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

面向广大受众的描述(必填) 这是一项介于纯数学、应用数学和计算机科学之间的跨学科建议。代数和微分几何中的先进方法被用来为一般的非线性偏微分方程组创建数学理论、算法和计算机实现。该算法识别并包括因区分这些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是至关重要的。 不完备系统是指缺少约束的系统，在几何分类问题中经常出现。这引起了卡尔坦等经典几何学家的极大关注，最近又引起了奥尔弗等人的注意。例如，它们出现在确定使对象保持不变(对称)的变换中，或将类的一个成员变换为另一个成员(等价变换)。自然地，系统被区分(延长)以获得这样的可积性条件。众所周知，制定一个标准来衡量一个非线性系统应该延长到什么程度，以包括所有这样的条件，一直是出了名的困难。卡尔坦猜想，但无法证明，他的对合标准导致了这样一个有限的延长。仓西证明了卡尔坦的猜想，尽管受到了一定的限制。几何分类的现代应用，如Olver和合作者的移动标架方法，导致了复杂的不完全非线性系统。这促使了符号算法的发展，以包括缺失的约束，包括PI的工作。 不完备系统通常以高指标微分和偏微分代数方程(DAE和PDAE)的形式出现，其中指标是包含缺失约束的微分次数。事实上，这种DAE的复杂性使得计算机在每个阶段都是必不可少的，从形成到完成，再到数值求解。这促使了强大的问题解决环境的开发，如MapleSim和SystemModeler。事实上，PI以前的学生Wittkopf是MapleSim数值引擎的主要设计师。这样的应用激发了当前的提议，即为包括不等式在内的近似实系统开发完备化算法。例如，我们可能需要指定未知密度是实数和正数，或者将机械手的位置约束为在圆柱体内移动。这一建议最重要的部分是建立在半定规划(SDP)和实数值代数几何的一些令人兴奋的突破的基础上来刻画这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应用。

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.