A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature
The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared erro...
Ausführliche Beschreibung
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| Autor*in: |
Mika, Michal L. [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021transfer abstract |
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| Schlagwörter: |
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| Übergeordnetes Werk: |
Enthalten in: Does enhanced hydration have impact on autogenous deformation of internally cued mortar? - Zou, Dinghua ELSEVIER, 2019, Amsterdam [u.a.] |
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| Übergeordnetes Werk: |
volume:379 ; year:2021 ; day:1 ; month:06 ; pages:0 |
| Links: |
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| DOI / URN: |
10.1016/j.cma.2021.113730 |
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| 245 | 1 | 0 | |a A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature |
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| 520 | |a The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. | ||
| 520 | |a The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. | ||
| 650 | 7 | |a Isogeometric analysis |2 Elsevier | |
| 650 | 7 | |a Fredholm integral eigenvalue problem |2 Elsevier | |
| 650 | 7 | |a Random fields |2 Elsevier | |
| 650 | 7 | |a Matrix-free solver |2 Elsevier | |
| 650 | 7 | |a Kronecker products |2 Elsevier | |
| 700 | 1 | |a Hughes, Thomas J.R. |4 oth | |
| 700 | 1 | |a Schillinger, Dominik |4 oth | |
| 700 | 1 | |a Wriggers, Peter |4 oth | |
| 700 | 1 | |a Hiemstra, René R. |4 oth | |
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10.1016/j.cma.2021.113730 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001569.pica (DE-627)ELV05359939X (ELSEVIER)S0045-7825(21)00066-9 DE-627 ger DE-627 rakwb eng 690 VZ 56.45 bkl Mika, Michal L. verfasserin aut A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature 2021transfer abstract nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. Isogeometric analysis Elsevier Fredholm integral eigenvalue problem Elsevier Random fields Elsevier Matrix-free solver Elsevier Kronecker products Elsevier Hughes, Thomas J.R. oth Schillinger, Dominik oth Wriggers, Peter oth Hiemstra, René R. oth Enthalten in Elsevier Science Zou, Dinghua ELSEVIER Does enhanced hydration have impact on autogenous deformation of internally cued mortar? 2019 Amsterdam [u.a.] (DE-627)ELV002113945 volume:379 year:2021 day:1 month:06 pages:0 https://doi.org/10.1016/j.cma.2021.113730 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U 56.45 Baustoffkunde VZ AR 379 2021 1 0601 0 |
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10.1016/j.cma.2021.113730 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001569.pica (DE-627)ELV05359939X (ELSEVIER)S0045-7825(21)00066-9 DE-627 ger DE-627 rakwb eng 690 VZ 56.45 bkl Mika, Michal L. verfasserin aut A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature 2021transfer abstract nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. Isogeometric analysis Elsevier Fredholm integral eigenvalue problem Elsevier Random fields Elsevier Matrix-free solver Elsevier Kronecker products Elsevier Hughes, Thomas J.R. oth Schillinger, Dominik oth Wriggers, Peter oth Hiemstra, René R. oth Enthalten in Elsevier Science Zou, Dinghua ELSEVIER Does enhanced hydration have impact on autogenous deformation of internally cued mortar? 2019 Amsterdam [u.a.] (DE-627)ELV002113945 volume:379 year:2021 day:1 month:06 pages:0 https://doi.org/10.1016/j.cma.2021.113730 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U 56.45 Baustoffkunde VZ AR 379 2021 1 0601 0 |
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10.1016/j.cma.2021.113730 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001569.pica (DE-627)ELV05359939X (ELSEVIER)S0045-7825(21)00066-9 DE-627 ger DE-627 rakwb eng 690 VZ 56.45 bkl Mika, Michal L. verfasserin aut A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature 2021transfer abstract nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. Isogeometric analysis Elsevier Fredholm integral eigenvalue problem Elsevier Random fields Elsevier Matrix-free solver Elsevier Kronecker products Elsevier Hughes, Thomas J.R. oth Schillinger, Dominik oth Wriggers, Peter oth Hiemstra, René R. oth Enthalten in Elsevier Science Zou, Dinghua ELSEVIER Does enhanced hydration have impact on autogenous deformation of internally cued mortar? 2019 Amsterdam [u.a.] (DE-627)ELV002113945 volume:379 year:2021 day:1 month:06 pages:0 https://doi.org/10.1016/j.cma.2021.113730 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U 56.45 Baustoffkunde VZ AR 379 2021 1 0601 0 |
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10.1016/j.cma.2021.113730 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001569.pica (DE-627)ELV05359939X (ELSEVIER)S0045-7825(21)00066-9 DE-627 ger DE-627 rakwb eng 690 VZ 56.45 bkl Mika, Michal L. verfasserin aut A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature 2021transfer abstract nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. Isogeometric analysis Elsevier Fredholm integral eigenvalue problem Elsevier Random fields Elsevier Matrix-free solver Elsevier Kronecker products Elsevier Hughes, Thomas J.R. oth Schillinger, Dominik oth Wriggers, Peter oth Hiemstra, René R. oth Enthalten in Elsevier Science Zou, Dinghua ELSEVIER Does enhanced hydration have impact on autogenous deformation of internally cued mortar? 2019 Amsterdam [u.a.] (DE-627)ELV002113945 volume:379 year:2021 day:1 month:06 pages:0 https://doi.org/10.1016/j.cma.2021.113730 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U 56.45 Baustoffkunde VZ AR 379 2021 1 0601 0 |
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10.1016/j.cma.2021.113730 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001569.pica (DE-627)ELV05359939X (ELSEVIER)S0045-7825(21)00066-9 DE-627 ger DE-627 rakwb eng 690 VZ 56.45 bkl Mika, Michal L. verfasserin aut A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature 2021transfer abstract nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. Isogeometric analysis Elsevier Fredholm integral eigenvalue problem Elsevier Random fields Elsevier Matrix-free solver Elsevier Kronecker products Elsevier Hughes, Thomas J.R. oth Schillinger, Dominik oth Wriggers, Peter oth Hiemstra, René R. oth Enthalten in Elsevier Science Zou, Dinghua ELSEVIER Does enhanced hydration have impact on autogenous deformation of internally cued mortar? 2019 Amsterdam [u.a.] (DE-627)ELV002113945 volume:379 year:2021 day:1 month:06 pages:0 https://doi.org/10.1016/j.cma.2021.113730 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U 56.45 Baustoffkunde VZ AR 379 2021 1 0601 0 |
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a matrix-free isogeometric galerkin method for karhunen–loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature |
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A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature |
| abstract |
The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. |
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The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. |
| abstract_unstemmed |
The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2 d dimensional domain, where d , in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C 0 -conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness. |
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A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature |
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