Symbolic-numeric algorithms and applications for systems of differential polynomial equations and inequalities

微分多项式方程组和不等式组的符号数值算法和应用

• 批准号: RGPIN-2016-06458
• 负责人: Reid, Gregory
• 金额: $ 1.09万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675666
• 关键词: Symbolic; numeric; algorithms; applications; systems

Description for a wide audience (required)***This is an interdisciplinary proposal between pure mathematics, applied mathematics and computer science. Advanced methods from algebraic and differential geometry are used to create mathematical theory, algorithms and computer implementations for general systems of nonlinear partial differential equations. The algorithms identify and include missing constraints (integrability conditions) resulting from differentiating such systems. Such constraints restrict initial and boundary data, and are crucial in determination of analytical features and numerical solutions of such systems.***Incomplete systems, those that have missing constraints, arise frequently in geometric classification problems. This drew much attention from classical geometers such as Cartan and more recently by Olver and others. For example they arise in determination of transformations which left objects invariant (symmetries), or transformed one member of a class to another (equivalence transformations). Naturally a system is differentiated (prolonged) to obtain such integrability conditions. Developing criteria for how far a nonlinear system should be prolonged to include all such conditions has been notoriously difficult. Cartan conjectured, but was unable to prove, that his involutivity criteria resulted in such a finite prolongation. Kuranishi proved Cartan's conjecture, albeit under certain restrictions. Modern applications of geometric classification, such as the moving frames approach of Olver and collaborators, have led to complicated incomplete nonlinear systems. This has prompted the development of symbolic algorithms to include missing constraints, including work by the PI.******Incomplete systems commonly arise as higher index differential and partial differential algebraic equations (DAE and PDAE) where the index is the number of differentiations to include the missing constraints. Indeed the complexity of such DAE makes computers essential at every stage, from their formation, to completion, to numerical solution. This has prompted the development of powerful problem solving environments, such as MapleSim and SystemModeler. Indeed the PI's former student Wittkopf, is the main architect of the numerical engine of MapleSim. Such applications have motivated the current proposal, that of developing completion algorithms for approximate real systems, including inequalities. For example, we may need to specify that an unknown density is real and positive or that the position of a robot hand is constrained to move inside a cylinder. The most important part of this proposal is to build on some exciting breakthroughs in Semi-Definite Programming (SDP) and real numerical algebraic geometry to characterize real solutions of such nonlinear systems. Computer programs with user-friendly interfaces will make the results widely available.**

面向广泛受众的描述（必填）* 这是一个纯数学、应用数学和计算机科学之间的跨学科建议。从代数和微分几何的先进方法被用来创建数学理论，算法和计算机实现的一般系统的非线性偏微分方程。 该算法识别并包括因区分此类系统而导致的缺失约束（可积性条件）。 这些约束限制了初始和边界数据，并且在确定此类系统的分析特征和数值解时至关重要。在几何分类问题中，经常会出现不完备系统，即缺少约束的系统。这引起了很多注意，从古典geometers，如嘉当和最近由奥尔弗和其他人。 例如，它们出现在确定的变换，使对象不变（对称性），或转换一个成员的一个类到另一个（等价变换）。 自然地，系统被微分（延拓）以获得这样的可积性条件。发展一个非线性系统应该延伸到什么程度以包括所有这些条件的标准是出了名的困难。 嘉当证明了，但无法证明，他的对合性准则导致了这样一个有限的延长。仓西证明了嘉当的猜想，尽管有一定的限制。 几何分类的现代应用，如Olver及其合作者的移动标架方法，导致了复杂的不完全非线性系统。 这促使符号算法的发展，包括缺失的约束，包括PI的工作。不完全系统通常以高指数微分和偏微分代数方程（DAE和PDAE）的形式出现，其中指数是包含缺失约束的微分的数量。 事实上，这种DAE的复杂性使得计算机在每个阶段都必不可少，从它们的形成，到完成，再到数值解。这促使了强大的问题解决环境的发展，如MapleSim和SystemModeler。 事实上，PI的前学生Wittkopf是MapleSim数值引擎的主要架构师。 这样的应用激发了目前的建议，即开发完成算法的近似真实的系统，包括不等式。 例如，我们可能需要指定一个未知的密度是真实的和正的，或者机器人手的位置被限制在圆柱体内移动。 这个建议的最重要的部分是建立在半定规划（SDP）和真实的数值代数几何的一些令人兴奋的突破，以表征这种非线性系统的真实的解决方案。 具有用户友好界面的计算机程序将广泛提供结果。