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Algorithms for numerical analysis in high dimensions
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DOI：10.1137/040604959.png)
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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.
Keyword:
curse of dimensionality
multidimensional function
multidimensional operator
algorithms in high dimensions
separation of variables
separated representation
alternating least squares
separation-rank reduction
separated solutions of linear systems
multiparticle Schrodinger equation
antisymmetric functions
期刊
IF:
2.6
论文数:
5.1K
被引数:
1.8W
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