The Applications Of MQ-RBF And Lasso Model In A Type Of Inverse Heat Conductive Problems

In this thesis, two forms of a type of inverse heat conductive problems (IHCP) have been considered. One is to reconstruct heat source and the partial initial value with Dirichlet boundary conditions, the other is to reconstruct heat source and the entire initial value with the Nuemann boundary conditions. There are two processes when we solve the heat conductive problems:the first is to transfer them into corresponding linear equation systems. Because of the ill-posedness of IHCPs, the corresponding linear equation systems are also ill-posed, that is, the relevant matrices are ill-conditioned. The second process is to solve the above ill-conditioned linear equation systems by the regularization method.This thesis is organized as follows:In chapter one, we introduce the ill-posed problem and inverse problem, including IHCP. We mainly use meshless methods to transfer IHCPs into linear equation systems. Therefore, two meshless methods will be introduced:the fundamental solution method (MFS) and the radial basis function method (RBF). Then, we briefly describe the idea of regularization method and the Tikhonov regularization method.In chapter two, two forms of the inverse heat conductive problem that we will considered are illustrated.The following are our main work:In chapter three, we discuss the first process. In this thesis, we employ MQ function as the basis function, and then use the collocation method based on it to transfer the heat problems into corresponding linear equation systems.In chapter four, we concretely introduce two methods which are used to solve the ill conditioned linear equation systems:the Tikhonov regularization method and the Lasso model. By the Lasso model, the ill-posed linear equation problem can be transferred into corresponding quadric optimization problem. It can be solved by the alternating direction method (ADM). The idea and the iterative fonnat of ADM are illustrated by a classic linear regression model as an example.In chapter five, several numerical experiments are performed as the exact solutions are known. In the first process, we use the MFS and the RBF collocation to converting IHCPs into corresponding linear equation systems, respectively; in the second process, we adopt the Tikhonov regularization method and the Lasso model, respectively. Through the combination of these methods in the two processes, there are four different ways to solve IHCP. From the numerical results, we can see that, the MFS combined with Tikhonov regularization method and the MQ-RBF combined with Lasso model, both can approximate the exact solutions well. Moreover, the results of the latter one are better. In addition, although the matrices obtained by the MQ collocation method are also ill-conditioned, the condition numbers are much less than these by the MFS. As a result, the MQ-RBF combined with Lasso model is a valid method to solve IHCP.