Sparse Grid Collocation for Uncertainty Quantification

When we're computing data for a system with uncertainty,
it's natural to worry about how the uncertain inputs 
affect the output.  The Monte Carlo method answers by 
sampling the input space and averaging the resulting 
outputs.  

This really amounts to estimating a weighted
integral, and a product rule would be an alternative,
and highly accurate alternative -- except that such rules
cannot be employed for high dimensional problems.

In this talk, we'll suggest how the accuracy of the product 
rule can be achieved at a much lower cost, using Smolyak's 
sparse grid definition.  We'll look at a common form of 
the sparse grid based on the Clenshaw Curtis rule, and
consider some numerical examples that illustrate this 
way of computing the statistics of an uncertain process.