Preconditioning Methods for Inverse Source Problem in Fractional Diffusion Equations

The time fractional diffusion equation can be used to describe the anomalous diffusion phenomena instead of the classical diffusion process. The main idea is to replace the standard time derivative by the time fractional derivative (the Caputo derivative). In some applications, time-space fractional diffusion equation is employed by replacing the standard spatial derivative by the Riemann-Liouville derivative or the Riesz fractional derivative. Because of nonlocal nature of these derivatives, their discretized systems of equations involve dense coefficient matrices. Preconditioning iterative methods have been developed for solving such very large linear systems.

In many practical applications, the source term in fractional diffusion equations is unknown. In this project, we are interested in solving this inverse source problem by employing additional measurement data. We study modified quasi-boundary value method and Tiknonov regularization method for solving this inverse source problem. The main challenge is to develop fast iterative solvers for such very large block linear systems arising from the modified quasi-boundary method and Tiknonov regularization method. We remark that the number of unknowns can be very large when the time and spatial discretization step-sizes are very small. The main objective of this proposal is to develop new preconditioning methods for solving such very large linear systems. Our idea is to construct Schur complement preconditioners based on block-circulant-type approximation and divide-and-conquer technique. We will investigate the spectra analysis of the proposed preconditioned matrices and demonstrate the fast convergence of the preconditioned iterative method.

On the other hand, we study Tikhonov-type regularization methods for inverse source problem in fractional diffusion equations. We investigate into the invertibility and the condition numbers of the regularized discretized matrices from this inverse source problem by using block matrix technique and eigen-decomposition. In addition, the convergence rate analysis for unknown sources in spatial and temporal domains is studied by employing a suitable regularization matrix and a priori and an a posteriori choice rule for the regularization parameter.