Dimensionality Reduction Moving Least Squares Data Fitting Problem And Its Application

In scientific experiments,in order to discover the internal connection between a large number of discrete data,data fitting is often performed.The moving least squares method has been widely used in data fitting problems due to its strong adaptability to the data distribution shape,flexible local fitting and high fitting accuracy.Combining the moving least square method with the boundary integral equation method,the boundary element-free method in the numerical simulation mesh-free method is produced.This method does not need to arrange the grid in advance,thereby reducing the computational cost in the simulation process and improving the simulation accuracy.The application of the research results in the moving least squares method to the boundary element-free method has become a research hotspot in recent years.This article explains the basic principles and methods of moving least squares and boundary element-free methods,and uses moving least squares to deal with the data fitting problem defined on spatial curves and surfaces.Coordinate transformation is used to transform the curves and surfaces during the whole process.The solution domain of is transformed into the solution domain of line and plane,which embodies the idea of dimensionality reduction.This method is also applied to the boundary element-free method,and a relatively ideal result is obtained.The organization structure of this paper is as follows.The first chapter mainly introduces the background of data fitting and numerical simulation research,briefly expounds the concepts and development of moving least squares method,element-free method and boundary element-free method,and summarizes its research status;The second chapter is the preliminary knowledge related to the moving least squares method,leading to the relevant theories and applications of the least squares method,the moving least squares method and its improvement methods;Chapter 3 gives solutions to the data fitting problem defined on the curve and surface.Explains the use of coordinate transformation methods such as polar coordinates and parametric equations to reduce the dimensions of high-dimensional data points defined on curves and surfaces,and then use moving least squares to fit and obtain satisfactory fitting accuracy;Chapter 4 summarizes the basic principles of the boundary integral equation and the boundary element-free method;Chapter 5 applies the dimensionality reduction ideas obtained in Chapter 3 to the boundary element-free method,that is,the boundary function is performed by the one-dimensional moving least square method.Fitting,and give a numerical example to illustrate.Finally,the thesis summarized the research work and looked forward to the next stage of work and put forward suggestions for improvement.