A new second-order dynamical method for solving linear inverse problems in Hilbert spaces

Abstract

A new second-order dynamic method (SODM) is proposed for solving ill-posed linear inverse problems in Hilbert spaces. The SODM can be viewed as a combination of Tikhonov regularization and second-order asymptotical regularization methods. As a result, a double-regularization-parameter strategy is adopted. The regularization properties of SODM are demonstrated under both a priori and a posteriori stopping rules. In the context of time discretization, we propose several iterative schemes with different choices of damping parameters. A truncated discrepancy principle is employed as the stop criterion. Finally, numerical experiments are performed to show the efficiency of the SODM: on the whole, compared with the classical Tikhonov method and the first-order dynamical-system method, the SODM leads to more-accurate approximate solutions while requiring fewer-iterative numbers.