Tensor Network Computation: Representations, Algebra, and Applications

现代数据收集和计算模拟产生了许多高维数据集。这种高维数据集代表了计算数学中的关键挑战之一。众所周知，主要困难在于维数灾难，即以传统方式表示和分析高维函数所需的自由度的数量随着维数呈指数增长。最近，张量网络出现了，主要来自理论和计算物理界，作为一个有前途的工具，用于表示和操纵高维函数和概率。 该项目对张量网络方法进行了系统的计算研究。调查作为第一步，提供了一个总体框架的工作与高维数据集。该项目侧重于张量网络的四个方面。(1)研究人员的目标是改进张量网络并行化的近期工作，其中的主要任务是为非齐次系统和一般张量网络开发有效的张量收缩算法，重点是并行实现。(2)研究人员计划开发新的算法，用于通过采样或从现有但冗余的张量表示构建张量网络。该方法将从随机数值代数、非线性优化和多尺度方法中汲取工具和思想。(3)该项目将开发基本代数运算的算法，用于操作张量网络表示中的高维函数。这些基本运算包括加法、减法、逐项乘法、逐项求逆、逐项函数应用以及张量网络算子形式的线性算子的应用。(4)最后一个目标是研究张量网络在传统统计和量子力学领域之外的新应用，例如在数值均匀化，非常高维的不确定性量化，和高-该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的评估来支持的。影响审查标准。

Modern data collection and computational simulations produce many high-dimensional data sets. Such high-dimensional data sets represent one of the key challenges in computational mathematics. As is well-known, the main difficulty lies in the curse of the dimensionality, that is, the number of degrees of freedom required to represent and analyze high-dimensional functions in the traditional way grows exponentially with the number of dimensions. Recently, tensor networks have emerged, mostly from the theoretical and computational physics community, as a promising tool for representing and manipulating high-dimensional functions and probabilities. This project undertakes a systematic computational study of the tensor network approach. The investigation serves as an initial step in providing a general framework for work with high-dimensional data sets. The project focuses on four aspects of tensor networks. (1) The investigator aims to improve on recent work in tensor network skeletonization, where the main task is to develop efficient tensor contraction algorithms for inhomogeneous systems and general tensor networks, with an emphasis on parallel implementation. (2) The investigator plans to develop novel algorithms for constructing a tensor network either via sampling or from an existing but redundant tensor representation. The approach will draw tools and ideas from randomized numerical algebra, nonlinear optimization, and multiscale methods. (3) The project will develop algorithms for basic algebraic operations for manipulating high-dimensional functions in the tensor network representation. These basic operations include addition, subtraction, entry-wise multiplication, entry-wise inversion, entry-wise function application, and application of linear operators in tensor network operator form. (4) The last objective is to investigate new applications of tensor networks outside the traditional realm of statistical and quantum mechanics, for example in numerical homogenization, uncertainty quantification in very high dimension, and high-dimensional partial differential equations in control and molecular dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.