In this paper we discuss general regularization estimators. This class includes Tikhonov type and spectral cut-off estimators as well as iterative methods, such as ν-methods and the Landweber iteration. The latter estimators achieve the same (optimal) convergence rates as spectral cut-off, but do not require explicit spectral information on the operator and are often much faster to compute than...

In this study, we first propose the use of edge-preserving regularization in optimizing an ill-conditioned problem in the reconstruction procedure for diffuse optical tomography to prevent unwanted edge smoothing, which usually degrades the attributes of images for distinguishing tumors from background tissues when using Tikhonov regularization. In the edge-preserving regularization method pres...

Truncated Singular Value Decomposition (TSVD) regularization method have been used by Zhao et al. [ " Kronecker product approximations for image restoration with new mean boundary conditions " (2011), Applied Mathematical Modelling, Vol. 36, pp. 225-237]. In this report, I propose an alternative regularization the Tikhonov method. The new regularization method gives better relative error when a...

Estimating the generalization performance of learning algorithms is one of the main purposes of machine learning theoretical research. The previous results describing the generalization ability of Tikhonov regularization algorithm are almost all based on independent and identically distributed (i.i.d.) samples. In this paper we go far beyond this classical framework by establishing the bound on...

Inverse problems are typically ill-posed or ill-conditioned and require regularization. Tikhonov regularization is a popular approach and it requires an additional parameter called the regularization parameter that has to be estimated. The χ method introduced by Mead in [8] uses the χ distribution of the Tikhonov functional for linear inverse problems to estimate the regularization parameter. H...

We describe two regularization techniques based on optimal control for solving two types of ill-posed problems. We include convergence proofs of the regularization method and error estimates. We illustrate our method through problems in signal processing and parameter identification using an efficient Riccati solver. Our numerical results are compared to the same examples solved using Tikhonov ...

An iterative method is introduced for solving noisy, ill-conditioned inverse problems. Analysis of the semi-convergence behavior identifies three error components iteration error, noise error, and initial guess error. A derived expression explains how the three errors are related to each other relative to the number of iterations. The Standard Tikhonov regularization method is just the first it...

We consider a new idea for solving Tikhonov regularized discretized ill-posed problems. The optimization problem is formulated as a nonlinear least squares problems containing the Tikhonov regularization parameter λ. In order to find the size of the regularization parameter and attain good convergence in the optimization method we use the nonlinear Land a-curve. The nonlinear L-curve is a direc...