Sparse Pseudo Spectral Projection Methods with Directional Adaptation for Uncertainty Quantification

J. Winokur, D. Kim, Fabrizio Bisetti, O. P. Le Maître, Omar Knio

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22 Scopus citations


We investigate two methods to build a polynomial approximation of a model output depending on some parameters. The two approaches are based on pseudo-spectral projection (PSP) methods on adaptively constructed sparse grids, and aim at providing a finer control of the resolution along two distinct subsets of model parameters. The control of the error along different subsets of parameters may be needed for instance in the case of a model depending on uncertain parameters and deterministic design variables. We first consider a nested approach where an independent adaptive sparse grid PSP is performed along the first set of directions only, and at each point a sparse grid is constructed adaptively in the second set of directions. We then consider the application of aPSP in the space of all parameters, and introduce directional refinement criteria to provide a tighter control of the projection error along individual dimensions. Specifically, we use a Sobol decomposition of the projection surpluses to tune the sparse grid adaptation. The behavior and performance of the two approaches are compared for a simple two-dimensional test problem and for a shock-tube ignition model involving 22 uncertain parameters and 3 design parameters. The numerical experiments indicate that whereas both methods provide effective means for tuning the quality of the representation along distinct subsets of parameters, PSP in the global parameter space generally requires fewer model evaluations than the nested approach to achieve similar projection error. In addition, the global approach is better suited for generalization to more than two subsets of directions.
Original languageEnglish (US)
Pages (from-to)596-623
Number of pages28
JournalJournal of Scientific Computing
Issue number2
StatePublished - Dec 19 2015

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This work was supported by the US Department of Energy (DOE), Office of Science, Office of Advanced Scientific Computing Research, under Award Number DE-SC0008789. The authors wish to express their gratitude to Dr. Cosmin Safta for providing a pre-release version of TChem that enables the simulation of adiabatic combustion at constant volume.


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