Algorithms for Numerical Analysis in High Dimensions

Nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension. The separated representation, introduced previously, allows many operations to be performed with scaling that is formally linear in the dimension. In this paper we f...

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Bibliographic Details
Published in:SIAM journal on scientific computing Vol. 26; no. 6; pp. 2133 - 2159
Main Authors: Beylkin, Gregory, Mohlenkamp, Martin J.
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
Subjects:
Accuracy
Algorithms
Applied mathematics
Approximations and expansions
Engineering Sciences
Euclidean space
Exact sciences and technology
Linear algebra
Mathematical analysis
Mathematics
Mechanics
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Quantum physics
Sciences and techniques of general use
Structural mechanics
separation-rank reduction
separated representation
Direct method
Separation method
65Z05
Least squares method
41A63
Algorithm complexity
Multidimensional operator
41A29
Dimensionality
multiparticle Schrödinger equation
curse of dimensionality
Numerical linear algebra
Antisymmetric function
separated solutions of linear systems
65Y20
antisymmetric functions 65D15
Schrödinger equation
81-08
alternating least squares
Numerical analysis
Linear system
Scientific computation
Multidimensional space
Matrix inversion
Quantum mechanics
Multidimensional function
Condition number
algorithms in high dimensions
separation of variables
Algorithm analysis
multidimensional function
antisymmetric functions
multidimensional operator
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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Summary:Nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension. The separated representation, introduced previously, allows many operations to be performed with scaling that is formally linear in the dimension. In this paper we further develop this representation by (i) discussing the variety of mechanisms that allow it to be surprisingly efficient; (ii) addressing the issue of conditioning; (iii) presenting algorithms for solving linear systems within this framework; and (iv) demonstrating methods for dealing with antisymmetric functions, as arise in the multiparticle Schrodinger equation in quantum mechanics. Numerical examples are given.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:1064-8275
1095-7197
DOI:10.1137/040604959