the vorticity-velocity *** approximation is based on freezing the velocity on time subintervals resulting in a linear stochastic parabolic equation for *** each time step,the velocity is expressed via vorticity using a formula corresponding to the Biot-Savart-type *** prove the first mean-square convergence order of the vorticity approximation.

关键词： Navier-Stokes equations vorticity numerical method stochastic partial differential equations mean-square convergence

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Difference Approximation of stochastic Elastic Equation Driven by Infinite Dimensional Noise

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Numerical Mathematics(Theory,Methods and Applications) 2016年第1期9卷 123-146页

作者： Yinghan Zhang Xiaoyuan Yang Ruisheng Qi Department of Mathematics Beihang UniversityLMIB of the Ministry of EducationBeijing 100191China

An explicit differencescheme is described,analyzed and tested for numer-ically approximating stochastic elastic equation driven by infinite dimensional *** noise processes are approximated by piecewise constant random processes and the integral formula of the stochastic elastic equation is approximated by a truncated *** analysis of the numerical method yields estimate of convergence *** rate of convergence is demonstrated with numerical experiments.

关键词： stochastic partial differential equations difference scheme stochastic elastic equation infinite dimensional noise rate of convergence

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