Keywords: proper generalized decomposition, tensor product Hilbert space, best approximation.

The proper generalized decomposition (PGD) is a technique that reduces calculation and storage cost drastically and presents some similarities with the proper orthogonal decomposition (POD). It was initially introduced for the analysis and reduction of statistical and experimental data, the a posteriori decomposition techniques, also known as the Karhunen-Loeve Expansion, singular value decomposition or principal component analysis, are now used in the context of model reduction. They are also related to the so-called n-best term approximation problem.

The PGD method was first introduced under the name of "radial-type approximation" for the solution of time dependent partial differential equations (PDE), by separating space and time variables, and used in the context of the LATIN method in computational solid mechanics. It has been also introduced for the separation of coordinates in multidimensional PDEs, with many applications in kinetic theory of complex fluids, financial mathematics, computational chemistry, etc. It has also been introduced in the context of stochastic or parametrized PDEs by introducing a separation of physical variables (such as space and time) and (random) parameters. Still in the context of stochastic PDEs, a further separation of parameters has also been introduced, by exploiting the tensor product structure of stochastic function spaces.

The main goal of this paper is to use a separated representation of the solution of a class of elliptic problems. It allows a tensor product approximation basis to be defined as well as the numerical integration of a high dimensional model to be decoupled in each dimension. The milestone of this methodology is the use of shape functions given by a tensorial based construction. This fact has advantages as the manipulation of only one dimensional polynomials and their derivatives, that provide a better computational performance and simplified implementation and use one-dimensional integration rules. Moreover, it makes the solution of models defined in spaces of more than hundred dimensions possible in some specific applications. This problem is closely related to the decomposition of a tensor as a sum of rank-one tensors, that it can be considered as a higher order extension of the matrix singular value decomposition.

In this paper we study and analyze the different mathematical and computational problems appearing in the optimization procedures related to the PGD and its relative n-best term approximation problem. Some numerical examples will be given.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £125 +P&P)