Some matrix nearness problems suggested by Tikhonov regularization

The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for problems of small to moderate size are Tikhonov regularization and truncated singular value decomposition (TSVD). By considering matrix nearness problems related to Tikhonov regularization, several novel regularization methods are derived. These methods share properties with both Tikhonov regularization and TSVD, and can give approximate solutions of higher quality than either one of these methods

Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

In this article the concept of saturation of an arbitrary regularization method is formalized based upon the original idea of saturation for spectral regularization methods introduced by A. Neubauer in 1994. Necessary and sufficient conditions for a regularization method to have global saturation are provided. It is shown that for a method to have global saturation the total error must be optimal in two senses, namely as optimal order of convergence over a certain set which at the same time, must be optimal (in a very precise sense) with respect to the error. Finally, two converse results are proved and the theory is applied to find sufficient conditions which ensure the existence of global saturation for spectral methods with classical qualification of finite positive order and for methods with maximal qualification. Finally, several examples of regularization methods possessing global saturation are shown.Comment: 29 page

Fractional regularization matrices for linear discrete ill-posed problems

The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore the use of fractional powers of the matrices {Mathematical expression} (for Tikhonov regularization) and A (for Lavrentiev regularization) as regularization matrices, where A is the matrix that defines the linear discrete ill-posed problem. Both small- and large-scale problems are considered. © 2013 Springer Science+Business Media Dordrecht

Regularization Methods for Ill-Posed Problems in Multiple Hilbert Scales

Several convergence results in Hilbert scales under different source conditions are proved and orders of convergence and optimal orders of convergence are derived. Also, relations between those source conditions are proved. The concept of a multiple Hilbert scale on a product space is introduced, regularization methods on these scales are defined, both for the case of a single observation and for the case of multiple observations. In the latter case, it is shown how vector-valued regularization functions in these multiple Hilbert scales can be used. In all cases convergence is proved and orders and optimal orders of convergence are shown.Comment: 32 pages, 2 figure

REGULARIZATION PARAMETER SELECTION METHODS FOR ILL POSED POISSON IMAGING PROBLEMS

A common problem in imaging science is to estimate some underlying true image given noisy measurements of image intensity. When image intensity is measured by the counting of incident photons emitted by the object of interest, the data-noise is accurately modeled by a Poisson distribution, which motivates the use of Poisson maximum likelihood estimation. When the underlying model equation is ill-posed, regularization must be employed. I will present a computational framework for solving such problems, including statistically motivated methods for choosing the regularization parameter. Numerical examples will be included

Adaptive complexity regularization for linear inverse problems

We tackle the problem of building adaptive estimation procedures for ill-posed inverse problems. For general regularization methods depending on tuning parameters, we construct a penalized method that selects the optimal smoothing sequence without prior knowledge of the regularity of the function to be estimated. We provide for such estimators oracle inequalities and optimal rates of convergence. This penalized approach is applied to Tikhonov regularization and to regularization by projection.Comment: Published in at http://dx.doi.org/10.1214/07-EJS115 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification

We present a new approach to convexification of the Tikhonov regularization using a continuation method strategy. We embed the original minimization problem into a one-parameter family of minimization problems. Both the penalty term and the minimizer of the Tikhonov functional become dependent on a continuation parameter. In this way we can independently treat two main roles of the regularization term, which are stabilization of the ill-posed problem and introduction of the a priori knowledge. For zero continuation parameter we solve a relaxed regularization problem, which stabilizes the ill-posed problem in a weaker sense. The problem is recast to the original minimization by the continuation method and so the a priori knowledge is enforced. We apply this approach in the context of topology-to-shape geometry identification, where it allows to avoid the convergence of gradient-based methods to a local minima. We present illustrative results for magnetic induction tomography which is an example of PDE constrained inverse problem