On the computation of intrinsic Proper Generalized Decomposition modes of parametric symmetric elliptic problems on Grassmann manifolds
In this work, we introduce an iterative optimization algorithm to obtain the intrinsic Proper Generalized Decomposition modes of elliptic partial differential equations. The main idea behind this procedure is to adapt the general Gradient Descent algorithm to the algebraic version of the intrinsic P...
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| Vydané v: | Applied mathematics and computation Ročník 470; s. 128579 |
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01.06.2024
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| Abstract | In this work, we introduce an iterative optimization algorithm to obtain the intrinsic Proper Generalized Decomposition modes of elliptic partial differential equations. The main idea behind this procedure is to adapt the general Gradient Descent algorithm to the algebraic version of the intrinsic Proper Generalized Decomposition framework, and then to couple a one-dimensional case of the algorithm with a deflation algorithm in order to keep enriching the solution by computing further intrinsic Proper Generalized Decomposition modes. For this novel iterative optimization procedure, we present some numerical tests based on physical parametric problems, in which we obtain very promising results in comparison with the ones already presented in the literature. This supports the use of this procedure in order to obtain the intrinsic PGD modes of parametric symmetric problems.
•Computation of Proper Generalized Decomposition modes geometrically.•Adaptation of the Gradient Descent algorithm to the Proper Generalized Decomposition matrix framework.•Application to physical problems with very promising results compared to previous literature. |
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| AbstractList | In this work, we introduce an iterative optimization algorithm to obtain the intrinsic Proper Generalized Decomposition modes of elliptic partial differential equations. The main idea behind this procedure is to adapt the general Gradient Descent algorithm to the algebraic version of the intrinsic Proper Generalized Decomposition framework, and then to couple a one-dimensional case of the algorithm with a deflation algorithm in order to keep enriching the solution by computing further intrinsic Proper Generalized Decomposition modes. For this novel iterative optimization procedure, we present some numerical tests based on physical parametric problems, in which we obtain very promising results in comparison with the ones already presented in the literature. This supports the use of this procedure in order to obtain the intrinsic PGD modes of parametric symmetric problems.
•Computation of Proper Generalized Decomposition modes geometrically.•Adaptation of the Gradient Descent algorithm to the Proper Generalized Decomposition matrix framework.•Application to physical problems with very promising results compared to previous literature. |
| ArticleNumber | 128579 |
| Author | Fernández-García, Soledad Gómez-Mármol, Macarena Bandera, Alejandro |
| Author_xml | – sequence: 1 givenname: Alejandro orcidid: 0000-0002-5633-2333 surname: Bandera fullname: Bandera, Alejandro email: abandera@us.es organization: Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Calle Tarfia s/n, Seville, 41012, Spain – sequence: 2 givenname: Soledad orcidid: 0000-0001-6993-407X surname: Fernández-García fullname: Fernández-García, Soledad organization: Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Calle Tarfia s/n, Seville, 41012, Spain – sequence: 3 givenname: Macarena surname: Gómez-Mármol fullname: Gómez-Mármol, Macarena organization: Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Calle Tarfia s/n, Seville, 41012, Spain |
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| Cites_doi | 10.1007/s40324-019-00198-7 10.1007/s11831-010-9049-y 10.1137/S0895479895290954 10.1007/s11831-010-9053-2 10.1137/17M1137164 10.1007/s12289-009-0647-x 10.1016/j.jnnfm.2006.07.007 10.7151/dmdico.1078 10.1146/annurev.fl.25.010193.002543 10.1007/s002110100282 10.1007/s11831-011-9064-7 10.1051/m2an/2011053 10.1137/S0036142901389049 10.1016/j.jmaa.2010.12.003 |
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| Keywords | Symmetric elliptic problems Proper Generalized Decomposition Gradient descent Grassmann manifold Reduced order modeling |
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| Title | On the computation of intrinsic Proper Generalized Decomposition modes of parametric symmetric elliptic problems on Grassmann manifolds |
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