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.

我专注于创建和测试计算算法的近似解决连续非线性数学问题。这样的算法提供了基本上是唯一的方法来获得现代理论描述的物理，经济或工程现象，使详细的预测，并允许设计有效的设备和strategy. I有一个广泛的记录，显着的算法工作中的每一个三个主要的重叠研究领域：计算线性和多项式代数，计算动力系统，计算特殊功能。我的大部分工作都是抽象的，因为抽象提供了杠杆，而计算提供了力量。但我与应用保持联系，因为应用经常比内省带来更大的挑战。我的工作的应用包括模拟人类心脏的电行为的非线性模型，血液流动，金融期权的定价，流体中的传热，人眼暗适应的化学动力学，以及许多其他。 几乎每一位工程师、科学家和社会科学家都依赖于这些算法来完成他们的工作。在上个世纪，许多有用的、甚至是关键的算法被开发出来，大量的资源（金钱、时间、能源和碳成本）被投入到执行这些算法来模拟物理、生物或社会现象。 但是我们的算法知识是不完整的：例如，我们甚至不知道将矩阵相乘的最快/最经济的方法。我们对非线性问题的算法的知识甚至更不完整。本提案中所述的研究方案解决了这一领域的关键问题。具体而言，本申请提出了矩阵多项式特征值问题的新算法，求解连续动力系统的新算法，以及计算某些特殊函数（反Gamma函数和Mathieu函数的非规则情况）的新算法。我还提出了一个新的快速“分治”算法计算一类特殊矩阵的特征值。在几乎所有领域的振动研究中都需要近似特征值，而这类特殊的矩阵经常出现，因此需要快速算法来计算它们。 最近，我发现了一类新的离散问题，具有广泛的适用性，我称之为Bohemian问题，对于整数的有界高度矩阵（BOHEMI）。这个新的研究领域与图论，压缩感知，随机矩阵和组合学有关。我们已经创建了一个网页和GitHub展示我们的结果（搜索“Bohemian Matrices”）。我们将研究具有有用矩阵结构的特殊波希米亚族，例如Metzler矩阵和相关矩阵，它们在生物学和神经科学中有应用。