American Institute of Mathematical Sciences

Advanced
July  2006, 6(4): 941-955. doi: 10.3934/dcdsb.2006.6.941

Stochastic Galerkin method for elliptic spdes: A white noise approach

Luis J. Roman 1, and  Marcus Sarkis 1,

1. 

Worcester Polytechnic Institute, Department of Mathematical Sciences, 100 Institute Rd, Worcester, MA 01609-2280, United States, United States

Received  January 2005 Revised  October 2005 Published  April 2006

An equation that arises in mathematical studies of the transport of pollutants in groundwater and of oil recovery processes is of the form: $-\nabla_{x}\cdot(\kappa(x,\cdot)\nabla_{x}u(x,\omega))=f(x)$, for $x\in D$, where $\kappa(x,\cdot)$, the permeability tensor, is random and models the properties of the rocks, which are not know with certainty. Further, geostatistical models assume $\kappa(x,\cdot)$ to be a log-normal random field. The use of Monte Carlo methods to approximate the expected value of $u(x,\cdot)$, higher moments, or other functionals of $u(x,\cdot)$, require solving similar system of equations many times as trajectories are considered, thus it becomes expensive and impractical. In this paper, we present and explain several advantages of using the White Noise probability space as a natural framework for this problem. Applying properly and timely the Wiener-Itô Chaos decomposition and an eigenspace decomposition, we obtain a symmetric positive definite linear system of equations whose solutions are the coefficients of a Galerkin-type approximation to the solution of the original equation. Moreover, this approach reduces the simulation of the approximation to $u(x,\omega)$ for a fixed $\omega$, to the simulation of a finite number of independent normally distributed random variables.
Keywords: Stochastic Differential Equations, White Noise, Galerkin Method, Elliptic Partial Differential Equations..
Mathematics Subject Classification: Primary: 60H30, 60H35, 65C20; Secondary: 65C30, 65N3.
Citation: Luis J. Roman, Marcus Sarkis. Stochastic Galerkin method for elliptic spdes: A white noise approach. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 941-955. doi: 10.3934/dcdsb.2006.6.941
[1]

Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure and Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607
[2]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209
[3]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[4]

Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3651-3657. doi: 10.3934/dcdsb.2020077
[5]

Xiang Lv. Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2313-2323. doi: 10.3934/dcdsb.2021133
[6]

Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100
[7]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[8]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[9]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control and Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013
[10]

Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667
[11]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
[12]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
[13]

Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024
[14]

Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control and Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007
[15]

Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1565-1581. doi: 10.3934/dcdsb.2019240
[16]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021020
[17]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[18]

Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006
[19]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203
[20]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks and Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy