On the Convergence of a Greedy Rank-One Update Algorithm for a Class of Linear Systems

By Amine Ammar, Francisco Chinesta, Antonio Falcó. 2010

98 citations recorded by the source corpus.

In this paper we study the convergence of the well-known Greedy Rank-One Update Algorithm. It is used to construct the rank-one series solution for full-rank linear systems. The existence of the rank one approximations is also not new, but surprisingly the focus there has been more on the applications side more that in the convergence analysis. Our main contribution is to prove the convergence of the algorithm and also we study the required rank one approximation in each step. We also give some numerical examples and describe its relationship with the Finite Element Method for High-Dimensional Partial Differential Equations based on the tensorial product of one-dimensional bases. We illustrate this situation taking as a model problem the multidimensional Poisson equation with homogeneous Dirichlet boundary condition.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Numerical Analysis

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mathematics

1. Elías Cueto
2. Francisco Chinesta
3. Amine Ammar
4. Antonio Huerta
5. Roland Keunings
6. David González

In Computational Theory and Mathematics

1. Francisco Chinesta
2. Amine Ammar
3. Antonio Huerta
4. Adrien Leygue
5. Felipe Bordeu
6. Icíar Alfaro
7. Elías Cueto
8. Roland Keunings

In Numerical Analysis

1. Francisco Chinesta
2. Amine Ammar
3. Antonio Huerta
4. Adrien Leygue
5. Elías Cueto

Citations in the corpus

• A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Convergence results in this direction can be found in [3,39,40,84–86, A number of improvements to the simple scheme outlined above have been proposed to increase its convergence speed, see, e.g., [4, 95, 200, 202].”
• Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces (2011, unknown): “Progressive definitions of PGDs can thus be considered as Greedy algorithms [37] for constructing separated representations [1,6].”
• PGD-Based Computational Vademecum for Efficient Design, Optimization and Control (2013, methodology): “The calculation of these data representations was performed off-line and then used on-line in many branches of engineering sciences for design and optimization.”
• Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models (2010, background): “The reason of this lack of optimality was understood by performing a numerical analysis (see [21] in the present issue).”
• The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer (2013, background): “Other error estimators based on quantities of interest in the study of a particular problem are proposed in [1, 2].”