RUI: Commutativity in Numerical Computation

RUI：数值计算中的交换律

• 批准号: 2012216
• 负责人: Brian Sutton
• 金额: $ 6.46万
• 依托单位: Randolph-Macon College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012216&HistoricalAwards=false
• 关键词: RUI; Commutativity; Numerical; Computation

This project concerns consistency in scientific inferences. Multiple observations of the same phenomenon rarely produce identical results. Statistical noise may be the first complication that comes to mind, but there are plenty of other possible discrepancies as well: different antenna arrangements; different camera angles; different modalities, e.g., images and text; and more. Inferring the true underlying phenomenon requires sophisticated mathematics. This project focuses on matrix methods. Hiding within the simple numerical entries of a matrix are intricate patterns called eigenvectors that are connected to the harmonics of musical instruments. Even though different kinds of measurements may produce different numerical values, they may share the same eigenvectors when observing the same objects. One of the goals of the project is to pull consistency out of inconsistent data. In theory, measurements on the same object from different instruments should agree in certain ways, but because of statistical noise and other sources of error and uncertainty, each instrument provides an inexact view of the world, and the two instruments will not agree perfectly with each other. The goal is to bring the measurements into alignment using mathematics and therefore likely into better agreement with the true state of the universe. The project will be completed at a Primarily Undergraduate Institution, and introducing undergraduate students to research in STEM fields is a primary objective.The mathematical focus of the project is simultaneous diagonalization of commuting Hermitian matrices. In theory, commuting Hermitian matrices share a common eigenvector matrix. However, numerically-computed eigenvectors may differ wildly in the presence of ill conditioning. A numerical method for simultaneous diagonalization seeks, either explicitly or implicitly, small perturbations of the input matrices that commute exactly. The project objectives include contributions to theory, method, and application. First, a reconsideration of "near commutativity" is planned, in which eigenvalues are fixed in place and attention shifts from the space of Hermitian matrices to the unitary group of eigenvector matrices. Second, the project will pursue new numerical methods for simultaneous diagonalization, motivated in part by relatively recent spectral divide-and-conquer methods. Third, the new approaches will be applied to applications such as independent component analysis and unsupervised machine learning with multimodal data.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.

这个项目涉及科学推论的一致性。对同一现象的多次观察很少会产生相同的结果。统计噪音可能是第一个想到的复杂因素，但也存在大量其他可能的差异：不同的天线排列;不同的摄像机角度;不同的模式，例如，图像和文本;以及更多。推断真正的潜在现象需要复杂的数学。这个项目的重点是矩阵方法。隐藏在矩阵的简单数字条目中的是称为本征向量的复杂模式，这些模式与乐器的谐波有关。即使不同种类的测量可能产生不同的数值，当观察相同的对象时，它们可能共享相同的特征向量。该项目的目标之一是从不一致的数据中提取一致性。从理论上讲，不同仪器对同一物体的测量在某些方面应该是一致的，但由于统计噪声和其他误差和不确定性来源，每台仪器都提供了一个不精确的世界观，两台仪器也不会完全一致。目标是使用数学方法使测量结果对齐，从而可能与宇宙的真实状态更好地吻合。该项目将在一所私立本科院校完成，主要目标是让本科生参与STEM领域的研究。该项目的数学重点是交换厄米特矩阵的同时对角化。在理论上，交换厄米特矩阵共享一个共同的特征向量矩阵。然而，数值计算的特征向量可能会有很大的不同，在病态的存在。同时对角化的数值方法寻求，显式或隐式，输入矩阵的小扰动，正好交换。该项目的目标包括理论，方法和应用的贡献。首先，重新考虑“近交换性”的计划，其中特征值是固定的，注意力从厄米特矩阵的空间转移到酉群的特征向量矩阵。第二，该项目将寻求新的数值方法，同时对角化，部分动机是相对较新的频谱分治方法。第三，新方法将应用于独立成分分析和多模态数据的无监督机器学习等应用。该奖项反映了NSF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。