CAREER: Solving Over-Constrained Systems of Non-Linear Equations by Symbolic-Numeric Methods

主要研究者：Szanto， Agnes题目：职业：用符号-数值方法求解非线性方程组的过约束系统非线性、过约束方程组经常出现在科学和工程中，这些科学和工程中使用数学模型进行分析。这些领域包括：地球物理现象、计算机图形学和视觉、流体和机械结构分析。过度约束方程很难使用，因为没有可靠和有效的方法来求解它们。传统的数值和符号解方法不适于处理过度约束系统或不精确系数。因此，本研究正在探索求解非线性、过度约束方程组的理论、算法和软件。目标是奠定理论框架，将符号和数值技术扩展到过度约束方程系统。特别注意的是，通过课程计划的发展，为学生参与拟议的研究提供必要的基础，将研究与研究生教育相结合。强调广泛的应用领域。本文研究了两类系统：(1)非精确系数代数方程的过约束系统；(2)具有对称解的微分方程的过约束系统。研究的计划是：系数不精确的代数方程的过约束系统。首先，通过有意义和可验证的解概念以及向后误差和条件分析，将为可靠的计算奠定理论框架。接下来，将努力通过利用解决集的可能的小基数来大幅提高现有符号-数值方法的复杂性。我们的目标是设计一个算法，它是输入加上输出大小的多项式。具有对称解的微分方程的过约束系统：最近关于移动框架方法的结果允许根据对称群的不变量来编写微分系统，并通过不变有限差分方程来设计不变数值近似。为了将这些技术应用于过约束微分系统，研究者将代数消元技术扩展到不变的非线性微分系统。然而，不变微分算子的应用在相关微分代数中引入了非交换性。目的是证明消去算法的终止性，并给出消去算法的阶限。这两个问题实际上是密切相关的，因此研究者也将探讨它们之间的深层联系，目的是为统一的方法奠定数学基础。

ABSTRACTPROPOSAL: 0347506INSTITUTION: North Carolina State UPRINCIPAL INVESTIGATOR: Szanto, Agnes TITLE: CAREER: Solving Over-Constrained Systems of Non-Linear Equations by Symbolic-Numeric MethodsNon-linear, over-constrained systems of equations often arise in science and engineering where mathematical models are used for analysis. Such areas include: geo-physical phenomenon, computer graphics and vision, analysis of fluid and mechanical structures. Over-constrained equations are difficult to use, as there are no reliable and efficient methods for solving them. Traditional numerical and symbolic solution methods are not designed to handle either over-constrained systems or inexact coefficients. Thus, this research is exploring the theory, algorithms and software for solving systems of non-linear, over-constrained systems of equations. The goal is to lay the theoretical framework to extend both symbolic and numeric techniques to over-constrained equation systems. Special attention is paid to the integration of the research with graduate education through the development of course plans that give the required foundations for the students to participate in the proposed research. A wide range of application areas is being emphasized.This research considers two types of systems: (1) Over-constrained systems of algebraic equations with inexact coefficients; and (2) Over-constrained systems of differential equations with symmetric solutions. The plan for investigation is:Over-constrained systems of algebraic equations with inexact coefficients. First, a theoretical framework will be laid for reliable computation via meaningful and verifiable notions of solutions together with backward error and conditioning analysis. Next, effort will be made to substantially improve the complexity of existing symbolic-numeric methods by exploiting the possible small cardinality of the solution set. The goal is to devise an algorithm that is polynomial in the input plus the output size.Over-constrained systems of differential equations with symmetric solutions: Recent results on moving frame methods allow to write a differential system in terms of the invariants of a symmetry group, and to design invariant numerical approximations via invariant finite difference equations. In order to apply these techniques on over-constrained differential systems, the investigator will extend algebraic elimination techniques to invariantized non-linear differential systems. However, the application of the invariant differential operators introduces non-commutativity in the associated differential algebra. The goal is to prove termination and give degree bounds for the elimination algorithms. These two problems are actually closely related and thus the investigator will also explore their deep interconnection with the goal of laying the mathematical bases of a unified approach.