Abstract
This article presents a study of the Stochastic Galerkin Method (SGM) applied to the Darcy flow problem with a log-normally distributed random material field given by a mean value and an autocovariance function. We divide the solution of the problem into two parts. The first one is the decomposition of a random field into a sum of products of a random vector and a function of spatial coordinates; this can be achieved using the Karhunen-Loeve expansion. The second part is the solution of the problem using SGM. SGM is a simple extension of the Galerkin method in which the random variables represent additional problem dimensions. For the discretization of the problem, we use a finite element basis for spatial variables and a polynomial chaos discretization for random variables. The results of SGM can be utilised for the analysis of the problem, such as the examination of the average flow, or as a tool for the Bayesian approach to inverse problems.
Highlights
Introduction and Problem SettingThe standard mathematical modelling approach usually works with deterministic parameters of solved Fig. 1: Illustration of physical domain with boundary.c 2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING equals one on ΓD1, zero on ΓD2, and the flow equals zero on the rest of the boundary.∀ω ∈ Ω : −div (k (x; ω) ∇u (x; ω)) = 0, ∀x ∈ D, u (x; ω) = 1, ∀x ∈ ΓD1, (1) u (x; ω) = 0, ∂u(x;ω) ∂n(x) =∀x ∈ ΓD2, ∀x ∈ ΓN, where ω represents an event from the sample space Ω
We focus on a model problem with a very high uncertainty in input parameters, for which we demonstrate the application of Stochastic Galerkin Method (SGM)
That means that the function K0 = k (x0; ω) : Ω → R is a random variable for every fixed x0 ∈ D and the function k0 (x) = k (x; ω0) : D → R is a function on the spatial domain for every fixed ω0 ∈ Ω
Summary
The standard mathematical modelling approach usually works with deterministic parameters of solved Fig. 1: Illustration of physical domain with boundary. The problem Eq (1) is very complex because the random field k represents an infinite number of random variables. To solve this kind of problems, we need to reduce the number of random variables to some finite, ideally small, number We can achieve this using the Karhunen-Loeve (KL) decomposition, which will be discussed . In the case of the log-normal random material field, it is simpler to decompose the underlying GRF, which lies in space L2 Ω, L2 (D) , see for example [1]: corollary 4.41. In the case of the KL decomposition of the GRF, the random variables ξi will be Gaussian ξi ∼ N (0; 1). The scale parameter affects the amplitude (equivalent samples with different values of σ differ only in scale) and the correlation length parameter changes the correlation/distance ratio between two points
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