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Local and Dimension Adaptive Stochastic Collocation for Uncertainty Quantification

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Sparse Grids and Applications

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 88))

Abstract

In this paper we present a stochastic collocation method for quantifying uncertainty in models with large numbers of uncertain inputs and non-smooth input-output maps. The proposed algorithm combines the strengths of dimension adaptivity and hierarchical surplus guided local adaptivity to facilitate computationally efficient approximation of models with bifurcations/discontinuties in high-dimensional input spaces. A comparison is made against two existing stochastic collocation methods and found, in the cases tested, to significantly reduce the number of model evaluations needed to construct an accurate surrogate model. The proposed method is then used to quantify uncertainty in a model of flow through porous media with an unknown permeability field. A Karhunen–Loève expansion is used to parameterize the uncertainty and the resulting mean and variance in the speed of the fluid and the time dependent saturation front are computed.

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Acknowledgements

The authors would like to thank Jaideep Ray of Sandia National Laboratories, Livermore, CA, USA for providing us with the model discussed in Sect. 6.1.

Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    John D. Jakeman

  2. Mathematical Sciences Institute, Australian National University, Canberra, ACT, 0200, Australia

    Stephen G. Roberts

Authors
  1. John D. Jakeman
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  2. Stephen G. Roberts
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Corresponding author

Correspondence to John D. Jakeman .

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Editors and Affiliations

  1. , Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, Bonn, 53115, Germany

    Jochen Garcke

  2. Inst. Numerische Simulation, Universität Bonn, Wegelerstr. 6, Bonn, 53115, Germany

    Michael Griebel

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© 2012 Springer-Verlag Berlin Heidelberg

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Jakeman, J.D., Roberts, S.G. (2012). Local and Dimension Adaptive Stochastic Collocation for Uncertainty Quantification. In: Garcke, J., Griebel, M. (eds) Sparse Grids and Applications. Lecture Notes in Computational Science and Engineering, vol 88. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31703-3_9

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