Keywords: proper generalized decomposition, model reduction, separated representations, tensor product approximation, Galerkin proper generalized decomposition, minimal residual proper generalized decomposition, minimax proper generalized decomposition.

A family of a priori model reduction techniques, called proper generalized decomposition (PGD) methods, are receiving a growing interest in scientific computing [1,2,3]. These methods rely on the a priori construction of separated variables representation of the solution of models defined in tensor product spaces. They can be interpreted as generalizations of proper orthogonal decomposition (or singular value decomposition or Karhunen-Loève decompositon, depending on the context) for the a priori construction of separated representations. In this paper, we recall and illustrate the behaviour of several definitions of PGD, based on Galerkin or minimal residual formulations. Basic PGDs are based on a progressive construction of separated representations [4]. On one hand, the progressive PGD based on a minimal residual formulation is robust in the sense that the decomposition monotically converges with respect to the residual norm. However, the convergence with respect to useful norms can be very poor. On another hand, the progressive PGD based on a Galerkin formulation is not robust since monotonic convergence is not guaranteed for general non symmetric problems. Moreover, it may diverge in some situations. However, Galerkin formulations should be preferred since they require less computational effort and since they classically lead to better convergence properties with respect to useful metrics, when it converges! A possible improvement of these progressive decompositions consists in introducing some updating steps in order to capture an approximation of the optimal decomposition [3], obtained by defining the whole set of functions simultaneously (and not progressively). For many applications, it allows good convergence properties of separated representations to be recovered. However, for some large scale applications, these updating steps may be computationally prohibitive. We then introduce a new definition of PGD which can be interpreted as a Petrov-Galerkin model reduction technique, where test and trial reduced basis functions are related by an adjoint problem. This new definition can significantly improve convergence properties of separated representations with respect to a chosen metric. For the construction of this decomposition, a basic algorithm is proposed, which has almost the same numerical complexity as the classical algorithm used for the construction of the Galerkin PGD. Numerical examples illustrate and compare the different definitions of PGDs for different applications: separation of spatial coordinates for partial differential equation and separation of physical variables and parameters in stochastic partial differential equations.

1

P. Ladevèze, J.C. Passieux, D. Néron, "The LATIN multiscale computational method and the Proper Generalized Decomposition", Comp. Meth. Appl. Mech. Eng., 199(21-22), 1287-1296, 2010. doi:10.1016/j.cma.2009.06.023

2

F. Chinesta, A. Ammar, E. Cueto, "Recent advances in the use of the Proper Generalized Decomposition for solving multidimensional models", Arch. Comp. Meth. Eng., In press, 2010. doi:10.1007/s11831-010-9049-y

3

A. Nouy, "Proper Generalized Decompositions and separated representations for the numerical solution of high dimensional stochastic problems", Arch. Comp. Meth. Eng., In press, 2010. doi:10.1007/s11831-010-9054-1

4

A. Nouy, "A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations", Comp. Meth. Appl. Mech. Eng., 196(45-48), 4521-4537, 2007. doi:10.1016/j.cma.2007.05.016

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