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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=663151
• 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）***这是纯数学、应用数学和计算机科学之间的跨学科提案。代数和微分几何的先进方法用于创建非线性偏微分方程一般系统的数学理论、算法和计算机实现。 该算法识别并包括因区分此类系统而导致的缺失约束（可积条件）。 这些约束限制了初始数据和边界数据，对于确定此类系统的分析特征和数值解至关重要。***不完整的系统（缺少约束的系统）在几何分类问题中经常出现。这引起了嘉当等经典几何学家以及最近奥尔弗等人的广泛关注。 例如，它们出现在确定使对象保持不变（对称）的变换中，或者将类的一个成员变换为另一个成员（等价变换）。 自然地，系统被微分（延长）以获得这样的可积条件。制定非线性系统应延长多远以包含所有这些条件的标准是出了名的困难。 嘉当推测，但无法证明，他的对合性准则导致了如此有限的延长。仓西证明了嘉当猜想，尽管有一定的限制。 几何分类的现代应用，例如 Olver 及其合作者的移动框架方法，导致了复杂的不完整非线性系统。 这促使符号算法的发展包括缺失的约束，包括 PI 的工作。*****不完整系统通常出现为更高指数的微分和偏微分代数方程（DAE 和 PDAE），其中索引是包含缺失的约束的微分数。 事实上，这种 DAE 的复杂性使得计算机在从形成、完成到数值求解的每个阶段都至关重要。这促使了强大的问题解决环境的开发，例如 MapleSim 和 SystemModeler。 事实上，PI 的前学生 Wittkopf 是 MapleSim 数值引擎的主要架构师。 此类应用激发了当前的提议，即为近似真实系统（包括不等式）开发完成算法。 例如，我们可能需要指定未知密度是真实且正的，或者机器人手的位置被限制在圆柱体内移动。 该提案最重要的部分是建立在半定规划（SDP）和实数值代数几何方面的一些令人兴奋的突破的基础上，以表征此类非线性系统的实解。 具有用户友好界面的计算机程序将使结果广泛可用。**