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MATHICSE Technical Report : Multi-index stochastic collocation for random PDEs

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Title: MATHICSE Technical Report : Multi-index stochastic collocation for random PDEs
Authors: Haji Ali, Abdul Lateef, Nobile, Fabio, Tamellini, Lorenzo, Tempone, Raùl
Contributors: MATHICSE-Group
Publisher Information: MATHICSE
Écublens
Publication Year: 2019
Collection: Ecole Polytechnique Fédérale Lausanne (EPFL): Infoscience
Subject Terms: Uncertainty Quantiffication, Random PDEs, Multivariate approximation, Sparse grids, Stochastic Collocation methods, Multilevel methods, Combination technique
Description: In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most eective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more eective than other multi-level collocation methods already available in literature. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi- Index and Multilevel Monte Carlo, Multilevel Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods. ; CSQI ; MATHICSE Technical Report Nr. 22.2015 September 2015
Document Type: report
Language: unknown
Relation: https://infoscience.epfl.ch/record/263551/files/22.2015_AH-FN-LT-RT.pdf; #PLACEHOLDER_PARENT_METADATA_VALUE#; https://infoscience.epfl.ch/handle/20.500.14299/154076
DOI: 10.5075/epfl-MATHICSE-263551
Availability: https://doi.org/10.5075/epfl-MATHICSE-263551
https://infoscience.epfl.ch/handle/20.500.14299/154076
https://hdl.handle.net/20.500.14299/154076
Accession Number: edsbas.3BAC183D
Database: BASE
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  – Url: https://doi.org/10.5075/epfl-MATHICSE-263551#
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  Data: MATHICSE Technical Report : Multi-index stochastic collocation for random PDEs
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  Data: <searchLink fieldCode="AR" term="%22Haji+Ali%2C+Abdul+Lateef%22">Haji Ali, Abdul Lateef</searchLink><br /><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+Raùl%22">Tempone, Raùl</searchLink>
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  Data: 2019
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  Data: Ecole Polytechnique Fédérale Lausanne (EPFL): Infoscience
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  Data: <searchLink fieldCode="DE" term="%22Uncertainty+Quantiffication%22">Uncertainty Quantiffication</searchLink><br /><searchLink fieldCode="DE" term="%22Random+PDEs%22">Random PDEs</searchLink><br /><searchLink fieldCode="DE" term="%22Multivariate+approximation%22">Multivariate approximation</searchLink><br /><searchLink fieldCode="DE" term="%22Sparse+grids%22">Sparse grids</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+Collocation+methods%22">Stochastic Collocation methods</searchLink><br /><searchLink fieldCode="DE" term="%22Multilevel+methods%22">Multilevel methods</searchLink><br /><searchLink fieldCode="DE" term="%22Combination+technique%22">Combination technique</searchLink>
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  Label: Description
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  Data: In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most eective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more eective than other multi-level collocation methods already available in literature. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi- Index and Multilevel Monte Carlo, Multilevel Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods. ; CSQI ; MATHICSE Technical Report Nr. 22.2015 September 2015
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  Data: https://infoscience.epfl.ch/record/263551/files/22.2015_AH-FN-LT-RT.pdf; #PLACEHOLDER_PARENT_METADATA_VALUE#; https://infoscience.epfl.ch/handle/20.500.14299/154076
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.5075/epfl-MATHICSE-263551
– Name: URL
  Label: Availability
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  Data: https://doi.org/10.5075/epfl-MATHICSE-263551<br />https://infoscience.epfl.ch/handle/20.500.14299/154076<br />https://hdl.handle.net/20.500.14299/154076
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    Subjects:
      – SubjectFull: Uncertainty Quantiffication
        Type: general
      – SubjectFull: Random PDEs
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      – SubjectFull: Multivariate approximation
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      – SubjectFull: Sparse grids
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      – SubjectFull: Stochastic Collocation methods
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      – SubjectFull: Multilevel methods
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      – SubjectFull: Combination technique
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      – TitleFull: MATHICSE Technical Report : Multi-index stochastic collocation for random PDEs
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            NameFull: Haji Ali, Abdul Lateef
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              Y: 2019
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