Abstract
A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best rank-one approximation. Motivated by this fact, we are going to discuss some of the geometrical aspects of the best rank-one approximation procedure. More precisely, our main result is to construct explicitly a vector field over a low-dimensional vector space and to prove that we can identify its stationary points with the critical points of the best rank-one optimization problem. To obtain this result, we endow the set of tensors with fixed rank-one with an explicit geometric structure.
Highlights
ESI International Chair@CEU-UCH, Departamento de Matemáticas, Física y Ciencias Tecnológicas, Universidad Cardenal Herrera-CEU, CEU Universities San Bartolomé 55, 46115 Alfara del Patriarca, Spain; Departamento de Ingeniería Mecánica y Construcción, Universitat Jaume I, Avd
Despite the improvements in techniques used in high dimensional problems, some challenging questions remain unresolved due to the efficiency of our computers
The first one is the possibility of managing high dimensional problems, and the second is the possibility to include the model’s parameters as extra-coordinates. This last fact gives a powerful strategy to deal with classical problems because the Proper Generalized Decomposition (PGD) framework facilitates an efficient design and a real-time decision-making [5,6]
Summary
We introduce some notation used along this paper. We denote by R N × M the set of N × M-matrices and by A T the transpose of a given matrix A. The goal is to use (5) to approximate the solution of (4) To this end, for each n ∈ N, we define the set. S1 is a closed set in any norm-topology This fact implies (see Lemma 1 in [8]), that given. Given A ∈ GL( N1 N2 · · · Nd ) and f ∈ R N1 ··· Nd , we can construct for each n, by using (8) and (9), a vector n un =. The above theorem allows for us to construct a procedure, which we give in the pseudo-code form in Algorithm 1, under the assumption that we have a numerical method in order to find a y solving (7) (see the step 5 in Algorithm 1) and that we introduce below.
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