Galerkin Solution

The solution of Equation (14) involves two steps. First, Equation (14) is approximated by a stochastic algebraic equation. Second, optimal and sub-optimal Galerkin solutions are developed for this stochastic algebraic equation and some of their statistics are calculated.

The stochastic differential equation for W in Equation (14) with K in Equation (15) depends on the random variable Z in the definition of the random field K. The finite difference approximation of Equation (14) at an interior node has the form

(20 + Ku) Wij + 2 (Wt-u-1 + Wi-hj+1 + Wi+1J-1 + Wi+1J+1)

– 8 (Wu-1 + Wtj+1 + Wt-1j + Wt+1j)

+ Wij-2 + Wij+2 + Wi-2j + Wi+2j = Qij, (19)

where Wij, Kij, and Qij denote the values of W(x, y), K(x, y), and Q(x, y) and the node (i, j) of the finite difference mesh. The above relation written at all nodes with adequate modifications to account for boundary conditions yields a stochastic algebraic equation of the type in Equation (1) with coefficients depending on the random variables Z and solution X with entries the displacements Wj.

Since Z in Equation (17) is a standard Gaussian vector in Rn, n = 12, it has the representation

Z = U R,

where U is uniformly distributed on the unit sphere Sn(1) in R" centered at the origin of this space and R is a real-valued random variable following a chi distribution with n degrees of freedom. The random variable R is independent of U and has the distribution

P1(R < r) = I(r2/2,n/2), (21)

where I(x, q) = J0X tq-1 e-t dt/T(q) and T(q) denote the incomplete and the complete gamma functions, respectively.

We begin the construction of a partition [Aq} of ^1 by dividing Rn in rings of radii 0 = r0 < r 1 < … < rnr =ro such that each ring has the same probability content, that is,

P1(r„-1 <R < ru) = I (r2/2, n/2) – I (rl-1/2, n/2) = 1/nr, u = 1,…,nr. (22)

We then divide each ring in subsets of equal volume, and these subsets define the partition {Aq} of ^1 used to construct the optimal Galerkin solution. The resulting sets Aq have the same probability content, that is, P1(Aq) = 1/(nr ni) for all q, where щ > 1 denotes the number of subsets of equal volume in each ring. The conditional expectations E1[P(Z) | Aq] in the definition of the optimal Galerkin solution can be estimated from

1 "s

Ei[P(Z) І Л?] ~ («!,„)) 1Аі(й>і, и) (23)

"s u=1

where {Z(aj, u)}, u = 1,… ,ns, are ns independent samples of Z.