微分代数方程中的误差可控计算理论与算法

项目名称： 微分代数方程中的误差可控计算理论与算法

项目编号： No.11471307

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 吴文渊

作者单位： 中国科学院重庆绿色智能技术研究院

项目金额： 65万元

中文摘要： 微分代数方程（DAE）是由代数方程与微分方程耦合在一起的一类特殊方程，在工业设计、动力系统、控制论以及工程物理中有着广泛的应用。本课题针对其非线性性、规模大、稀疏和源于设计的模块化结构等特点，围绕方程解的描述和高精度求解这一大数学难题，采用符号数值混合计算理论与工具，深入探索DAE的多层次结构分析理论、高效指标约化算法以及误差可控的计算理论。其中，重点研究实代数几何的符号数值混合计算方法，微分方程的数值延拓与投影方法，进而用于DAE约化检验、退化情形处理、奇异点扰动等技术的开发。不同于符号计算方法，我们提出用点来描述DAE解空间；在数值代数几何witness point概念基础之上附加上数值性态的选点标准将更好地控制误差；而且在算法设计中更关注DAE系统结构性质的挖掘和利用。这项研究将弥补目前在非线性DAE方程求解方面研究的不足。同时，研究结果有望广泛应用于工程物理建模、控制论以及先进制造

中文关键词： 符号数值混合计算；多项式系统；复杂度分析；同伦方法；微分代数方程

英文摘要： A special type of differential equations combining with algebraic constraints, called differential-algebraic equations (DAE), has various applications in many areas such as industrial modelling, dynamical systems, control theory, engineering physics, etc. For nonlinear and large scale DAE, this project will focus on the description and high accuracy solving of solutions. By using symbolic-numeric computation tools, we will explore multi-structured analysis of DAE, efficient index reduction technique and error-controllable algorithms. The key step is to study the hybrid methods for real algebraic geometry, numerical version of prolongation and projection operations to differential equations. This enables us to do Jacobian test at consistent points, handle the degeneration of the fast prolongation method (Pryce method) and develop the perturbation techniques for singularities. Different from symbolic equation type of description, we propose point type of description for the solution space of DAE, which is a generalization of witness point concept in Numerical Algebraic Geometry imposed an extra condition well-conditionedess for better error control. In addition, we aim to design algorithms taking advantages of structures from background applications. By this project, it will initiate the study of error controllable computation for nonlinear DAE. Moreover, this work is very promising to be applied in areas such as multi-domain uniformed modelling,control theory and advanced manufacturing.

英文关键词： symbolic numeric hybrid computation;polynomial system;complexity analysis;homotopy continuation methods;differential algebraic equations