TY - JOUR

T1 - Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures

AU - Agarwal, Nitin

AU - Aluru, N. R.

N1 - Funding Information:
This work was funded by ONR grants N00014-11-1-0783, N00014-10-1-0200, N00014-04-1-0118, N00014-10-1-0811, and N00014-16-1-2567 and NSF grant AGS-1008848. Field campaign data used for this study can be obtained at u.arizona.edu/~armin, and questions about those data should be directed to the corresponding author (armin@email.arizona.edu). The NAAPS reanalysis data are available at http://usgodae.org/cgi-bin/datalist.pl?dset=nrl_naaps_reanalysis&summary=Go; the data on that server are updated as model improvements are made and reruns are completed. CALIOP data are available at the following website: https://eosweb.larc.nasa.gov/.

PY - 2011/3/18

Y1 - 2011/3/18

N2 - This paper deals with numerical solution of differential equations with random inputs, defined on bounded random domain with non-uniform 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 multi-dimensional 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 non-uniform probability measures, during the construction of sparse grids. The basic idea is to construct piecewise linear univariate interpolation formulas, where the support nodes are specially chosen based on the marginal probability distribution. These weighted univariate interpolation formulas are then used to construct weighted sparse grid interpolants, using the standard Smolyak algorithm. This algorithm results in sparse grids with higher number of support nodes in regions of the random domain with higher probability density. Several numerical examples are presented to demonstrate that the proposed approach results in a more efficient algorithm, for the purpose of computation of moments of the stochastic solution, while maintaining the accuracy of the approximation of the solution.

AB - This paper deals with numerical solution of differential equations with random inputs, defined on bounded random domain with non-uniform 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 multi-dimensional 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 non-uniform probability measures, during the construction of sparse grids. The basic idea is to construct piecewise linear univariate interpolation formulas, where the support nodes are specially chosen based on the marginal probability distribution. These weighted univariate interpolation formulas are then used to construct weighted sparse grid interpolants, using the standard Smolyak algorithm. This algorithm results in sparse grids with higher number of support nodes in regions of the random domain with higher probability density. Several numerical examples are presented to demonstrate that the proposed approach results in a more efficient algorithm, for the purpose of computation of moments of the stochastic solution, while maintaining the accuracy of the approximation of the solution.

KW - Non-uniform distributions

KW - Smolyak algorithm

KW - Sparse grid

KW - Stochastic collocation method

KW - Stochastic differential equations

KW - Uncertainty quantification

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U2 - 10.1002/nme.3019

DO - 10.1002/nme.3019

M3 - Article

AN - SCOPUS:79951545866

SN - 0029-5981

VL - 85

SP - 1365

EP - 1389

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

IS - 11

ER -