Dynamical Systems on Tensor Approximations

张量近似的动力系统

• 批准号: 1418787
• 负责人: Martin Mohlenkamp
• 金额: $ 21万
• 依托单位: Ohio University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418787&HistoricalAwards=false
• 关键词: Dynamical; Systems; Tensor; Approximations

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.

在许多数学、统计和科学问题中出现了多变量函数;一个特别值得注意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，以简单的方式使用这些函数进行计算所需的工作量增长非常迅速，很快就会变得令人望而却步。数学方法已经被开发出来，在某些情况下允许人们在没有这种快速增长的情况下进行计算，但该方法的关键部分却知之甚少，也不可靠。这个项目旨在了解并修复这些关键部分。学生将积极参与该项目，学习数学和如何进行数学研究;他们还将发展写作技能，举办研讨会和海报，以及软件开发和使用。 数学研究将进行张量的近似使用可分离张量的总和和多元函数的近似使用可分离函数的总和。 目标是理解（1）这种近似的行为和（2）如何有效地计算这种近似。该方法将迭代张量近似算法视为动力系统，以探测可分张量之和的集合，并了解该集合内算法的行为。 通过可分离张量的和来近似张量使得在处理多变量函数时绕过维数灾难的有希望的计算范例成为可能。该项目解决了理解和计算方面的瓶颈，这阻碍了计算范式充分发挥其潜力。