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

Abstract

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 a 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 Hölder-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 with a posteriori stopping rule to the diffusionbased bioluminescence tomography, which is modeled as an inverse source problem in elliptic partial differential equations with both Dirichlet and Neumann boundary data. 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. ‡ Corresponding Author

On Fractional Asymptotical Regularization of Linear Ill-Posed Problems in Hilbert Spaces

“…However, all of these issues are not addressed here since it is out of the scope of our current paper. Rather we refer to a similar result in [7] for the second order asymptotical regularization. Now, by employing the newly developed iterative regularization method (5.1), we present some numerical results for the following integral equation In this context, we choose X = Y := L 2 [0, 1] such that the operator A is compact, selfadjoint and injective.…”

Optimal-order convergence of Nesterov acceleration for linear ill-posed problems*

“…Conversely, given a sequence of orthogonal polynomials defined by the recurrence relation (10) with given sequences c n , d n . Then there exists a sequence α k (defined via (11)) such that the corresponding Nesterov iteration (1) has a residual function as in (9).…”

“…We note that although the main field of application of Nesterov acceleration lies in nonlinear optimization, in the paper we only treat the case of linear operator equations and the acceleration properties of the method for linear ill-posed problems. Other recent acceleration schemes proposed in the literature use, e.g., Hilbert scale preconditioning [6], the continuous version of Nesterov's scheme [4,9], or fractional asymptotical regularization [17].…”

Stochastic asymptotical regularization for linear inverse problems

“…(4), to some classical methods from data assimilation, namely the Kalman-Bucy filter and 3DVAR. Moving in a different direction, the authors in [3,9,30,31] extended (4) to second-order and fractional-order gradient flows. They proved that the developed high-order flows are accelerated optimal regularization methods, i.e.…”