Researches On Numerical Methods For A Class Of Inverse Heat Conduction Problems

Inverse heat conduction problems(called IHCPs)have a wide range of applications in the modern science,technology and engineering fields,such as heat fluid,the remote sensing technology,signal processing,continuum mechanics and industrial controlling etc.,which have attracted many researchers to study in this work.Since the external measured data are inevitably contaminated by noise errors,in the sense of Hadamard,IHCP is ill-posed,that is,any small disturbance in the data can result in an enormous change of numerical solution.Thus it can not keep the stability of solution.In order to overcome the ill-posedness of problem,it is key to find a stable numerical method for solving inverse problem.In this dissertation,a class of heat conduction problems,with the time-dependent heat source term(or the source control parameter term),are considered for two cases of different boundary conditions:(1)the case of Dirichlet conditions,(2)the case of Neumann conditions.The proposed approaches are further studied,including the discussion of some error estimations.The numerical experiment results show that the proposed methods have good accuracy,and are effective for some strong noises.The main contributions of this dissertation are summarized as follows.1.In Chapter 1,firstly,the applications of inverse problem,and the relation between inverse problem and ill-posed problem are introduced.Secondly,some basic theories of regulariza-tion method are presented.2.In Chapter 2,a recursion numerical scheme is considered to solve a class of IHCPs by using the Gaussian radial basis functions,with time-dependent heat source term f(t)and Neumann boundary conditions.The details of algorithms in the one-dimensional and two-dimensional cases,involving the global or partial initial conditions,are proposed respec-tively.Under the measurement data containing noise,A Tikhonov regularization method for solving ill-posed linear equations,based on the Generalized Cross-Validation criterion,is used to ensure the stability of numerical solutions.In addition,in the process of solving the numerical differentiation on discrete data with errors,it can be obtained more accurate numerical results by using regularization method of the smooth spline approximation model.Furthermore,a heuristic criterion for selecting regularization parameter is presented when the noise is unknown.Meanwhile,some conclusions of the condition number estimations,to a class of positive define matrices constructed by the Gaussian radial basis functions,are obtained.3.In Chapter 3,we use another numerical method to solve the same problem in Chapter 2 with Neumann boundary conditions and the global initial condition.A meshless numerical method based on PDE-constrained optimization is proposed,which does not require any discrete domain or boundary.By using the collocation technique in the given physical do?main,it can obtain a global approximation solution in both the spatial and temporal domains.Since the resulting matrix is ill-conditioned,it can obtain more stable and accurate numeri-cal results by applying Tikhonov regularization method,with Generalized Cross-Validation criterion.Finally,by comparing with the numerical results of other methods,the proposed method has better accuracy.4.In Chapter 4,a class of linear multistep difference methods are considered to solve one-dimensional IHCP,with the source control parameter p(t)and Dirichlet boundary condi-tions.The proposed scheme combining with Lagrange interpolation is applied to developed two different numerical difference schemes,namely,3-point difference scheme and 5-point difference scheme.Moreover,the truncation error orders,and the convergence conclusions are proposed for the above difference methods respectively.Since the problem of numerical differentiation involving noise is ill-posed,this chapter uses a smooth spline model based on the Tikhonov regularization method to solve such numerical derivatives containing noise.In the theoretical analysis,there are two cases to be discussed as:(1)the case of E(t)without noise,(2)the case of E(t)with noise,respectively.Further,the ill-posedness and convergence rates of the original problem are studied clearly.