Inverse Problem Of Stochastic Diffuse Equations

In recent years,there have been many studies on Cherenkov optical tomography,and the core problem of research is to improve the quality and speed of imaging at a lower cost.And the research on the radiative transfer equation of Cherenkov may be useful from the theoretical level.Cherenkov tomography is the practical problem of rebuilding light source,which corresponds to the inverse problem of diffuse equation.This paper discusses how to find the approximate numerical solution of the initial value of the random parabolic equation.Based on the mathematical model of stochastic partial differential equations for Cherenkov optical imaging proposed by us,this paper simplifies the model into one-dimensional space and simplifies Robin boundary condition to Dirichlet boundary condition.This paper focus on how to find the desirable approximate numerical solution.According to whether the coefficients of random items in the equation are related to function values,this paper divides the stochastic parabolic equations into two types.The accompanying equation and the numerical iteration code corresponding to the two kinds of equations are also very different.Firstly,the inverse reconstruction problem of stochastic parabolic equations is equivalent to solve the optimization problem based on the method of total variation regularization,Then,according to the variational adjoint method,the problem of the first type of regularized optimization function gradient is equivalent to the problem of the function value of a related partial differential equation at end time.Meanwhile,the gradient of the regular function of the second type of equation is equivalent to the problem of finding the end time function value of a stochastic partial differential equation.Finally,according to the optimization theory,an iterative algorithm for extremum based on BFGS method is designed to solve the above optimization problems.The iterative numerical solution for the two types of equations is respectively simulated by MATLAB in this paper.The results show that when the initial value of the equation is continuous,two types of equations,based on the total variation regularization method,can obtain a good numerical solution.But when the initial values of the equation are not smooth,the total variation regularization method will polish the initial value,and cannot obtain a good numerical solution.Therefore,when the initial value is not smooth and Want to get a non-smooth numerical solution is expected,the total variation regularization method is not suitable to find the approximate numerical solution of the initial value of the equation.