Abstract
The approximation of solutions to partial differential equations by tensorial separated representations is one of the most efficient numerical treatment of high dimensional problems. The key step of such methods is the computation of an optimal low-rank tensor to enrich the obtained iterative tensorial approximation. In variational problems, this step can be carried out by alternating minimization (AM) technics, but the convergence of such methods presents a real challenge. In the present work, the convergence of rank-one AM algorithms for a class of variational linear elliptic equations is studied. More precisely, we show that rank-one AM-sequences are in general bounded in the ambient Hilbert tensor space and are compact if a uniform non-orthogonality condition between iterates and the reaction term is fulfilled. In particular, if a rank-one AM-sequence is weakly convergent then it converges strongly and the common limit is a solution of the rank-one optimization problem.
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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 Eng. 17(4), 473–486 (2010)
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 modeling of complex fuids. J. Non Newton. Fluid Mech. 139(3), 153–176 (2006)
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)
Cances, E., Ehrlacher, V., Lelièvre, T.: Greedy algorithms for high-dimensional non-symmetric linear problems. ESAIM Proc. 41(12), 95–131 (2013)
Chinesta, F., Cueto, E.: PGD-Based Modeling of Materials. Structuresand Processes. Springer, Berlin (2014)
Chinesta, F., Keunings, R., Leygue, A.: The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer. Springer, Berlin (2014)
Denis de Senneville, B., El Hamidi, A., Moonen, C.: A direct pca-based approach for real-time description of physiological organ deformations. Trans. Med. Imaging 34(4), 974–982 (2015)
Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1(3), 211–218 (1936)
Espig, M., Hackbusch, W., Khachatryan, A.: On the convergence of alternating least squares optimisation in tensor format representations. Preprint-No 423 of Institut für Geometrie und Praktische Mathematik, submitted (2015)
Falcó, A., Hackbusch, W., Nouy, A.: On the dirac-frenkel variational principle on tensor banach spaces. arXiv:1610.09865v1 (submitted) (2015)
Falcó, A., Nouy, A.: A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional eckart-young approach. J. Math. Anal. Appl. 376(15), 469–480 (2011)
Falcó, A., Nouy, A.: Proper generalized decomposition for nonlinear convex problems in tensor banach spaces. Numerische Mathematik 121(3), 503–530 (2012)
Figueroa, L.-E., Suli, E.: Greedy approximation of high-dimensional ornstein-uhlenbeck operators. Found. Comput. Math. 12(5), 573–623 (2012)
Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Berlin (2012)
Ladevèze, P.: Nonlinear computational structural mechanics: New approaches and non-incremental methods of calculation. Springer, Berlin (1999)
Nouy, A.: A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Eng. 199(23–24), 1603–1626 (2010)
Schmidt, E.: Zur theorie der linearen und nichtlinearen integralgleichungen. i. teil: Entwicklung willkürlicher funktionen nach systemen vorgeschriebener. Math. Annalen 63(2), 433–476 (1907)
De Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3), 1084–1127 (2008)
Troltzsch, F., Volkwein, S.: Pod a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44(1), 83–115 (2009)
Uschmajew, A.: Local convergence of the alternating least squares algorithm for canonical tensor approximation. SIAM J. Matrix Anal. Appl. 33(2), 639–652 (2012)
Zhang, T., Golub, G.H.: Rank-one approximation to high order tensors. SIAM J. Matrix Anal. Appl. 23(2), 534–550 (2001)
Acknowledgements
The work of A. El Hamidi was supported in part by the GDRI—CNRS FSI (Fluid Structure Interaction). The authors are very grateful to the anonymous referee for his interesting remarks and suggestions that improve the quality of the manuscript.
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El Hamidi, A., Ossman, H. & Jazar, M. On the convergence of alternating minimization methods in variational PGD. Comput Optim Appl 68, 455–472 (2017). https://doi.org/10.1007/s10589-017-9920-y
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DOI: https://doi.org/10.1007/s10589-017-9920-y