Efficient symbolic-numeric algorithms for structured constrained differential polynamial systems

符号计算是研究处理数学表达式的算法和软件系统的学科。直到最近，这些算法还只处理精确的表达式和方程。在数据是近似的应用程序的需求的驱动下，该学科已经发展到包括多项式方程的符号-数字计算。这是一项广泛努力的一部分，目的是将精确符号计算的普遍性与数值计算的现实性和效率结合起来。 在我以前结果的基础上，我的建议将这个主题进一步扩展到用于近似微分多项式系统的有效的符号-数值算法。这样的系统表达了科学的基本定律，应用程序已经产生了越来越复杂的系统，例如，涉及医疗机器人复杂电路中的电流和部件运动的100‘S微分方程组。为了分析和求解如此复杂的模型，从公式到求解的每个阶段都需要计算机。 对这类系统的区分可以揭示隐藏的约束，这些约束在它们的几何和解的属性中是至关重要的。提议为这类系统创建和分析新的算法，使其能够：以稳定和有效的符号-数字算法确定其约束；以及在分析和求解过程中利用所产生的几何图形。还将开发算法，以利用应用程序中通常出现的系统结构。它建立在微分几何和微分代数的强大结果的基础上，结合新的区域数值代数几何，首次给出了一种数值稳定的方法来求解近似多项式方程的更简单的子类。现有的符号技术可以检测全局约束结构，但仅限于精确系统；而用于更真实的近似系统的现有数值技术可以检测局部结构，但不能检测全局结构。研究的重点是针对包括全局结构在内的结构化系统的符号-数值算法。

Symbolic Computation is the study of algorithms and software systems to manipulate mathematical expressions. Until recently the algorithms have processed exact expressions and equations only. Driven by the needs of applications, in which data is approximate, the subject has grown to include symbolic-numeric computation for polynomial equations. This is part of a broad effort to combine the generality of exact symbolic computation with the realism and efficiency of numeric computation. Building on my previous results my proposal extends this subject further to efficient symbolic-numeric algorithms for approximate differential polynomial systems. Such systems express fundamental laws of science and applications have yielded ever more complicated systems involving, for example 100's of differential equations for the currents in complex circuits and the motions of components of medical robots. To analyze and solve such complicated models, computers are required at every stage: from formulation to solution. Differentiation of such systems can reveal hidden constraints which are crucial in their geometric and solution properties. The proposal is to create and analyze new algorithms for such systems which enable: the stable and efficient symbolic-numeric algorithmic determination of their constraints; and the computer exploitation of the resulting geometry in the analysis and solution process. Algorithms will also be developed for exploiting the structure of systems typically arising in applications. This builds on powerful results in differential geometry and differential algebra, combined with the new area Numerical Algebraic Geometry, that gives a numerically stable approach for the first time to the simpler subclass of approximate polynomial equations. Existing symbolic techniques can detect global constraint structure but are limited to exact systems; while existing numeric techniques for more realistic approximate systems detect local, but not global structure. The research focuses on symbolic-numeric algorithms for structured systems, including global structure.