Dynamical Systems on Tensor Approximations

多变量函数出现在许多数学、统计和科学问题中；一个特别显著的例子是量子力学中的多粒子薛定谔方程。随着变量数量的增加，以一种直接的方式对这些函数进行计算所需的工作量会迅速增加，并且很快就会变得令人望而却步。数学方法已经发展起来，在某些情况下，人们可以在没有这种快速增长的情况下进行计算，但这种方法的关键部分人们理解得很差，而且不可靠。本项目旨在理解并修复这些关键部分。学生将积极参与项目，从而学习数学和如何进行数学研究；他们还将培养写作技能，展示研讨会和海报，以及软件开发和使用。本文将对用可分离张量和逼近张量和用可分离函数和逼近多元函数进行数学研究。目标是理解(1)这种近似是如何表现的，(2)这种近似是如何有效计算的。该方法是将迭代张量逼近算法作为动态系统来探索可分离张量和集合，并了解算法在该集合中的行为。通过可分离张量的和逼近张量，在处理多变量函数时，为绕过维数诅咒提供了一种有前途的计算范式。这个项目解决了一个瓶颈，在理解和计算，阻止计算范式实现其全部潜力。

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.