A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach
By Antonio Falcó, Anthony Nouy. 2011
84 citations recorded by the source corpus.
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.
Subfields: Computational Mathematics, Statistical and Nonlinear Physics, Statistics and Probability
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
- Mathilde Chevreuil
- Grégory Legrain
- Loïc Giraldi
- P. Cartraud
- Anthony Nouy
- André Uschmajew
- Reinhold Schneider
In Statistical and Nonlinear Physics
- Mathilde Chevreuil
- Grégory Legrain
- P. Cartraud
- Loïc Giraldi
- Anthony Nouy
- Sondipon Adhikari
- Antonio Falcó
- George Stefanou
In Statistics and Probability
- Mathilde Chevreuil
- Grégory Legrain
- P. Cartraud
- Loïc Giraldi
- Anthony Nouy
- Sondipon Adhikari
- George Stefanou
- Antonio Falcó
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, 40, 41, 85–87, 180].”
- Tensor-based methods for numerical homogenization from high-resolution images (2013, methodology): “In order to circumvent the above difficulties, a progressive definition of PGD is classically introduced [10, 15], which consists in building the sequence (vi)1≤i≤m term by term (in a greedy fashion).”
- Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, background): “φ∥(2)α (61) can be found in [16] in an abstract setting, for problems de ned in tensor product spaces.”
- Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics (2012, methodology): “PGD methods have been recently introduced for the a priori construction of separated representations of the solution of variational problems defined in tensor product spaces [14, 3, 4, 29, 8, 24].”
- Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces (2011, unknown): “Another family of methods, called Proper Generalized Decomposition (PGD) methods [9,16, 27,33,34], have been introduced for the direct construction of representations of type (1).”