MULTI-ELEMENT GENERALIZED POLYNOMIAL CHAOS FOR ...

926 XIAOLIANG WAN AND GEORGE EM KARNIADAKIS
Fig. 4.1. Cost comparison between ME-gPC and the standard Monte Carlo method for fixed
accuracy. Here p is the order of gPC in each element, N is the number of elements and d is the
number of dimensions. The symbol-lines are iso-cost lines.
Carlo method due to the k-p convergence as shown in cases (i) and (ii) of section 3.2.2.
In Figure 4.1, we present a comparison between the k-convergence O(N −2(p+1) )of
ME-gPC and the convergence O(n −1/2 ) of the standard Monte Carlo method; see [1].
For the same accuracy and different random dimension numbers, the lines show the
cases where the cost of the standard Monte Carlo method is equal to that of ME-gPC.
For a certain random dimension number (denoted by “d”), the region below the line
is where MC is more efficient; the region above the line is where ME-gPC is more
efficient.
In this work we use heuristically the decay rate of relative error of variance as the
indicator for the k-type refinement. A more rigorous a posteriori error estimate is still
needed for the kp-adaptivity. These issues will be addressed in future publications.
Appendix (application of ME-gPC to a stochastic elliptic problem).
Here we briefly elaborate on how to apply ME-gPC to solve differential equations with
stochastic coefficients using the following stochastic linear boundary value problem:
Find a stochastic function, u : D × Ω → R, such that almost surely the following
equation holds:
(4.1)
−∇ · (a(x; ω)∇u(x; ω)) = f(x; ω) on D,
u(x; ω) =0 on∂D,
where D is an open domain in the physical space with Lipschitz boundaries, a(x; ω)
and f(x; ω) are second-order random processes. We assume 0