A new class of accelerated regularization methods, with application to bioluminescence tomography
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.