Efficient Preconditioning for Time Fractional Diffusion Inverse Source Problems

Abstract

In this paper, we consider an inverse problem with quasi-boundary value regularization for recovering a source term of the time fractional diffusion equations from the final observation data. In particular, a two-by-two block linear system arising from the problem is studied. We propose a fast preconditioning technique by approximating the Schur complement in the system using a product of some factors, motivated by an approximate diagonalization of one of the blocks. The eigenvalues of the preconditioned system are shown to be clustered around 1, and the fast convergence of the methods is guaranteed theoretically. We also present an approach for selecting the regularization parameter of the quasi-boundary value method. Numerical experiments are carried out to demonstrate the effectiveness of our method.

A Matching-Strategy-Inspired Preconditioning for Elliptic Optimal Control Problems

“…The efficient solution of the PDE-constrained optimization problem is of great importance in many areas of science and engineering. The performance of iterative methods for these problems depends highly on the choice of preconditioners, which are drawing more and more attention in various fields [1][2][3][4][5][6][7][8][9][10]. Extensive attention has been drawn to preconditioning the saddle point problem arising from the PDE-constrained optimization problem with different PDE constraints, boundary conditions and control types.…”

A Direct Parallel-in-Time Quasi-Boundary Value Method for Inverse Space-Dependent Source Problems

“…The majority of these contributions focuses on the convergence properties of the proposed regularization techniques without discussing fast algorithms for their numerical implementations. For solving related time-fractional diffusion inverse source problems, the authors in [20] proposed a fast structured preconditioner based on approximate Schur complement and block -circulant matrix. However, as an iterative solver, the underlying convergence analysis for preconditioned GMRES in [20] is a daunting task (due to nonsymmetric systems) and the preconditioner can not be easily parallelized in time.…”

“…For solving related time-fractional diffusion inverse source problems, the authors in [20] proposed a fast structured preconditioner based on approximate Schur complement and block -circulant matrix. However, as an iterative solver, the underlying convergence analysis for preconditioned GMRES in [20] is a daunting task (due to nonsymmetric systems) and the preconditioner can not be easily parallelized in time. Our proposed direct PinT solver does not have such limitations for its practical use.…”