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 & 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 & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607
[2]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[3]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[4]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control & Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013
[5]

Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667
[6]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
[7]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
[8]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[9]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203
[10]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
[11]

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
[12]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061
[13]

Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete & Continuous Dynamical Systems - A, 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 & Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007
[15]

Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1
[16]

Tianlong Shen, Jianhua Huang, Caibin Zeng. Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1523-1533. doi: 10.3934/dcdsb.2018056
[17]

Ying Hu, Shanjian Tang. Nonlinear backward stochastic evolutionary equations driven by a space-time white noise. Mathematical Control & Related Fields, 2018, 8 (3&4) : 739-751. doi: 10.3934/mcrf.2018032
[18]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481
[19]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[20]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences