Regularization Numerical Method For Parameter Identification Of Parabolic Equations

The researches on inverse problems for partial differential equations have been hot topics in computational and applied mathematics for a long time.Especially,the parameters iden-tifications for parabolic equation system are of great importance,which appear in detections of groundwater pollution,chemical reaction diffusion and other engineering fields.So it is significant to study both theory and numerical solution of the inverse problems governed by parabolic equations.In this thesis,we consider three kinds of inverse problems for parabolic equations.Both theoretical analysis and numerical implementations are carried out.In the first chapter,the basic concepts of the linear ill-posed problems and the usual regularization methods are introduced.Then we recall the research states of the ill-posed problems for parabolic systems.The research topics and innovation points of our work are explained.In the second chapter,a backward conduction problem aiming to the identifications of the temperature distribution in the past time is considered.The homotopy-based iterative regularizing scheme for noisy input data is proposed for this linear ill-posed problem.The advantage of the proposed scheme is that,under general assumptions on the exact initial distribution,the convergence of the homotopy sequence using exact final data as initial guess can always be ensured.For noisy input data,the error analysis for the regularizing solution with noisy measurement data as the initial guess is also established.The algorithm is easy to implement with very low computational costs,while the error bound is still comparable to other regularization methods.Finally,numerical implementations are presented.In the third chapter,an inverse problem for the heat equation,which recovers the spa-tial dependent heat source and the initial temperature field simultaneously,is studied using measurement data specified at two final time moments.Firstly,the uniqueness and condi-tional stability for this inverse problem are established by the properties of parabolic equation and the representation of solution after reforming the equation.Then this inverse problem is reformulated as an optimization problem with some regularizing term,for which we propose an iteration algorithm in terms of the variational adjoint method.Two adjoint systems are constructed to construct the alternative iteration scheme.The negative gradient direction is selected as the first search direction.For the succeeding iterations,two optimization methods are used for the initial temperature inversion and the heat source identification respectively.The efficiency of the proposed scheme is tested by several numerical examples in two dimen-sion.The results show that the alternative iteration scheme can improve the computational efficiency greatly.We also compare the numerical behaviors of the iteration process for two different penalty terms,namely,the standard L2 penalty term and the gradient-based penalty term in the whole domain,which have a explicit engineering motivation.In the fourth chapter,we consider a nonlinear inverse problem for the parabolic equation with nonlinear source,identifying the thermal conductivity coefficient in the heat conduction equation from extra final measurement data.By establishing the cost functional with regular-ization terms,the inverse problem is reformulated as an optimization problem.The existence of the minimizers of the cost functional,with a rigorous analysis on the convergence property of the minimizing sequence,is proved.An iterative algorithm solving the optimization problem is proposed and numerical implementations are presented.Finally,in the fifth chapter,the research of this dissertation is summarized,and the future works are discussed.