Chongjun LI,Linlin XIE,Haidong LI.Reconstruction of the Linear Ordinary Differential System Based on Discrete Points[J].数学研究及应用,2017,37(1):73~89 
Reconstruction of the Linear Ordinary Differential System Based on Discrete Points 
Reconstruction of the Linear Ordinary Differential System Based on Discrete Points 
投稿时间：20161123 修订日期：20161219 
DOI：10.3770/j.issn:20952651.2017.01.007 
中文关键词: differential system discrete data normal vector method least square method parameterization 
英文关键词:differential system discrete data normal vector method least square method parameterization 
基金项目:Supported by the National Natural Science Foundation of China (Grant Nos.11290143; 11471066; 11572081), the Fundamental Research of Civil Aircraft (Grant No.MJF201204) and the Fundamental Research Funds for the Central Universities (Grant No.DUT15LK44). 

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中文摘要: 
In this paper, we discuss an inverse problem, i.e., the reconstruction of a linear differential dynamic system from the given discrete data of the solution. We propose a model and a corresponding algorithm to recover the coefficient matrix of the differential system based on the normal vectors from the given discrete points, in order to avoid the problem of parameterization in curve fitting and approximation. We also give some theoretical analysis on our algorithm. When the data points are taken from the solution curve and the set composed of these data points is not degenerate, the coefficient matrix $A$ reconstructed by our algorithm is unique from the given discrete and noisefree data. We discuss the error bounds for the approximate coefficient matrix and the solution which are reconstructed by our algorithm. Numerical examples demonstrate the effectiveness of the algorithm. 
英文摘要: 
In this paper, we discuss an inverse problem, i.e., the reconstruction of a linear differential dynamic system from the given discrete data of the solution. We propose a model and a corresponding algorithm to recover the coefficient matrix of the differential system based on the normal vectors from the given discrete points, in order to avoid the problem of parameterization in curve fitting and approximation. We also give some theoretical analysis on our algorithm. When the data points are taken from the solution curve and the set composed of these data points is not degenerate, the coefficient matrix $A$ reconstructed by our algorithm is unique from the given discrete and noisefree data. We discuss the error bounds for the approximate coefficient matrix and the solution which are reconstructed by our algorithm. Numerical examples demonstrate the effectiveness of the algorithm. 
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