Research On Some Problems Of Moving Least Square Method In The Data Fitting

Scientific computing involves a large number of experimental data, and fitting these data has an important significance. Moving least square method has many extensive applications in the data fitting, and recently has gotten the great development in the curve and surface modeling. In terms of the curve and surface fitting, moving least square method can overcome many shortcomings of the least square method, and has more incomparable advantages than other fitting methods. But moving least square still needs to be improved in some aspects. We can get better curve and surface fitting effect by discussing and analyzing some important contents of moving least square method. In some cases, we need to increase the interpolation and derivative conditions, and improve the moving least square method to get the required fitting effect.This paper mainly studies some problems of moving least square method. Then we verify and extend these related theories in the fitting data. The details are as follows:In the first part, the problem of the curve and surface fitting based on moving least square method is introduced. We compare the fitting effects between moving least square method and least square method, and apply two methods to estimate geometric properties of surfaces. We can estimate the geometric property of the fitting surface to approximately get the required differential geometric property of any point in the given surface.In the second part, we discuss how to select the influence radius of each knot when using moving least square method. We commonly get different fitting computation and effect if we select different radius of influence, and we discuss how to select the radius when the discrete points are scattered uniformly. In addition, for some complex distribution of data points, a new algorithm for searching key points is introduced. We can get the desired fitting effect by firstly select these key points, and then fit them.In the third part, we discuss how to select the weight function when using moving least square method. We can get different fitting effects if we choose different spline weight function when using moving least square method. By comparing these fitting results, we find that we can get good fitting effect by selecting the high order spline weight function. The error of the fitting is relatively small, but it needs the huge calculation. Usually we can use the low order spline weight function to get the satisfying fitting effect.In the forth part, a constructional method of moving least square with interpolation conditions is introduced. We firstly propose a new least square method with interpolation conditions. It has more advantages including that the degree of fitting function is low and the construction computation is convenient. Then this method is extended for the moving least square fitting with interpolation conditions. And it can also obtain the better fitting effect in the curve and surface fitting. Finally, we introduce the least square method with derivative conditions. We can get the fitting function by calculating coefficients with Lagrange's method of multipliers. Then we deduce the constructional method of moving least square with derivative conditions.At last, we summary the research work of the paper and point out the future work.