On a Perturbation Method for Stochastic Parabolic PDE

Abstract

In this article, we address two issues related to the perturbation method introduced by Zhang and Lu (J Comput Phys 194:773–794, 2004), and applied to solving linear stochastic parabolic PDE. Those issues are the construction of the perturbation series, and its convergence.

Keywords

Stochastic parabolic differential equation / Karhunen–Loève expansion

References

[1] Courant R, Hilbert D. Methods of Mathematical Physics. 1953 New York: Interscience

[2] Cramér H. Mathematical methods of statistics, Princeton Landmarks in Mathematics. 1999 Princeton: Princeton University Press

[3] Evans LC. Partial Differential Equations. 1998 Providence: American Mathematical Society

[4] Ghanem RG, Spanos PD. Stochastic Finite Elements: A Spectral Approach. 1991 New York: Springer

[5] Lenth, K.: Application of a perturbation method to nonlinear stochastic PDEs. PhD dissertation, University of Wyoming, (2013)

[6] Loève M. Probability Theory II, Graduate Texts in Mathematics 46. 1978 New York: Springer

[7] Xiu D. Numerical Methods for Stochastic Computations: A Spectral Method Approach. 2010 Princeton: Princeton University Press

[8] Zhang D, Lu Z. An efficient, high-order perturbation approach for flow in random porous media via Karhunen-Loève and polynomial expansions. J. Comput. Phys.. 2004, 194 773-794

Funding

U.S. Department of Energy(NEUP)