Identification Of The Potential And Initial Temperature In A Parabolic Equation

Parabolic equation,as a classical partial differential equation,is often used to describe the heat and diffusion phenomena in the microcosmic world.We can know the properties of the microcosmic world by studying the corresponding parabolic equation,which has been widely used in many fields such as geophysics,biochemistry,industrial control and population dynamics.However,some parameters in the equation derived from the microscopic system are unknown in practice,which cannot be directly measured or the measurement cost is too high.For that reason,additional observable data are needed to identify the unknowns in the equation,and then the solution of the equation is studied.Therefore,how to identify parameters in parabolic equations by local observable measurement has important research significance.This thesis is devoted to simultaneously reconstruct the potential and initial temperature in a parabolic equation,which is an ill-possed and nonlinear inverse problem.Compared to the previous works of identifying a single parameter,it is more complicated to reconstruct two parameters simultaneously on stability analysis or numerical reconstruction.Because this problem is ill-posed,that is,small errors in data will lead to a large deviations in results.Therefore,in this thesis,the priori bounds of potential and initial value are first given,and then the Lipschitz type stability of potential function and logarithmic type stability of initial value are established respectively by Carleman estimation.Numerically,the advantage of this thesis is that an alternate iterative optimization algorithm combined with Tikhonov regularization and the best perturbation method is proposed to reconstruct two unknown functions simultaneously.However,the choice of initial guesses for the best perturbation method has a crucial inffuence to the numerical results.in order to avoid the huge error caused by improper selection of initial iteration value,genetic algorithm is introduced to ensure the integrity of the whole numerical reconstruction process.Finally,numerical examples are given to verify the effectiveness of the proposed algorithm.