Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains
► We propose a tensor-based method for the solution of PDEs defined on uncertain parameterized domains. ► We use a fictitious domain approach to obtain a formulation in a tensor product space. ► We use a PGD algorithm for the construction of a tensor approximation of the solution. ► We introduce a c...
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| Vydáno v: | Computer methods in applied mechanics and engineering Ročník 200; číslo 45; s. 3066 - 3082 |
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| Jazyk: | angličtina |
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Kidlington
Elsevier B.V
01.01.2011
Elsevier |
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| ISSN: | 0045-7825, 1879-2138 |
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| Abstract | ► We propose a tensor-based method for the solution of PDEs defined on uncertain parameterized domains. ► We use a fictitious domain approach to obtain a formulation in a tensor product space. ► We use a PGD algorithm for the construction of a tensor approximation of the solution. ► We introduce a constrained SVD of the parameterized indicator function which preserves positivity. ► We analyze errors due to fictitious domain formulations and approximations of indicator functions.
A tensor-based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains. It provides an accurate solution which is explicit with respect to parameters defining the shape of the domain, thus allowing efficient
a posteriori probabilistic or parametric analyses. In the proposed method, a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain, yielding a weak formulation in a tensor product space (product of space functions and parametric functions). The paper is limited to the case of Neumann conditions on uncertain parts of the boundary. The Proper Generalized Decomposition method is then introduced for the construction of a tensor product approximation (separated representation) of the solution. It can be seen as an
a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution. This tensor-based method is made computationally tractable by introducing separated representations of variational forms, resulting from separated representations of the parameterized indicator function of the uncertain domain. For this purpose, a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well-posedness of problems associated with approximate indicator functions. Moreover, a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations. |
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| AbstractList | ► We propose a tensor-based method for the solution of PDEs defined on uncertain parameterized domains. ► We use a fictitious domain approach to obtain a formulation in a tensor product space. ► We use a PGD algorithm for the construction of a tensor approximation of the solution. ► We introduce a constrained SVD of the parameterized indicator function which preserves positivity. ► We analyze errors due to fictitious domain formulations and approximations of indicator functions.
A tensor-based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains. It provides an accurate solution which is explicit with respect to parameters defining the shape of the domain, thus allowing efficient
a posteriori probabilistic or parametric analyses. In the proposed method, a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain, yielding a weak formulation in a tensor product space (product of space functions and parametric functions). The paper is limited to the case of Neumann conditions on uncertain parts of the boundary. The Proper Generalized Decomposition method is then introduced for the construction of a tensor product approximation (separated representation) of the solution. It can be seen as an
a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution. This tensor-based method is made computationally tractable by introducing separated representations of variational forms, resulting from separated representations of the parameterized indicator function of the uncertain domain. For this purpose, a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well-posedness of problems associated with approximate indicator functions. Moreover, a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations. A tensor-based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains. It provides an accurate solution which is explicit with respect to parameters defining the shape of the domain, thus allowing efficient a posteriori probabilistic or parametric analyses. In the proposed method, a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain, yielding a weak formulation in a tensor product space (product of space functions and parametric functions). The paper is limited to the case of Neumann conditions on uncertain parts of the boundary. The Proper Generalized Decomposition method is then introduced for the construction of a tensor product approximation (separated representation) of the solution. It can be seen as an a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution. This tensor-based method is made computationally tractable by introducing separated representations of variational forms, resulting from separated representations of the parameterized indicator function of the uncertain domain. For this purpose, a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well-posedness of problems associated with approximate indicator functions. Moreover, a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations. A tensor-based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains. It provides an accurate solution which is explicit with respect to parameters defining the shape of the domain, thus allowing efficient a posteriori probabilistic or parametric analyses. In the proposed method, a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain, yielding a weak formulation in a tensor product space (product of space functions and parametric functions). The paper is limited to the case of Neumann conditions on uncertain parts of the boundary. The Proper Generalized Decomposition method is then introduced for the construction of a tensor product approximation (separated representation) of the solution. It can be seen as an a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution. This tensor-based method is made computationally tractable by introducing separated representations of variational forms, resulting from separated representations of the parameterized indicator function of the uncertain domain. For this purpose, a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well-posedness of problems associated with approximate indicator functions. Moreover, a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations. |
| Author | Nouy, A. Safatly, E. Chevreuil, M. |
| Author_xml | – sequence: 1 givenname: A. surname: Nouy fullname: Nouy, A. email: anthony.nouy@ec-nantes.fr – sequence: 2 givenname: M. surname: Chevreuil fullname: Chevreuil, M. – sequence: 3 givenname: E. surname: Safatly fullname: Safatly, E. |
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| Keywords | Uncertainty quantification Proper Generalized Decomposition Fictitious domain method Spectral stochastic methods Random domain Tensor product approximation Function space Tensor product Decomposition method Positivity Modeling Partial differential equation Uncertain system System reduction Variational calculus Reduced order systems Neumann problem Spectral method Probabilistic approach Well posed problem Indicator Reduction method Boundary value problem Weak solution Regularization Tensor method |
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| Snippet | ► We propose a tensor-based method for the solution of PDEs defined on uncertain parameterized domains. ► We use a fictitious domain approach to obtain a... A tensor-based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains. It provides an accurate... A tensor-based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains. It provides an accurate solution... |
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| SubjectTerms | Approximation Construction Engineering Sciences Exact sciences and technology Fictitious domain method Indicators Mathematical analysis Mathematical models Mathematics Mechanics Methods of scientific computing (including symbolic computation, algebraic computation) Numerical Analysis Numerical analysis. Scientific computation Partial differential equations Preserves Proper Generalized Decomposition Random domain Representations Sciences and techniques of general use Spectral stochastic methods Tensor product approximation Tensors Uncertainty quantification |
| Title | Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains |
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