Algorithms for Numerical Analysis in High Dimensions
Résumé
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 Schrödinger equation in quantum mechanics. Numerical examples are given.
Mots clés
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 Schrödinger equation
antisymmetric functions
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