 
Summary: INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng 2011; 85:13651389
Published online 2 September 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3019
Weighted Smolyak algorithm for solution of stochastic differential
equations on nonuniform probability measures
Nitin Agarwal and N. R. Aluru,
Department of Mechanical Science and Engineering, Beckman Institute for Advanced Science and Technology,
University of Illinois at UrbanaChampaign, 405 N. Mathews Avenue, Urbana, IL 61801, U.S.A.
SUMMARY
This paper deals with numerical solution of differential equations with random inputs, defined on bounded
random domain with nonuniform probability measures. Recently, there has been a growing interest in
the stochastic collocation approach, which seeks to approximate the unknown stochastic solution using
polynomial interpolation in the multidimensional random domain. Existing approaches employ sparse
grid interpolation based on the Smolyak algorithm, which leads to orders of magnitude reduction in the
number of support nodes as compared with usual tensor product. However, such sparse grid interpolation
approaches based on piecewise linear interpolation employ uniformly sampled nodes from the random
domain and do not take into account the probability measures during the construction of the sparse grids.
Such a construction based on uniform sparse grids may not be ideal, especially for highly skewed or
localized probability measures. To this end, this work proposes a weighted Smolyak algorithm based
on piecewise linear basis functions, which incorporates information regarding nonuniform probability
