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Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)

Authors :
Antonio Falcó
Lucía Hilario
Nicolás Montés
Marta C. Mora
Enrique Nadal
Source :
Mathematics, Vol 9, Iss 1, p 34 (2020)
Publication Year :
2020
Publisher :
MDPI AG, 2020.

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.

Details

Language :
English
ISSN :
22277390
Volume :
9
Issue :
1
Database :
Directory of Open Access Journals
Journal :
Mathematics
Publication Type :
Academic Journal
Accession number :
edsdoj.0d82797e7b3f4ce5a3b064b6cd3c66be
Document Type :
article
Full Text :
https://doi.org/10.3390/math9010034