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A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach. (English) Zbl 1210.65009

Summary: The proper generalized decomposition (PGD) is a methodology initially proposed for the solution of partial differential equations (PDE) defined in tensor product spaces. It consists in constructing a separated representation of the solution of a given PDE. In this paper we consider the mathematical analysis of this framework for a larger class of problems in an abstract setting. In particular, we introduce a generalization of Eckart and Young theorem [C. Eckart, G. Young, Psychometrika, Chicago, 1, 211–218 (1936; JFM 62.1075.02)] which allows to prove the convergence of the so-called progressive PGD for a large class of linear problems defined in tensor product Hilbert spaces.

MSC:

65C20 Probabilistic models, generic numerical methods in probability and statistics
35J99 Elliptic equations and elliptic systems

Citations:

JFM 62.1075.02
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References:

[1] Ammar, A.; Mokdad, B.; Chinesta, F.; Keunings, R., A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids, J. Non-Newtonian Fluid Mech., 139, 3, 153-176 (2006) · Zbl 1195.76337
[2] Ammar, A.; Mokdad, B.; Chinesta, F.; Keunings, R., A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part II: Transient simulation using space-time separated representations, J. Non-Newtonian Fluid Mech., 144, 2-3, 98-121 (2007) · Zbl 1196.76047
[3] Ammar, A.; Chinesta, F.; Falcó, A., On the convergence of a greedy rank-one update algorithm for a class of linear systems, Arch. Comput. Methods Engrg., 17, 4, 473-486 (2010) · Zbl 1269.65120
[4] Chen, J.; Saad, Y., On the tensor SVD and the optimal low rank orthogonal approximation of tensors, SIAM J. Matrix Anal. Appl., 30, 4, 1709-1734 (2008) · Zbl 1184.65043
[5] Chinesta, F.; Ammar, A.; Cueto, E., Recent advances in the use of the proper generalized decomposition for solving multidimensional models, Arch. Comput. Methods Engrg., 17, 4, 327-350 (2010) · Zbl 1269.65106
[6] De Lathauwer, L.; De Moor, B.; Vandewalle, J., A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21, 4, 1253-1278 (2000) · Zbl 0962.15005
[7] De Lathauwer, L.; De Moor, B.; Vandewalle, J., On the best rank-1 and rank-\((R_1, R_2, \ldots, R_N)\) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21, 4, 1324-1342 (2000) · Zbl 0958.15026
[8] de Silva, V.; Lim, L.-H., Tensor rank and ill-posedness of the best low-rank approximation problem, SIAM J. Matrix Anal. Appl., 30, 3, 1084-1127 (2008) · Zbl 1167.14038
[9] Doostan, A.; Iaccarino, G., A least-squares approximation of partial differential equations with high-dimensional random inputs, J. Comput. Phys., 228, 12, 4332-4345 (2009) · Zbl 1167.65322
[10] Dureisseix, D.; Ladevèze, P.; Schrefler, B. A., A computational strategy for multiphysics problems — application to poroelasticity, Internat. J. Numer. Methods Engrg., 56, 10, 1489-1510 (2003) · Zbl 1106.74425
[11] Eckart, Carl; Young, Gale, The approximation of one matrix by another of lower rank, Psychometrika, 1, 3, 211-218 (1936) · JFM 62.1075.02
[12] Ghanem, R. G.; Spanos, P. D., Stochastic Finite Elements: A Spectral Approach (2002), Dover · Zbl 0953.74608
[13] Kolda, T. G., Orthogonal tensor decompositions, SIAM J. Matrix Anal. Appl., 23, 1, 243-255 (2001) · Zbl 1005.15020
[14] Kolda, T. G., A counterexample to the possibility of an extension of the Eckart-Young low-rank approximation theorem for the orthogonal rank tensor decomposition, SIAM J. Matrix Anal. Appl., 24, 3, 762-767 (2003) · Zbl 1044.15020
[15] Ladevèze, P., Nonlinear Computational Structural Mechanics - New Approaches and Non-Incremental Methods of Calculation (1999), Springer-Verlag · Zbl 0912.73003
[16] Ladevèze, P.; Nouy, A., On a multiscale computational strategy with time and space homogenization for structural mechanics, Comput. Methods Appl. Mech. Engrg., 192, 28-30, 3061-3087 (2003) · Zbl 1054.74701
[17] Ladevèze, P.; Passieux, J. C.; Néron, D., The LATIN multiscale computational method and the proper generalized decomposition, Comput. Methods Appl. Mech. Engrg., 199, 21-22, 1287-1296 (2010) · Zbl 1227.74111
[18] Le Bris, C.; Lelievre, T.; Maday, Y., Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations, Constr. Approx., 30, 3, 621-651 (2009) · Zbl 1191.65156
[19] Nouy, A., A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg., 196, 45-48, 4521-4537 (2007) · Zbl 1173.80311
[20] Nouy, A., Generalized spectral decomposition method for solving stochastic finite element equations: invariant subspace problem and dedicated algorithms, Comput. Methods Appl. Mech. Engrg., 197, 4718-4736 (2008) · Zbl 1194.74458
[21] Nouy, A., Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations, Arch. Comput. Methods Engrg., 16, 3, 251-285 (2009) · Zbl 1360.65036
[22] Nouy, A., Proper generalized decompositions and separated representations for the numerical solution of high dimensional stochastic problems, Arch. Comput. Methods Engrg., 17, 4, 403-434 (2010) · Zbl 1269.76079
[23] Nouy, A., A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations, Comput. Methods Appl. Mech. Engrg., 199, 23-24, 1063-1626 (2010) · Zbl 1231.76219
[24] Nouy, A.; Ladevèze, P., Multiscale computational strategy with time and space homogenization: a radial-type approximation technique for solving micro problems, Int. J. Multiscale Comput. Eng., 170, 2, 557-574 (2004)
[25] Nouy, A.; Le Maître, O. P., Generalized spectral decomposition method for stochastic non linear problems, J. Comput. Phys., 228, 1, 202-235 (2009) · Zbl 1157.65009
[26] Soize, C.; Ghanem, R., Physical systems with random uncertainties: chaos representations with arbitrary probability measure, SIAM J. Sci. Comput., 26, 2, 395-410 (2004) · Zbl 1075.60084
[27] Xiu, D.; Karniadakis, G. E., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24, 2, 619-644 (2002) · Zbl 1014.65004
[28] Wiener, N., The homogeneous chaos, Amer. J. Math., 60, 897-936 (1938) · JFM 64.0887.02
[29] Witteveen, J. A.S.; Bijl, H., Effect of randomness on multi-frequency aerolastic responses resolved by unsteady adaptive stochastic finite elements, J. Comput. Phys., 228, 18, 7025-7045 (2009) · Zbl 1175.65011
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