Mar 27, 2007 - models which become more and more popular in many fields of science and .... International Applied Mechanics, 2000, Vol. ... 3-4, 143-166. 5.
Monte Carlo methods for stochastic PDEs K. Sabelfeld Weierstrass Institute for Applied Analysis and Stochastics Mohrenstrasse 39, D – 10117 Berlin, Germany sabelfeld@@wias-berlin.de and Institute of Computational Mathematics and Mathem. Geophysics Russian Acad. Sci., Lavrentieva str., 6 630090 Novosibirsk, Russia
Extended Abstract
It is well known that boundary value problems with random parameters are very interesting models which become more and more popular in many fields of science and technology, especially in problems where the data are highly irregular, for instance, like in the case of turbulent transport in atmosphere and ocean [8], [11], [15], flows in porous medium [2], [7], or evaluation of elastic properties of polymers and composites containing fibers which are randomly oriented in a plane [6], [12], in elastography imaging [13], and in elastic amorphous media where the so-called Boson peak has been found [3]. We mention also a stochastic model for the wind loads that are generated by thunderstorm downbursts [10], a random field approach to the term structure of interest rates with applications to credit risk [5], and an interesting stochastic model for polyphonic music modelling with random fields [9]. In conventional deterministic numerical methods, the stochastic PDEs are numerically solved as follows: first one constructs a synthesized sample of the input random parameter; then the obtained deterministic equation is solved numerically, say, by finite element method calculating the solution in all points of the grid domain. These two steps are repeated many times, so that the obtained statistics is reach enough to calculate the desired averages accurately. This approach is used in a stochastic finite element method, e.g., see [1], [4], [12], [14], [20]. This technique is however generally very time consuming, and to solve practically interesting problems one needs supercomputers to extract sufficient statistical information. In the Monte Carlo approach, the algorithms are designed so that the solution is calculated only in the desired set of points without constructing the solution in the whole domain which is important when calculating different statistical characteristics like correlation functions, e.g., see [16], [17], [18], [19]. To evaluate different statistical characteristics of random boundary value problems we use the Double Randomization technique (e.g., see [16]). This approach is ⋆ Supported partly by the RFBR Grant N 06-01-00498, the program of Leading Scientific Schools under Grant N 4774.2006.1, and NATO Linkage Grant ESP Nr CLG 981426
Preprint submitted to Elsevier Science
27 March 2007
possible if the desired statistical characteristics (e.g., the mean or the correlation tensor) can be represented in the form of a double expectation over the input random parameters, and over the trajectories of a Markov process used in a stochastic estimator for solving the deterministic equation. The advantage of this method is that we have no need to solve the equation many times, hence, the cost of this method is drastically decreased compared to the stochastic finite element method. Assume we have to solve a PDE which includes a random field σ, say in the right-hand side, in coefficients, or in the boundary conditions: Lu(x, σ) = f (x, σ),
u|Γ = uγ .
