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ADAPTIVE SMOLYAK PSEUDOSPECTRAL APPROXIMATIONS.

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Titel: ADAPTIVE SMOLYAK PSEUDOSPECTRAL APPROXIMATIONS.
Autoren: CONRAD, PATRICK R., MARZOUK, YOUSSEF M.
Quelle: SIAM Journal on Scientific Computing; 2013, Vol. 35 Issue 6, pA2643-A2670, 28p
Schlagwörter: ALGORITHMS, POLYNOMIAL approximation, SPARSE graphs, ORTHOGONAL polynomials, APPROXIMATION theory, CHEMICAL kinetics
Abstract: Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for nonintrusive pseudospectral approximation, based on Smolyak's algorithm with generalized sparse grids. We rigorously analyze and extend the nonadaptive method proposed in [P. G. Constantine, M. S. Eldred, and E. T. Phipps, Comput. Methods Appl. Mech. Engrg., 229-232 (2012), pp. 1-12], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that O(1) errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal aliasing and makes far more effective use of sparse function evaluations. These results are applicable to broad choices of quadrature rule and generalized sparse grids. Exploiting this flexibility, we introduce a greedy heuristic for adaptive refinement of the pseudospectral approximation. We numerically demonstrate convergence of the algorithm on the Genz test functions, and illustrate the accuracy and efficiency of the adaptive approach on a realistic chemical kinetics problem. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: ADAPTIVE SMOLYAK PSEUDOSPECTRAL APPROXIMATIONS.
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  Data: SIAM Journal on Scientific Computing; 2013, Vol. 35 Issue 6, pA2643-A2670, 28p
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  Data: <searchLink fieldCode="DE" term="%22ALGORITHMS%22">ALGORITHMS</searchLink><br /><searchLink fieldCode="DE" term="%22POLYNOMIAL+approximation%22">POLYNOMIAL approximation</searchLink><br /><searchLink fieldCode="DE" term="%22SPARSE+graphs%22">SPARSE graphs</searchLink><br /><searchLink fieldCode="DE" term="%22ORTHOGONAL+polynomials%22">ORTHOGONAL polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22APPROXIMATION+theory%22">APPROXIMATION theory</searchLink><br /><searchLink fieldCode="DE" term="%22CHEMICAL+kinetics%22">CHEMICAL kinetics</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for nonintrusive pseudospectral approximation, based on Smolyak's algorithm with generalized sparse grids. We rigorously analyze and extend the nonadaptive method proposed in [P. G. Constantine, M. S. Eldred, and E. T. Phipps, Comput. Methods Appl. Mech. Engrg., 229-232 (2012), pp. 1-12], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that O(1) errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal aliasing and makes far more effective use of sparse function evaluations. These results are applicable to broad choices of quadrature rule and generalized sparse grids. Exploiting this flexibility, we introduce a greedy heuristic for adaptive refinement of the pseudospectral approximation. We numerically demonstrate convergence of the algorithm on the Genz test functions, and illustrate the accuracy and efficiency of the adaptive approach on a realistic chemical kinetics problem. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label:
  Group: Ab
  Data: <i>Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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      – Type: doi
        Value: 10.1137/120890715
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 28
        StartPage: A2643
    Subjects:
      – SubjectFull: ALGORITHMS
        Type: general
      – SubjectFull: POLYNOMIAL approximation
        Type: general
      – SubjectFull: SPARSE graphs
        Type: general
      – SubjectFull: ORTHOGONAL polynomials
        Type: general
      – SubjectFull: APPROXIMATION theory
        Type: general
      – SubjectFull: CHEMICAL kinetics
        Type: general
    Titles:
      – TitleFull: ADAPTIVE SMOLYAK PSEUDOSPECTRAL APPROXIMATIONS.
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            NameFull: CONRAD, PATRICK R.
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            NameFull: MARZOUK, YOUSSEF M.
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              M: 09
              Text: 2013
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              Y: 2013
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