The Inverse Source Problem Of Cherenkov Radiation Model

In this paper,we first propose a new model for Cerenkov radiation.Because the establishment of the optimized radiative transfer model is helpful to improve the reconstruction quality and speed of Cerenkov optical tomography,on the basis of stochastic partial differential equation model driven by Brownian motion,we propose stochastic partial differential equation driven by Poisson jump for Cerenkov radiation,By approximating the initial value and dividing the time interval,the average of the solution is taken in each cell,and the average value and the approximation sequence of the initial value are brought into the solution of the equation to obtain a corresponding expression under different initial values.Then,by using the relevant theorems in stochastic analysis,it is proved that the limit of this group of expressions exists,thus indicating the existence of the solution of the equation.Secondly,the stochastic partial differential equation with Poisson jump is numerically solved.The backward Euler method is used to discretize the equation in time and the finite element method is used to discretize the equation in space.The numerical solution of the equation is obtained.By making a difference between the semigroup representation of the solution and the numerical solution,it is divided into several parts.The properties of the semigroup and the corresponding compensation for the Poisson process are used,By martingale isomorphism,the error analysis of numerical solution and weak solution is obtained.Finally,we discuss the inverse problem of the equation when the initial value is 0,that is,we know the expectation of the solution of the equation at the terminal time to solve the source term.When we take two special terminal values,and the difference between them is as small as possible,we find that the corresponding source term solutions are very different,which indicates that the inverse problem is ill posed.Tikhonov regularization method is used to get the regularization solution,and the proof of error analysis is given for the exact solution and regularization solution of the source term.The error comparison between the two is realized by MATLAB.The experiment shows that when the number of terms is more and the regularization parameter is smaller,the error of the exact solution and regularization solution of the source term is smaller.