  For uncertainty propagation of highly complex and/or nonlinear problems, one
must resort to sample-based non-intrusive approaches [1]. In such cases,
minimizing the number of function evaluations required to evaluate the response
surface is of paramount importance. Sparse grid approaches have proven
effective in reducing the number of sample evaluations. For example, the
discrete projection collocation method has the notable feature of exhibiting
fast convergence rates when approximating smooth functions; however, it lacks
the ability to accurately and efficiently track response functions that exhibit
fluctuations, abrupt changes or discontinuities in very localized regions of
the input domain. On the other hand, the piecewise linear collocation
interpolation approach can track these localized variations in the response
surface efficiently, but it converges slowly in the smooth regions. The
proposed methodology, building on an existing work on adaptive hierarchical
sparse grid collocation algorithm [2], is able to track localized behavior
while also avoiding unnecessary function evaluations in smoother regions of the
stochastic space by using a finite difference based one-dimensional derivative
evaluation technique in all the dimensions. This derivative evaluation
technique leads to faster convergence in the smoother regions than what is
achieved in the existing collocation interpolation approaches. Illustrative
examples show that this method is well suited to high-dimensional stochastic
problems, and that stochastic elliptic problems with stochastic dimension as
high as 100 can be dealt with effectively.
