A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach

• Published in: Journal of mathematical analysis and applications Vol. 376; no. 2; pp. 469 - 480
• Main Authors: Falcó, A., Nouy, A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.04.2011; Elsevier
• Subjects: Exact sciences and technology; Functional analysis; Mathematical analysis; Mathematics; Numerical Analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Proper Generalized Decomposition; Sciences and techniques of general use; Singular values; Tensor product Hilbert spaces; Proper Generalized Decomposition; Singular values; Tensor product Hilbert spaces; Tensor product; Singular value; Decomposition method; Representation; Partial differential equation; Convergence; Functional; Elliptic equation; Mathematical analysis; Hilbert space; Singular Values; Tensor Product Hilbert Spaces; Separated representation
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2010.12.003

The Proper Generalized Decomposition (PGD) is a methodology initially proposed for the solution of partial differential equations (PDE) defined in tensor product spaces. It consists in constructing a separated representation of the solution of a given PDE. In this paper we consider the mathematical analysis of this framework for a larger class of problems in an abstract setting. In particular, we introduce a generalization of Eckart and Young theorem which allows to prove the convergence of the so-called progressive PGD for a large class of linear problems defined in tensor product Hilbert spaces.