Collaborative Research: A Tensor-Based Computational Framework for Model Reduction and Structured Matrices

科学计算中的许多任务涉及到本质上多维的数据或运算符：例如，当每个图像以标准二维数组格式存储时，灰度图像数据库构成三维数组。然而，许多标准数值方法将数据和相关运算符视为二维数组或矩阵。这表明可以用于计算增益的额外结构可能未被发现和未充分利用。最近的研究表明，张量（多维数组）和几种相应的分解方法可以用于揭示高维空间中数据和算子的潜在相关性。实际上，张量分解可以证明在表示某些类型的数据时优于基于矩阵的对应。本研究项目解决了两个重要问题：(1)如何使用多维张量分解来揭示数据和运算符中的潜在结构，以及(2)如何使用这些揭示的结构来开发一个强大的计算框架，从而获得这种结构的好处。研究生课程的动手教学材料将开发随机矩阵方法和张量分解。这种教学材料，以Python笔记本的形式，以及作为该项目一部分开发的代码，将在开源许可下作为软件库免费提供。研究人员的目标是在科学计算中具有重要意义和深远影响的两种应用的背景下回答这些问题：模型简化，一种用于降低与复杂物理现象的高保真模拟相关的计算成本的数学框架，以及结构矩阵近似，这在参数偏微分方程和图像去模糊等应用中很重要。这项工作将通过一种全新的、多维的视角来解决这些问题，其优势是提供了新的计算效率和结构，而这些只有通过转移到更高的维度才能获得。该项目的两个标志性特征是：(1)通过使用随机矩阵方法和张量分解的代数结构，设计和分析在计算和内存访问方面效率高的结构化张量分解；(2)利用张量模型和相应的结构化分解，在模型约简和结构化矩阵近似中挖掘潜在的多维结构。这项研究预计也将有利于科学和工程领域的许多其他应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many tasks in scientific computing involve either data or operators that are inherently multidimensional: for example, a database of gray-scale images constitutes a three-dimensional array when each image is stored in a standard two-dimensional array format. Yet many standard numerical methods treat the data and associated operators as two-dimensional arrays, or matrices. This suggests that additional structure that could be leveraged for computational gain may be going undiscovered and underutilized. Recent research has shown that tensors (multidimensional arrays) and several types of corresponding decomposition methods can be instrumental in revealing latent correlations of both data and operators residing in high-dimensional spaces. Indeed, tensor decompositions can be provably superior to matrix-based counterparts in representation of certain types of data. This research project tackles two important questions: (1) how to uncover latent structure in data and operators using multidimensional tensor factorizations, and (2) how to use these revealed structures to develop a powerful computational framework that can harvest the benefits of this structure. Hands-on teaching material for graduate courses will be developed on randomized matrix methods and tensor decompositions. This teaching material, in the form of Python notebooks, along with the code developed as a part of this project, will be freely available as a software library under an open source license.The investigators aim to answer these questions in the context of two applications in scientific computing of major importance and far-reaching consequences: model reduction, a mathematical framework for reducing the computational cost associated with high-fidelity simulations of complicated physical phenomena, and structured matrix approximation, which is important in applications such as parametric partial differential equations and image deblurring. The work will approach these questions through an entirely new, multidimensional lens with the advantages of providing new computational efficiencies and structure that can only be obtained by moving to a higher-dimensional regime. Two signature features of the project are: (1) design and analysis of structured tensor decompositions that are efficient in terms of computations and memory accesses, by using randomized matrix methods and the algebraic structure of tensor decompositions; and (2) use of tensor models, and a corresponding suite of structured decompositions, to exploit latent multidimensional structure in model reduction and structured matrix approximations. The research is expected to benefit numerous other applications in science and engineering as well.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.