Estimating High Dimensional Integrals using Sparse Grids,
John Burkardt
Department of Scientific Computing
Florida State University

Workshop on Numerical Methods for Stochastic PDE's,
Ajou University, Suwon, Korea, 
24 September 2011.

The collocation method for analyzing stochastic partial differential
equations assumes that the uncertainty in the problem can be modeled
using a probability density function associated with the uncertain
parameters.  The method can then represent the expected value 
of the solution, or higher moments, as integrals in a high dimensional
probability space.  It is not uncommon for these spaces to have
dimension of 20, 50, or 100.

To numerically evaluate such estimates requires a quadrature approach
that can efficiently handle integrands that are smooth, but high
dimensional.  Product rules cannot be used for such problems, because
their expense grows exponentially with the spatial dimension.  However,
a technique known as the sparse grid approach can produce integral
estimates (and also interpolants to the solution) that are accurate
and efficient over a surprisingly high range of dimensions.

We will discuss the definition and implemention of sparse grids,
and compare this approach to the standard Monte Carlo method, which
can also produce estimates of quantities associated with stochastic
processes.