Numerical Solutions Of Several Types Of Parabolic Inverse Problems

Many complex physical phenomena and engineering systems,such as heat exchangers,reflux condensers,combustion chambers,nuclear vessels,etc.,have certain unmeasurable properties due to the high temperature/pressure harsh environment involved,so using inverse analysis to affect/determine them is very important and desirable.Examples of this also occur in bioheat transfer,where information on blood perfusion is crucial for calculating the temperature of the blood flowing through the tissue.In this thesis,we consider the problem of inverse bioheat transfer to determine a space-dependent blood perfusion coefficient from temperature measurements,which is of great interest for biomedical engineering applications,e.g.,hyperthermic cancer treatment.Furthermore,in this application the initial temperature of the diffusion process is often not known.In the direct problem,the cause(perfusion coefficient)is given and the effect(temperature field)is determined.Unlike the direct problem,in the inverse problem,the cause is often estimated from the known effect.In recent years,such parabolic inverse problems have attracted the attention of some scholars because they tell us how,with appropriate additional measurements,we can determine the unknown physical properties of the given medium under examination.However,most models of inverse problems of parabolic equations are usually difficult to obtain analytical solutions,and can only be solved by numerical methods.Therefore,this thesis studies the numerical solutions of several types of inverse problems of parabolic equations.First,this thesis considers the inverse problem of reconstructing coefficients from external observation conditions.We divide the method to solve this inverse coefficient problem into two stages.In the first stage,starting from the original equation,through appropriate variable substitution,the nonlinear coefficient inverse problem is linearized into an inverse source problem,and the derivative equation of the solution of the parabolic equation relative to the time variable is established;in the second stage,the quasi-inverse regularization method is used to numerically solve this equation,turning the equation into an optimization problem.Therefore,iteratively solving the inverse source problem can provide a numerical solution to the coefficient inverse problem.In the two-dimensional case,a detailed implementation of the algorithm is given,and numerical tests are performed for the one-and two-dimensional cases,respectively,to give numerical results for the considered nonlinear coefficient inverse problems and the corresponding inverse source problems.Then,this thesis considers the inverse problem of simultaneously reconstructing the space-dependent coefficient and initial temperature from two linearly independent weighted time-integrated temperature observations.A quasi-solution of the inverse problem is obtained by minimizing the least-squares objective functional,and the Frechet gradient with respect to two unknown space-dependent quantities is derived.We apply the conjugate gradient method for the numerical implementation and derive the detailed steps for the implementation of the algorithm.To ensure the stability of the algorithm,the discrepancy principle is applied to stop the iterations.The accuracy and stability of the numerical results of the proposed algorithm are revealed by numerical tests on one-and two-dimensional cases.