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Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs

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Titel: Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs
Autoren: Nobile, Fabio, Tamellini, Lorenzo, Tempone, Raul
Verlagsinformationen: Springer
Publikationsjahr: 2014
Bestand: Ecole Polytechnique Fédérale Lausanne (EPFL): Infoscience
Schlagwörter: Uncertainty Quantification, random PDEs, linear elliptic equations, multivariate polynomial approximation, best M-terms polynomial approximation, Smolyak approximation, Sparse grids, Stochastic Collocation method
Beschreibung: In this work we provide a convergence analysis for the quasi-optimal version of the sparse-grids stochastic collocation method we presented in a previous work: “On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods” (Beck et al., Math Models Methods Appl Sci 22(09), 2012). The construction of a sparse grid is recast into a knapsack problem: a profit is assigned to each hierarchical surplus and only the most profitable ones are added to the sparse grid. The convergence rate of the sparse grid approximation error with respect to the number of points in the grid is then shown to depend on weighted summability properties of the sequence of profits. This is a very general argument that can be applied to sparse grids built with any uni-variate family of points, both nested and non-nested. As an example, we apply such quasi-optimal sparse grids to the solution of a particular elliptic PDE with stochastic diffusion coefficients, namely the “inclusions problem”: we detail the convergence estimates obtained in this case using polynomial interpolation on either nested (Clenshaw-Curtis) or non-nested (Gauss-Legendre) abscissas, verify their sharpness numerically, and compare the performance of the resulting quasi-optimal grids with a few alternative sparse-grid construction schemes recently proposed in the literature. ; CSQI
Publikationsart: article in journal/newspaper
Sprache: unknown
ISSN: 0029-599X
0945-3245
Relation: https://infoscience.epfl.ch/record/196966/files/2016_Nobile_Tamellini_Tempone_NM_Convergence.pdf; Numerische Mathematik; #PLACEHOLDER_PARENT_METADATA_VALUE#; https://infoscience.epfl.ch/handle/20.500.14299/100963; WOS:000382146600005
DOI: 10.1007/s00211-015-0773-y
Verfügbarkeit: https://doi.org/10.1007/s00211-015-0773-y
https://infoscience.epfl.ch/handle/20.500.14299/100963
https://hdl.handle.net/20.500.14299/100963
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  Data: Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs
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  Data: <searchLink fieldCode="AR" term="%22Nobile%2C+Fabio%22">Nobile, Fabio</searchLink><br /><searchLink fieldCode="AR" term="%22Tamellini%2C+Lorenzo%22">Tamellini, Lorenzo</searchLink><br /><searchLink fieldCode="AR" term="%22Tempone%2C+Raul%22">Tempone, Raul</searchLink>
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  Data: 2014
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  Data: Ecole Polytechnique Fédérale Lausanne (EPFL): Infoscience
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  Data: <searchLink fieldCode="DE" term="%22Uncertainty+Quantification%22">Uncertainty Quantification</searchLink><br /><searchLink fieldCode="DE" term="%22random+PDEs%22">random PDEs</searchLink><br /><searchLink fieldCode="DE" term="%22linear+elliptic+equations%22">linear elliptic equations</searchLink><br /><searchLink fieldCode="DE" term="%22multivariate+polynomial+approximation%22">multivariate polynomial approximation</searchLink><br /><searchLink fieldCode="DE" term="%22best+M-terms+polynomial+approximation%22">best M-terms polynomial approximation</searchLink><br /><searchLink fieldCode="DE" term="%22Smolyak+approximation%22">Smolyak approximation</searchLink><br /><searchLink fieldCode="DE" term="%22Sparse+grids%22">Sparse grids</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+Collocation+method%22">Stochastic Collocation method</searchLink>
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  Data: In this work we provide a convergence analysis for the quasi-optimal version of the sparse-grids stochastic collocation method we presented in a previous work: “On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods” (Beck et al., Math Models Methods Appl Sci 22(09), 2012). The construction of a sparse grid is recast into a knapsack problem: a profit is assigned to each hierarchical surplus and only the most profitable ones are added to the sparse grid. The convergence rate of the sparse grid approximation error with respect to the number of points in the grid is then shown to depend on weighted summability properties of the sequence of profits. This is a very general argument that can be applied to sparse grids built with any uni-variate family of points, both nested and non-nested. As an example, we apply such quasi-optimal sparse grids to the solution of a particular elliptic PDE with stochastic diffusion coefficients, namely the “inclusions problem”: we detail the convergence estimates obtained in this case using polynomial interpolation on either nested (Clenshaw-Curtis) or non-nested (Gauss-Legendre) abscissas, verify their sharpness numerically, and compare the performance of the resulting quasi-optimal grids with a few alternative sparse-grid construction schemes recently proposed in the literature. ; CSQI
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  Data: https://infoscience.epfl.ch/record/196966/files/2016_Nobile_Tamellini_Tempone_NM_Convergence.pdf; Numerische Mathematik; #PLACEHOLDER_PARENT_METADATA_VALUE#; https://infoscience.epfl.ch/handle/20.500.14299/100963; WOS:000382146600005
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      – SubjectFull: Uncertainty Quantification
        Type: general
      – SubjectFull: random PDEs
        Type: general
      – SubjectFull: linear elliptic equations
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      – SubjectFull: best M-terms polynomial approximation
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      – SubjectFull: Sparse grids
        Type: general
      – SubjectFull: Stochastic Collocation method
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      – TitleFull: Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs
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            NameFull: Tempone, Raul
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