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An Adaptive Wavelet Stochastic Collocation Method for Irregular Solutions of Partial Differential Equations with Random Input Data

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Sparse Grids and Applications - Munich 2012

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 97))

Abstract

A novel multi-dimensional multi-resolution adaptive wavelet stochastic collocation method (AWSCM) for solving partial differential equations with random input data is proposed. The uncertainty in the input data is assumed to depend on a finite number of random variables. In case the dimension of this stochastic domain becomes moderately large, we show that utilizing a hierarchical sparse-grid AWSCM (sg-AWSCM) not only combats the curse of dimensionality but, in contrast to the standard sg-SCMs built from global Lagrange-type interpolating polynomials, maintains fast convergence without requiring sufficiently regular stochastic solutions. Instead, our non-intrusive approach extends the sparse-grid adaptive linear stochastic collocation method (sg-ALSCM) by employing a compactly supported wavelet approximation, with the desirable multi-scale stability of the hierarchical coefficients guaranteed as a result of the wavelet basis having the Riesz property. This property provides an additional lower bound estimate for the wavelet coefficients that are used to guide the adaptive grid refinement, resulting in the sg-AWSCM requiring a significantly reduced number of deterministic simulations for both smooth and irregular stochastic solutions. Second-generation wavelets constructed from a lifting scheme allows us to preserve the framework of the multi-resolution analysis, compact support, as well as the necessary interpolatory and Riesz property of the hierarchical basis. Several numerical examples are given to demonstrate the improved convergence of our numerical scheme and show the increased efficiency when compared to the sg-ALSCM method.

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Acknowledgements

Max Gunzburger was supported by the US Air Force Office of Scientific Research (AFOSR) under grant number FA9550-11-1-0149. Clayton G. Webster was supported by the US AFOSR under grant number 1854-V521-12. Also supported by the Laboratory Directed Research and Development (LDRD) Program at the Oak Ridge National Laboratory (ORNL). The ORNL is operated by UT-Battelle, LLC, for the United States Department of Energy under Contract DE-AC05-00OR22725. Guannan Zhang was supported by the US AFOSR under grant number FA9550-11-1-0149. Also supported by the Advanced Simulation Computing Research (ASCR), Department of Energy, through the Householder Fellowship at ORNL. The ORNL is operated by UT-Battelle, LLC, for the United States Department of Energy under Contract DE-AC05-00OR22725.

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Authors and Affiliations

  1. Department of Scientific Computing, 400 Dirac Science Library, Florida State University, Tallahassee, FL, 32306-4120, USA

    Max Gunzburger

  2. Computer Science and Mathematics Division, Oak Ridge National Laboratory, One Bethel Valley Road, 2008, MS-6164, Oak Ridge, TN, 37831-6164, USA

    Clayton G. Webster & Guannan Zhang

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  1. Max Gunzburger
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  2. Clayton G. Webster
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  3. Guannan Zhang
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Correspondence to Max Gunzburger .

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Editors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Jochen Garcke

  2. Institute for Parallel and Distributed Systems, Universität Stuttgart, Stuttgart, Germany

    Dirk Pflüger

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Gunzburger, M., Webster, C.G., Zhang, G. (2014). An Adaptive Wavelet Stochastic Collocation Method for Irregular Solutions of Partial Differential Equations with Random Input Data. In: Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Munich 2012. Lecture Notes in Computational Science and Engineering, vol 97. Springer, Cham. https://doi.org/10.1007/978-3-319-04537-5_6

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