Hybrid Symbolic-Numeric Algorithms for Complex Nonlinear Systems

复杂非线性系统的混合符号数值算法

• 批准号: RGPIN-2020-06438
• 负责人: Corless, Robert
• 金额: $ 2.11万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=743670
• 关键词: Hybrid; Symbolic; Numeric; Algorithms; Complex

I specialize in the creation and testing of computational algorithms for the approximate solution of continuous nonlinear mathematical problems. Such algorithms provide essentially the only way to get modern theoretical descriptions of physical, economic, or engineering phenomena to make detailed predictions and to allow the design of effective devices and strategies.  I have an extensive record of significant algorithmic work in each of three major overlapping research areas: computational linear and polynomial algebra, computational dynamical systems, and computational special functions. Much of my work is abstract, because abstraction gives leverage, while computation gives power.  But I maintain contact with applications, because applications frequently give greater challenges than introspection does. The applications of my work include simulation of nonlinear models of the electrical behaviour of the human heart, blood flow, pricing of financial options, heat transfer in fluids, the chemical kinetics of dark adaptation in the human eye, and many others.       Nearly every engineer, scientist, and social scientist relies on such algorithms, implemented in software for them to use, in order to do their work.  Many useful, indeed critical, algorithms have been developed in the last century, and enormous resources (money, time, energy and carbon cost) are devoted to executing these algorithms in simulations of physical or biological or social phenomena.    But our algorithmic knowledge is incomplete: e.g. we do not even know the fastest/most economical way to multiply matrices together. Our knowledge of algorithms for nonlinear problems is even less complete. The research program described in this proposal addresses critical issues in this area. Specifically, this application proposes work on new algorithms for matrix polynomial eigenvalue problems, new algorithms for solving continuous dynamical systems, and new algorithms for computing certain special functions (the inverse Gamma function, and the irregular case of the Mathieu functions). I also propose a new fast "divide-and-conquer" algorithm for computing eigenvalues of a special class of matrices. Approximate eigenvalues are needed in studies of vibration in nearly all fields, and this special class of matrices occurs often enough that fast algorithms for their computation is desirable.       Recently, I have identified a new class of discrete problems, that has wide applicability, which I have called Bohemian problems, for BOunded Height Matrices of Integers (BOHEMI).  This new field of study has connections to graph theory, to compressed sensing, to random matrices, and combinatorics.  We have created a web page and a GitHub showcasing our results (search for "Bohemian Matrices"). We will study special Bohemian families with useful matrix structures, such as Metzler matrices and correlation matrices, which have applications in biology and in neuroscience, for example.

我专门研究计算算法的创建和测试，用于连续非线性数学问题的近似解决方案。这种算法实质上提供了获取物理，经济或工程现象的现代理论描述以做出详细预测并允许设计有效设备和策略的唯一方法。我在三个主要重叠的研究领域中的每个领域中的每一个：计算线性和多项式代数，计算动态系统和计算特殊功能中都有大量算法工作的广泛记录。我的大部分工作都是抽象的，因为抽象会产生杠杆作用，而计算赋予了力量。但是我与应用程序保持联系，因为应用程序经常带来比内省更大的挑战。我的工作的应用包括模拟人心的电气行为的非线性模型，血液流动，财务选择的定价，烟道中的传热，人眼中黑暗适应的化学动力学以及其他许多。几乎每个工程师，科学家和社会科学家都依赖于此类算法，该算法在软件中实施了供他们使用，以便完成他们的工作。上个世纪已经开发了许多有用的算法，并且巨大的资源（金钱，时间，能源和碳成本）专门用于在模拟物理或生物学或社会现象的模拟中执行这些算法。但是我们的算法知识是不完整的：例如我们甚至都不知道将物品繁殖的最快/最经济方式。我们对非线性问题算法的了解甚至不那么完整。该提案中描述的研究计划解决了该领域的关键问题。具体而言，该申请提案在用于矩阵多项式特征问题问题，用于求解连续动态系统的新算法以及用于计算某些特殊功能的新算法的新算法上工作。我还提出了一种新的快速“分裂和争议”算法，用于计算特殊类别的特征值。在几乎所有领域的振动研究中都需要近似特征值，并且这种特殊类别的物质经常足够多，以至于需要快速算法来计算其计算。最近，我确定了一个新的离散问题，这些问题具有广泛的适用性，我称其为波西米亚问题，用于整数的有限高度矩阵（Bohemi）。这个新的研究领域与图形论，压缩传感器，随机矩阵和组合学具有联系。我们创建了一个网页和一个展示结果的GitHub（搜索“ Bohemian矩阵”）。例如，我们将研究具有有用矩阵结构的特殊波西米亚家庭，例如Metzler矩阵和相关矩阵，这些矩阵在生物学和神经科学中都有应用。