The Inverse Problem Of Coefficients Reconstruction Of Differential Equation And Its Application In Curve And Surface Representation

The scope of inverse problems has existed for a long time in various branches of mathemat-ics,physics,engineering and so on.Due to the importance of the inverse problem in application,its theory has been widely developed in the past decades.The coefficients of differential equa-tions are generally related to the physical properties of the system to be modeled.In simple cases,these physical properties can be identified directly from data obtained by some kinds of experiments.In complicated situations it is hard or impossible to measure the physical proper-ties associated with coefficients in a model equation.In such cases it may be necessary to pro-ceed indirectly by the data or additional information which relates to the solution of the inverse problem,we call this problem as the reconstruction problem of coefficients in the differential equation.In this thesis,we study two types of the reconstruction problem of coefficients in the differ-ential equation:One type of this problem is that we reconstruct coefficients of the differential equation satisfying the requirement by discrete data with regarding the curve or surface as the solution of this differential equation.The conventional method in the modeling and design of curves and surfaces can construct smooth and high-precision curves and surfaces.However,in many applications,curves and surfaces not only need to meet the geometric design requirements,but also need to meet some physical characteristics related to tangent vectors.In order to meet these requirements,this thesis reconstructs the first-order linear differential equation based on discrete data points or discrete data points and tangent vectors,and the solution of this equation is used to represent the curve or surface.The another type of this problem is the reconstruction problem of coefficients in the parabolic differential equation based on additional conditions.Although the reconstruction prob-lem of the parabolic differential equation has become one of the hot research field in recent years,there are many problems that need to be studied in depth on theory and numerical algorithms.In particular,there are not sufficient study on the reconstruction problem of multiple coefficients in parabolic differential equation.This thesis considers two kinds of the reconstruction problems of two coefficients in a parabolic differential equation simultaneously.The main contributions of this thesis are as follows:(1)The first-order differential equation can be reconstructed based on discrete data points or discrete data points and tangent vectors so that the solution curve or surface of the differential equation can fit these data points or data points and tangent vectors well.To facilitate the rep-resentation of curves or surfaces,it is necessary to consider the differential system with explicit solution.In chapter 2,we discuss an inverse problem,i.e.,the reconstruction of a linear differential equation from the given discrete data points of the solution.We propose a model and its cor-responding algorithm to recover the coefficient matrix of the differential system based on the normal vectors of 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 exact on the solution curve and the set composed of these data points is not degenerate,the coefficient matrix of the differential equation reconstructed by our algorithm is unique.We discuss the error bound of the approximate coefficient matrix which is recon-structed by our algorithm compared with the exact ones.Numerical examples demonstrate the effectiveness of our algorithm.In chapter 3,the method of reconstructing curves and surfaces by the homogeneous variable-coefficient differential equations is proposed,in order to adapt to the discrete data points and tangent vectors of general curves or surfaces.First,the differential equation with specific form which has an explicit solution is considered,and its solution curve or surface satisfies the end-interpolation conditions.The method for fitting curves or surfaces based on the diagonal differential equation is proposed and its corresponding algorithm is given.The curves or sur-faces reconstructed by this algorithm can satisfy the end-interpolation conditions.Furthermore,a fitting model based on the homogeneous variable coefficient differential equations is proposed for discrete data points and tangent vectors.Numerical experiments verify the effectiveness of the algorithm and the model in this chapter.In chapter 4,the algorithms for reconstructing non-homogeneous differential equation-s based on discrete data points are proposed based on the relationship between the non-homogeneous differential equations and the exponential representation of curves or surfaces.The curves and surfaces reconstructed by these algorithms satisfy the end-interpolation condi-tions.Numerical experiments verify the effectiveness of the algorithm in this chapter.(2)The study on the reconstruction problem of coefficients in the parabolic differential equation based on additional information.In chapter 5,determining the coefficient at the higher-order derivative and the coefficient multiplying the temperature with flux and point conditions that depend on time t is considered.We establish conditions for the existence and uniqueness of the solution for this problem.We also propose a numerical method by a difference method and an optimization method after a transformation for this problem.The results of the numerical experiments demonstrate that the proposed method has good approximation accuracy and robustness to noisy data.In chapter 6,determining two coefficients of a parabolic equation with Dirichlet and inte-gral conditions that depend on time t is considered.We establish conditions for the existence and uniqueness of the solution for this problem.A numerical method using an optimization method and a differential method iteratively is proposed.In our numerical method,B-splines are used to approximate the unknown coefficients,and a method to adaptively select the knots of two B-splines functions in the objective function to be optimized simultaneously is proposed.The results of the numerical experiments demonstrate that the proposed method has good approxi-mation accuracy and robustness to noisy data.