Sine Transform Based Preconditioning for an Inverse Source Problem of Time-Space Fractional Diffusion Equations

We investigate an inverse problem with quasi-boundary value regularization for reconstructing a source term of time-space fractional diffusion equations from the final observation. A sine transform based preconditioning technique is developed for the linear system which arises from the finite difference discretization of the regularized problem. By making use of the special structure, the proposed preconditioner can be inverted efficiently by the fast sine transform and fast Fourier transform. Theoretically, we show that the preconditioned matrix can be written as the sum of two matrices. The eigenvalues of one matrix are located within a rectangular domain which is uniformly bounded away from zero. Moreover, the boundaries of the domain are independent of grid numbers, regularization parameter, and the noise level. The other matrix has rank less than twice the number of spatial grids but is independent of the number of temporal grids. Numerical experiments are performed to verify the effectiveness of the proposed preconditioner.