  In this paper we propose a new class of iterative regularization methods for
solving ill-posed linear operator equations. The prototype of these iterative
regularization methods is in the form of second order evolution equation with a
linear vanishing damping term, which can be viewed not only as an extension of
the asymptotical regularization, but also as a continuous analog of the
Nesterov's acceleration scheme. New iterative regularization methods are
derived from this continuous model in combination with damped symplectic
numerical schemes. The regularization property as well as convergence rates and
acceleration effects under the H\"older-type source conditions of both
continuous and discretized methods are proven.
  The second part of this paper is concerned with the application of the newly
developed accelerated iterative regularization methods to the diffusion-based
bioluminescence tomography, which is modeled as an inverse source problem in
elliptic partial differential equations with both Dirichlet and Neumann
boundary data. A relaxed mathematical formulation is proposed so that the
discrepancy principle can be applied to the iterative scheme without the usage
of Sobolev embedding constants. Several numerical examples, as well as a
comparison with the state-of-the-art methods, are given to show the accuracy
and the acceleration effect of the new methods.
