Tensor-product approximation to operators and functions in high dimensions
Journal of Complexity
International Journal of Computer Mathematics - Fast Iterative and Preconditioning Methods for Linear and Non-Linear Systems
A least-squares approximation of partial differential equations with high-dimensional random inputs
Journal of Computational Physics
Tensor decomposition in electronic structure calculations on 3D Cartesian grids
Journal of Computational Physics
Digital predistortion for power amplifiers using separable functions
IEEE Transactions on Signal Processing
Classification with sums of separable functions
ECML PKDD'10 Proceedings of the 2010 European conference on Machine learning and knowledge discovery in databases: Part I
Mathematics and Computers in Simulation
Stochastic algorithms in linear algebra: beyond the Markov chains and von Neumann-Ulam scheme
NMA'10 Proceedings of the 7th international conference on Numerical methods and applications
Krylov Subspace Methods for Linear Systems with Tensor Product Structure
SIAM Journal on Matrix Analysis and Applications
Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition
SIAM Journal on Matrix Analysis and Applications
SIAM Review
Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs
SIAM Journal on Scientific Computing
Numerical Solution of the Hartree-Fock Equation in Multilevel Tensor-Structured Format
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
A tensor decomposition approach to data compression and approximation of ND systems
Multidimensional Systems and Signal Processing
The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format
SIAM Journal on Scientific Computing
Learning to Predict Physical Properties using Sums of Separable Functions
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
Journal of Computational Physics
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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 Schrödinger equation in quantum mechanics.Numerical examples are given.