Multiscale Iterative Regularization Method For Inverse Problem

As an interdisciplinary research, the research for inverse problems has expanded to many fields, such as medical imaging, geophysics exploration, and signal de-tection. It is well-known that inverse problems are always ill-posed, so the regu-larization strategy should be applied to obtain the stable regularized solution. In this manuscript, we mainly focused on some latest regularized iterated algorithms: multiscale analysis in Hilbert space and iterated analysis in Banach space.In the first part of this manuscript, for the purpose to examine the application to the solution of linear moderately ill-posed linear inverse problem, we first consider the multiscale semi-discrete Tikhonov regularization method in Hilbert space, which can be seen as a connection of multiscale method in scattered data approximating and general semi-discrete Tikhnov regularization method in inverse problem. We discuss the convergence of the algorithm for noise-free data, and the error estimates for the noisy data. Both a priori and a posteriori regularization parameter choice rules are derived.We then consider the extension of the multiscale Tikhonov regularization iter-ated method in the first part. Utilizing the well-known support vector regression method in learning theory and multiscale analysis, we discuss a so-called multi-scale support vector approach iterated method to solving linear moderately ill-posed problem. In order to avoid the over-fitting by noisy data which destroys the general-ization property of the approximant, regularization technique is performed by using the cut-off parameter and Vapnik intensive function, the standard l2 loss function is replaced. Compared with the multiscale method in the first part, for the case of noisy data, we show that a priori parameter strategy can be derived, so it is unnecessary to use discrepancy principle to calculate the regularization parameter iteratively at each step, thus reduce the computing cost.In the last part of the manuscript, we consider the nonstationary iterated Tikhnov regularization method in Banach space. By the Bregman distance induced by a general proper, lower weakly semi-continuous and uniformly convex function, the penalty term is allowed to be non-smooth to include L1 and total variation like penalty functionals, which are significant in reconstructing special features of solutions such as sparsity and discontinuities in practical applications. We present the detailed convergence analysis and obtain the regularization property when the method is terminated by the discrepancy principle.