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

科学计算中的许多任务要么涉及数据，要么涉及本质上是多维的运算符：例如，当每幅图像以标准的二维数组格式存储时，灰度图像的数据库构成三维数组。然而，许多标准的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，可以用于计算收益的额外结构可能没有被发现，也没有得到充分利用。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来揭示高维空间中数据和算子之间的潜在相关性。事实上，在表示某些类型的数据方面，张量分解可以被证明优于基于矩阵的对应物。这个研究项目解决了两个重要的问题：(1)如何利用多维张量分解来发现数据和算子中的潜在结构，以及(2)如何利用这些揭示的结构来开发一个强大的计算框架，以获取这种结构的好处。研究生课程的实践教材将以随机矩阵方法和张量分解为基础。这本教材以Python笔记本的形式，以及作为该项目一部分开发的代码，将在开放源码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两个具有重要意义和深远影响的应用程序的背景下回答这些问题：模型简化，一个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，以及结构化矩阵近似，它在参数偏微分方程和图像去模糊等应用程序中非常重要。这项工作将通过一种全新的多维透镜来解决这些问题，其优点是提供了新的计算效率和结构，只有通过转移到更高维的区域才能获得。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存储器访问方面高效的结构化张量分解；(2)使用张量模型和相应的一套结构化分解，在模型降阶和结构化矩阵近似中利用潜在的多维结构。这项研究预计也将惠及科学和工程领域的许多其他应用。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.