An efficient adaptive sparse grid collocation method through derivative estimation

Abstract

For uncertainty propagation of highly complex and/or nonlinear problems, one must resort to sample-based non-intrusive approaches (Le Maître and Knio, 2010). 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 (Ma and Zabaras, 2009), 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.