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

Antonio Falcó, Lucía Hilario, Nicolás Montés, Marta C. Mora and Enrique Nadal
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Antonio Falcó: 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
Lucía Hilario: 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
Nicolás Montés: 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
Marta C. Mora: Departamento de Ingeniería Mecánica y Construcción, Universitat Jaume I, Avd. Vicent Sos Baynat s/n, 12071 Castellón, Spain
Enrique Nadal: Departamento de Ingeniería Mecánica y de Materiales, Universitat Politècnica de València Camino de Vera, s/n, 46022 Valencia, Spain

Mathematics, 2020, vol. 9, issue 1, 1-14

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.

Keywords: proper generalised decomposition; alternating least squares; greedy rank one update algorithm; tensor numerical methods (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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