Numerical solution of high-dimensional problems in quantum mechanics via 'standard' mathematical tools inevitably leads to the curse of dimensionality  . To circumvent the curse of dimensionality, all involved objects (functions, solutions, operators) have to be represented in some low-parametric formats. One of the few universal tools to represent multivariate functions is the classical idea of separation of variables , but surprisingly, very little was known in mathematics how to do separation of variables numerically: one has to use suitable factorization of multiway arrays (tensors). The most common approach is the canonical format, used by Beylkin and Mohlenkamp, but it can not be computed in a robust way. This was the motivation for the development of novel tensor formats.In this talk I will present a brief review on recent results on the numerical tools for separation of variables based on the so-called Tensor Train (TT)-format (now it is clear that a similar representation Matrix Product States (MPS) was known for a long time in other research areas). The TT-format comes with all basic arithmetic and approximation algorithms, and can be used to solve complex high-dimensional problems (eigenvalue and dynamical problems, multivariate interpolation). I will present numerical results in quantum molecular dynamics and for the computation of vibrational spectra in high dimensions.

Solving high-dimensional problems via stable and efficient tensor factorization techniques

246th National Meeting (2013)

The official archive and repository for all ACS Meetings & Expositions abstracts and select posters or presentations.