Artículo
A new algorithm of proper generalized decomposition for parametric symmetric elliptic problems
Autor/es | Azaïez, Majdi
Ben Belgacem, Faker Casado Díaz, Juan Chacón Rebollo, Tomás Murat, François |
Departamento | Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico |
Fecha de publicación | 2018 |
Fecha de depósito | 2019-01-15 |
Publicado en |
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Resumen | We introduce a new algorithm of proper generalized decomposition (PGD) for parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most ... We introduce a new algorithm of proper generalized decomposition (PGD) for parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation---in the mean parametric norm associated to the elliptic operator---of the error between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the proper orthogonal decomposition (POD) subspaces, except that in our case the norm is parameter-dependent. We apply a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, directly in the online step, and we prove that the partial sums converge to the continuous solution in the mean parametric elliptic norm. We show that the standard PGD for the considered parametric problem is strongly related to the deflation algorithm introduced in this paper. This opens the possibility of computing the PGD expansion by directly solving the optimization problems that yield the optimal subspaces. |
Identificador del proyecto | MTM2015-64577-C2-1-R
51661135011 - PHASEFIELD MTM2014-53309-P |
Cita | Azaïez, M., Ben Belgacem, F., Casado Díaz, J., Chacón Rebollo, T. y Murat, F. (2018). A new algorithm of proper generalized decomposition for parametric symmetric elliptic problems. SIAM Journal on Mathematical Analysis, 50 (5), 5426-5445. |
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