A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach

Journal of Mathematical Analysis and Applications

Volumn 376, Issue 2, 2011, Pages 469-480

  Falco A ^a   Nouy, A ^b  

^a UNIVERSIDAD CEU CARDENAL HERRERA   (Spain)

^b UNIVERSITÉ DE NANTES   (France)

Author keywords

Proper Generalized Decomposition; Singular values; Tensor product Hilbert spaces

Indexed keywords

EID: 78650698308     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2010.12.003     Document Type: Article

References (29)

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. 2006, 139(3):153-176.
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. 2007, 144(2-3):98-121.
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. 2010, 17(4):473-486.
4. Chen J., Saad Y. On the tensor SVD and the optimal low rank orthogonal approximation of tensors. SIAM J. Matrix Anal. Appl. 2008, 30(4):1709-1734.
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. 2010, 17(4):327-350.
6. De Lathauwer L., De Moor B., Vandewalle J. A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 2000, 21(4):1253-1278.
7. De Lathauwer L., De Moor B., Vandewalle J. On the best rank-1 and rank-(R1,R2,...,RN) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 2000, 21(4):1324-1342.
8. de Silva V., Lim L.-H. Tensor rank and ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 2008, 30(3):1084-1127.
9. Doostan A., Iaccarino G. A least-squares approximation of partial differential equations with high-dimensional random inputs. J. Comput. Phys. 2009, 228(12):4332-4345.
10. Dureisseix D., Ladevèze P., Schrefler B.A. A computational strategy for multiphysics problems - application to poroelasticity. Internat. J. Numer. Methods Engrg. 2003, 56(10):1489-1510.
11. Eckart Carl, Young Gale The approximation of one matrix by another of lower rank. Psychometrika 1936, 1(3):211-218.
12. Ghanem R.G., Spanos P.D. Stochastic Finite Elements: A Spectral Approach 2002, Dover. 2nd ed.
13. Kolda T.G. Orthogonal tensor decompositions. SIAM J. Matrix Anal. Appl. 2001, 23(1):243-255.
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. 2003, 24(3):762-767.
15. Ladevèze P. Nonlinear Computational Structural Mechanics - New Approaches and Non-Incremental Methods of Calculation 1999, Springer-Verlag.
16. Ladevèze P., Nouy A. On a multiscale computational strategy with time and space homogenization for structural mechanics. Comput. Methods Appl. Mech. Engrg. 2003, 192(28-30):3061-3087.
17. Ladevèze P., Passieux J.C., Néron D. The LATIN multiscale computational method and the proper generalized decomposition. Comput. Methods Appl. Mech. Engrg. 2010, 199(21-22):1287-1296.
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. 2009, 30(3):621-651.
19. Nouy A. A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 2007, 196(45-48):4521-4537.
20. Nouy A. Generalized spectral decomposition method for solving stochastic finite element equations: invariant subspace problem and dedicated algorithms. Comput. Methods Appl. Mech. Engrg. 2008, 197:4718-4736.
21. Nouy A. Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Arch. Comput. Methods Engrg. 2009, 16(3):251-285.
22. Nouy A. Proper generalized decompositions and separated representations for the numerical solution of high dimensional stochastic problems. Arch. Comput. Methods Engrg. 2010, 17(4):403-434.
23. Nouy A. A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Engrg. 2010, 199(23-24):1063-1626.
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. 2004, 170(2):557-574.
25. Nouy A., Le Maître O.P. Generalized spectral decomposition method for stochastic non linear problems. J. Comput. Phys. 2009, 228(1):202-235.
26. Soize C., Ghanem R. Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 2004, 26(2):395-410.
27. Xiu D., Karniadakis G.E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 2002, 24(2):619-644.
28. Wiener N. The homogeneous chaos. Amer. J. Math. 1938, 60:897-936.
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. 2009, 228(18):7025-7045.