Application Of Unconstrained Optimization Algorithm In Geometric Iterative Approximation

In recent years,the B-spline curve has received extensive attention and application in geometric modeling and design,and it inherits the excellent domination property of the Bézier curve.Meanwhile,it overcomes the disadvantage that Bézier curves do not have local property and solves the connection problems encountered when Bézier curves describe complex shapes.In engineering surveys and scientific experiments,the data obtained is usually discrete.To obtain the values of points other than these discrete points,interpolation or fitting is required based on known data.Traditional data points interpolation and fitting problems generally require solving large-scale linear equations.As the scale of data points increases,the process is not only time-consuming,but also an extremely large project.Progressive iterative approximation is favored by researchers for its many advantages such as no need to solve linear equations,intuitionistic geometric meaning,easy programming and so on.In view of the above research,this dissertation converts the data points fitting problem into an unconstrained optimization problem and does the following works with the help of unconstrained optimization algorithm:1.A stepwise geometric iterative fitting method(SGIF for short)is constructed to fit large-scale data points.SGIF starts with an initial blending curve,a set of fitting curves are produced by adjusting the control points iteratively.In each iteration,the negative gradient direction is chosen as the descent direction in order to minimize the distances between the initial points and the corresponding fitting points.Thus,iterations are carried out in these directions.Numerical examples show that SGIF method has explicit geometric meanings locally and globally.2.Combined with DFP method(proposed by Davidon,Fletcher and Powell),a large-scale data points fitting method is given.It is verified that the proposed method is convergent.It inherits all the nice properties of the classical least square progressive iterative approximation algorithm,such as intuitive geometric significance,flexible fitting of large-scale data points,and arbitrary choosing of initial control points.3.The singular value decomposition and orthogonal triangular decomposition are introduced to accelerate the traditional data points fitting methods.Numerical examples show that the two methods introduced in this dissertation can significantly accelerate the convergence rate of iteration.