A Meshless Method For Scattered Data Approximation And Its Applications

Scattered data approximation is defined as constructing the unknown func-tion using the scattered data, and digital geographic model, virtual reality, medical imaging process, digital graphic simulation, numerical solution of partial differen-tial equations all boil down to scattered data approximation. Parabolic partial differential equation is a kind of important equation to express the heat conduc-tion and diffusion. Many problems in engineering and natural sciences, such as, mechanics, physics, can be summed up in solving the parabolic partial differen-tial equations. The inverse problem of parabolic partial differential equation is an important issue in scientific research, and it has been widely used in physics and engineering technology.One of the effective methods to deal with scattered data approximation is the meshless method. In meshless method, the trial function is constructed based on the scattered data, and mesh generation on the spatial domain is not needed, so it brings great convenience to science and engineering calculation, and it is convenient to adaptive computing, at the same time, the computation accuracy is improved. Compared with the numerical method based on the grid, the numerical calculation process in meshless method is not depend on the grid, and it is suitable to scattered data; the approximating function is constructed by scattered data, and the structure is simple; and the approximate function has good continuity.In this paper, a meshless method for scattered data approximation and its application in parabolic partial differential equations and curve and surface fitting are studied. The main work are described as follows:The application of meshless method based on radial basis functions in two classes of inverse problems in parabolic partial differential equations is studied, the unknnown source term is about t, and the unknown source term is about x, t. The numerical method and numerical examples are given, and the numerical ex-periments illustrate that the error decreases with the decrease of the step length At and decreases with the increase of the number of data. The case that there is noise data in additional determination is discussed. In the inverse problem that the un-known source term is about x, t, we compare the radial basis function method with the finite difference method, the results show that radial basis functions method is superior to the finite difference method.The application of meshless method based on moving least squares approxi- mation in two classes of inverse problems in parabolic partial differential equations is studied, the unknown source term is about t, and the unknown source term is about x, t. The numerical method and numerical examples are given, and the numerical experiments illustrate that the error decreases with the decrease of the step length At and decreases with the increase of the number of data. The case that there is noise data in additional determination is discussed. In the inverse problem that the unknown source term is about t, we compare the moving least square method with the finite difference method, the results show that moving least squares method is much more stable than the finite difference method.The local meshless method based on the linear combination of moving least squares and local radial basis function is studied, the discrete scheme is collocation method, the error estimation in sobolev space Wk,p(Q) is given, and the error decreases with the decrease of the full space h. Using this method, we solve the parabolic partial differential equation and its inverse problem. The numerical method and the numerical experiments are given to demonstrate the validity of the theory and the feasibility of the method, and the conclusion is that the method in this paper is superior to the moving least squares method and the local radial basis functions method, and so on. In the inverse problems, the numerical method and the corresponding numerical experiments are given, and the numerical experiments illustrate that the error decreases with the decrease of the step length At and decreases with the increase of the number of data. The noise data is discussed, and in the inverse problem that the unknown source term is about x, t, we compare the method in this paper with the other four methods, and we obtain that our method is feasible, accurate and effective.The curve and surface fitting based on the moving least square and local radial basis function is studied, the numerical examples are given in two cases:regular data and scattered data, and the fitting results are also discussed in the presence of noise disturbance. The results show that noise disturbance has no obvious effect on fitting results, and fitting precision is consistent in the cases:regular data and scattered data, so the method in this paper is feasible and effective.