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Title: Geometry of Local Adaptive Galerkin Bases

Abstract

The local adaptive Galerkin bases for large-dimensional dynamical systems, whose long-time behavior is confined to a finite-dimensional manifold, are optimal bases chosen by a local version of a singular decomposition analysis. These bases are picked out by choosing directions of maximum bending of the manifold restricted to a ball of radius {epsilon} . We show their geometrical meaning by analyzing the eigenvalues of a certain self-adjoint operator. The eigenvalues scale according to the information they carry, the ones that scale as {epsilon}{sup 2} have a common factor that depends only on the dimension of the manifold, the ones that scale as {epsilon}{sup 4} give the different curvatures of the manifold, the ones that scale as {epsilon}{sup 6} give the third invariants, as the torsion for curves, and so on. In this way we obtain a decomposition of phase space into orthogonal spaces E {sub m} , where E{sub m} is spanned by the eigenvectors whose corresponding eigenvalues scale as {epsilon}{sup m} . This decomposition is analogous to the Frenet frames for curves. We also discover a practical way to compute the dimension and local structure of the invariant manifold.

Authors:
 [1]
  1. CIMAT, Apto. 402, Postal Guanajuato Gto. 36000 (Mexico)
Publication Date:
OSTI Identifier:
21064273
Resource Type:
Journal Article
Journal Name:
Applied Mathematics and Optimization
Additional Journal Information:
Journal Volume: 41; Journal Issue: 3; Other Information: DOI: 10.1007/s0024599110160; Copyright (c) 2000 Springer-Verlag New York Inc.; Article Copyright (c) Inc. 2000 Springer-Verlag New York; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EIGENVALUES; EIGENVECTORS; GALERKIN-PETROV METHOD; GEOMETRY; MANY-DIMENSIONAL CALCULATIONS; MATHEMATICAL MANIFOLDS; PHASE SPACE

Citation Formats

Solis, F. J. Geometry of Local Adaptive Galerkin Bases. United States: N. p., 2000. Web. doi:10.1007/S0024599110160.
Solis, F. J. Geometry of Local Adaptive Galerkin Bases. United States. doi:10.1007/S0024599110160.
Solis, F. J. Mon . "Geometry of Local Adaptive Galerkin Bases". United States. doi:10.1007/S0024599110160.
@article{osti_21064273,
title = {Geometry of Local Adaptive Galerkin Bases},
author = {Solis, F. J.},
abstractNote = {The local adaptive Galerkin bases for large-dimensional dynamical systems, whose long-time behavior is confined to a finite-dimensional manifold, are optimal bases chosen by a local version of a singular decomposition analysis. These bases are picked out by choosing directions of maximum bending of the manifold restricted to a ball of radius {epsilon} . We show their geometrical meaning by analyzing the eigenvalues of a certain self-adjoint operator. The eigenvalues scale according to the information they carry, the ones that scale as {epsilon}{sup 2} have a common factor that depends only on the dimension of the manifold, the ones that scale as {epsilon}{sup 4} give the different curvatures of the manifold, the ones that scale as {epsilon}{sup 6} give the third invariants, as the torsion for curves, and so on. In this way we obtain a decomposition of phase space into orthogonal spaces E {sub m} , where E{sub m} is spanned by the eigenvectors whose corresponding eigenvalues scale as {epsilon}{sup m} . This decomposition is analogous to the Frenet frames for curves. We also discover a practical way to compute the dimension and local structure of the invariant manifold.},
doi = {10.1007/S0024599110160},
journal = {Applied Mathematics and Optimization},
issn = {0095-4616},
number = 3,
volume = 41,
place = {United States},
year = {2000},
month = {5}
}