Simultaneous Inversion Of Perfusion Coefficient And Initial Temperature Based On Optimization Theory

Inverse problems have important applications in aerospace,medical imaging,biological heat transfer and other fields.In the classical sense,it is often ill-posed,which has become one of the difficulties of our research.In this paper,two types of coefficient inversion problems in the context of biological heat transfer are mainly studied,and we will use different additional conditions to invert the space-related perfusion coefficient and initial temperature at the same time.At present,with the rapid development of hyperthermia technology,the blood perfusion coefficient has an important influence on the temperature distribution in biological tissues.In addition,in many related applications,the initial temperature of the diffusion process is also unknown.Therefore,in this framework,we discuss the coefficient reconstruction of two-dimensional linear second-order parabolic equation and time-fractional diffusion equation.The practical significance of this problem is self-evident.The main contents of this paper are as follows:The first chapter mainly introduces the basic situation of the inverse problem and the research status based on the background of biological heat transfer.Then,the main work of this paper is introduced.In the second chapter,in the background of biological heat transfer,we study the inverse problem of reconstructing the perfusion coefficient and initial temperature of two-dimensional linear second-order parabolic equation with additional conditions.Unlike other literature,this chapter studies non-local and convective boundary conditions,which are more difficult and complex.Based on optimal control theory,a binary functional containing two independent variables and two independent regularization parameters is constructed,which transforms the original problem into an optimization problem,and discusses the existence of the optimal solution and the necessary conditions satisfied.Since the control functionals are non-convex and there is no globally unique solution,assuming that the terminal time T is relatively small,the uniqueness and stability of the minimal element can be successfully derived.The third chapter mainly studies a class of inverse problems that use additional conditions to reconstruct the perfusion coefficient and initial temperature in the time-fractional diffusion equation.In this case,the additional condition is not the terminal observation in the usual sense,but two integrated observation data with linear independent weight functions.First,we prove the uniqueness of the existence of a class of weak forms of positive problem solutions.Secondly,because the inverse problem is ill posed,the Tikhonov regularization method is used to transform the original problem into a variational problem,and the corresponding minimal strict convex functional is constructed by using the observation data and prior estimation,and the existence,stability and convergence of the regular solution of the variational problem are given.Chapter 4 briefly summarizes the conclusions and shortcomings of this paper,and further looks forward to the follow-up work.