Dynamical Systems on Tensor Approximations

Ohio University, Athens OH

Investigators

Abstract

Functions of many variables arise in numerous mathematical, statistical, and scientific problems; a particularly notable example is the multiparticle Schrodinger equation in quantum mechanics. The effort required to compute in a straightforward way with such functions grows extremely rapidly as the number of variables increases, and soon becomes prohibitive. Mathematical methods have been developed that in some cases allow one to compute without this rapid growth, but crucial parts of the method are poorly understood and unreliable. This project seeks to understand and then fix these crucial parts. Students will be actively involved in the project and so learn mathematics and how to conduct mathematical research; they will also develop skills in writing, presenting seminars and posters, and software development and usage. A mathematical study will be conducted on the approximation of tensors using sums of separable tensors and the approximation of multivariate functions using sums of separable functions. The objectives are to understand (1) how such approximations behave and (2) how such approximations can be effectively computed. The method is to consider iterative tensor approximation algorithms as dynamical systems to probe the set of sum-of-separable tensors and to understand the behavior of the algorithm within this set. The approximation of tensors by sums of separable tensors enables a promising computational paradigm for bypassing the curse of dimensionality when working with functions of many variables. This project addresses a bottleneck, in understanding and in computation, that prevents the computational paradigm from achieving its full potential.