| In this paper, we mainly consider a backward fractional diffusion with the initial value calculation of a ill-posed problem in the two-dimensional space. First, we obtain the analytic solution of the fractional diffusion equation, and deduce the relation between the solution and the initial time of the diffusion equation. Next, this is a ill-posed problem for using the observed value of the moment of T to solve the initial value a, so we use the total variation regularization and the double regularization method to turn the ill-posed problem into a well-posed problem to obtain the initial value, that is, solve this optimization problem:where λ,τ,, and β are positive regularized parameters, and then we prove the existence,uniqueness, stability and convergence of the regular solution with this optimization problem. Finally, we use the Bregman iteration to approximate the optimization problem solution, and give the choice strategy for the regularizing parameters in terms of the noise level of measurement data, which yields the error estimate on the iterative solution. |