Suppose we have constructed a stochastic method for solving this problem, for a fixed sample of σ. This implies, e.g., that an unbiased random estimator ξ(x| σ) is defined so that for a fixed σ, u(x, σ) = hξ(x| σ)i where h·i stands for averaging over the random trajectories of the stochastic method (e.g., a diffusion process, a Random Walk on Spheres, or a Random Walk on Boundary, e.g., see [16], [19]). Let us denote by Eσ the average over the distribution of σ. The double randomization method is based on the equality: Eσ u(x, σ) = Eσ hξ(x| σ)i . The algorithm for evaluation of Eσ u(x, σ) then reads: 1. Choose a sample of the random field σ. 2. Construct a sample of the random walk along which the random estimator ξ(x| σ) is calculated. 3. Repeat 1. and 2. N times, and take the arithmetic mean. Suppose one needs to evaluate the covariance of the solution. Let us denote the random trajectory by ω. It is not difficult to show [16] that hu(x, σ) u(y, σ)i = E(ω1 ,ω2 ,σ) [ξω1 (x|σ)ξω2 (y|σ)] . The algorithm for calculation of hu(x, σ) u(y, σ)i follows from this relation: 1. Choose a sample of the random field σ. 2. Having fixed this sample, construct two conditionally independent trajectories ω1 and ω2 , starting at x and y, respectively, and evaluate ξω1 (x|σ)ξω2 (y|σ) 3. Repeat 1. and 2. N times, and take the arithmetic mean. In this talk we deal with two types of PDEs with random field inputs: (1) PDEs with random right-hand sides, and (2) PDEs with random boundary functions. For some classes of PDEs, we derive closed equations for the correlation function or tensor. 2
Indeed, assume that we have a linear scalar PDE with random right-hand side and zero boundary values Lu = f , x ∈ D, u|Γ = 0, where the random field f (not necessarily homogeneous) has B f (x, y) as its correlation function. The solution u can be represented through the Green formula u(x) =
Z
G(x, y) f (y)dy
D
where G(x, y) is the Green function for the domain D. Under certain smoothness conditions we can prove that the correlation function of the solution Bu (x, y) = hu(x)u(y)i and the input correlation function B f (x, y) = h f (x) f (y)i are related by the iterated equation Lx Ly Bu (x, y) = B f (x, y) with boundary conditions Bu |y∈Γ = 0, Ly B(x, y)|x∈Γ = 0. Here Lx implies that the operator L acts with respect to the variable x, for fixed y. This can be derived as follows. First, starting from the definition Bu (x, y) = hu(x)u(y)i we use the above Green formula to evaluate this product for the points x and y, and change the product of integrals by the double integrals over the domain D; then we take the expectation under the double integral sign. This leads us to the representation Bu (x, y) = hu(x)u(y)i =
Z Z
G(x, y′) G(y, y′′) B f (y′ , y′′)i dy′ dy′′ .
D D
It now remains to notice that the same expression is obtained by applying the Green formula successively to the above iterated equation. These arguments work only in the case of scalar equations, e.g., in the case of Laplace operator L = ∆, is ∆x ∆y Bu (x, y) = B f (x, y) with boundary conditions Bu |y∈Γ = 0, ∆y B(x, y)|x∈Γ = 0. For systems of PDEs the relevant expressions are more complicated. Let us consider the Dirichlet problem of elasticity, i.e., the Lamé system of equations with the relevant boundary conditions for the vector of displacements u(x): ∆u + α grad div u = f
u|Γ = 0 .
We consider the correlation tensor of the solution B(u) (x, y) = hu(x)uT (y)i, and the correlation tensor of the body forces B( f ) (x, y) = hf(x)fT (y)i. In the case of a matrix W (x), the Lamé operator L acts on the matrix W column-wise. This means, the matrix equation LW = B (B is a matrix) is a pair of Lamé equations written for the relevant first and second columns of matrices W and B. 3
Analogously to the scalar case, after direct evaluations we arrive at B(u) (x, y) =
Z Z
G(x, y′) B( f ) (y′ , y′′ )GT (y, y′′)dy′ dy′′ .
D D
It is also possible to write down a differential relation between the input matrix B( f ) (y′ , y′′ ) and the correlation matrix of the solution B(u) (x, y). Indeed, introduce a tensor V (x, y), and write the following system of coupled systems Lx B(u) (x, y) = V T (x, y),
B(u) (x, y)|x∈Γ = 0,
Ly V (x, y) = [B( f ) (x, y)]T ,
V (x, y)|y∈Γ = 0 .
The derived system of equations can be written as one system of 4-th order. Indeed, using the ˆ = LV T we apply the operator Lˆ y to both sides of the first system in the above definition LV coupled system of equations. This yields Lˆ y Lx B(u) (x, y) = [B( f )(x, y)]T with boundary conditions B(u) (x, y)|x∈Γ = 0,
Lx B(u) (x, y)|y∈Γ = 0 .
We will present the same analysis for PDEs with random boundary functions. This much more difficult case is however well treated by the Monte Carlo approach.
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