  In this paper, we establish an initial theory regarding the Second Order
Asymptotical Regularization (SOAR) method for the stable approximate solution
of ill-posed linear operator equations in Hilbert spaces, which are models for
linear inverse problems with applications in the natural sciences, imaging and
engineering. We show the regularizing properties of the new method, as well as
the corresponding convergence rates. We prove that, under the appropriate
source conditions and by using Morozov's conventional discrepancy principle,
SOAR exhibits the same power-type convergence rate as the classical version of
asymptotical regularization (Showalter's method). Moreover, we propose a new
total energy discrepancy principle for choosing the terminating time of the
dynamical solution from SOAR, which corresponds to the unique root of a
monotonically non-increasing function and allows us to also show an order
optimal convergence rate for SOAR. A damped symplectic iterative regularizing
algorithm is developed for the realization of SOAR. Several numerical examples
are given to show the accuracy and the acceleration affect of the proposed
method. A comparison with other state-of-the-art methods are provided as well.
