On the acceleration of optimal regularization algorithms for linear ill-posed inverse problems

Accelerated regularization algorithms for ill-posed problems have received muchattention from researchers of inverse problems since the 1980s. The current optimaltheoretical results indicate that some regularization algorithms, e.g. the nu-method andthe Nesterov method, are such that under conventional source conditions the optimalconvergence rates can be obtained with approximately the square root of the iterationsof those needed for the benchmark (i.e. the Landweber iteration). In this paper, we pro-pose a new class of regularization algorithms with parametern, called the Acceleration Regularization of ordern (AR(n)). Theoretically, we prove that, for an arbitrary number n >-1, AR(n) can attach the optimal convergence rates with approximately then+1root of the iterations needed for the benchmark method. Moreover, unlike the existingaccelerated regularization algorithms, AR(n)s have no saturation restriction. Some sym-plectic iterative regularizing algorithms are developed for the numerical realization of AR(n). Finally, numerical experiments with integral equations and inverse problems inpartial differential equations demonstrate that, at least for n <= 2, the numerical behav-iorofARnmatches our theoretical findings, also breaking the practical accelerationcapability of all existing regularization algorithms.

Automated summary

Accelerated regularization algorithms for ill-posed problems have been a focus of research since the 1980s. This paper proposes a new class of regularization algorithms called AR(n), which can achieve optimal convergence rates with approximately the square root of the iterations required by benchmark methods. Unlike existing algorithms, AR(n) does not have a saturation restriction. Numerical experiments show that, for n <= 2, the practical acceleration capability of AR(n) matches the theoretical findings and exceeds existing regularization algorithms.