# 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:

- 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}

}