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;
D O I:
10.1137/040604959
中图分类号:
O29 [应用数学];
学科分类号:
070104 ;
摘要:
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.