Peter Mathé

Bernd Hofmann, Peter Mathé

We study the Tikhonov regularization for ill-posed non-linear operator equations in Hilbert scales. Our focus is on the interplay between the smoothness-promoting properties of the penalty and the smoothness inherent in the solution. The objective is to study the situation when the unknown solution fails to have a finite penalty value, hence when the penalty is oversmoothing. By now this case was only studied for linear operator equations in Hilbert scales. We extend those results to certain classes of non-linear problems. The main result asserts that under appropriate assumptions order optimal reconstruction is still possible. In an appendix we highlight that the non-linearity assumption underlying the present analysis is met for specific applications.

Bernd Hofmann, Stefan Kindermann, Peter Mathé

The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.

Steven Bürger, Peter Mathé

Lavrent'ev regularization for the autoconvolution equation was considered by J. Janno in {\itshape Lavrent'ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation}, Inverse Problems, 16(2):333--348, 2000. Here this study is extended by considering discretization of the Lavrent'ev scheme by splines. It is shown how to maintain the known convergence rate by an appropriate choice of spline spaces and a proper choice of the discretization level. For piece-wise constant splines the discretized equation allows for an explicit solver, in contrast to using higher order splines. This is used to design a fast implementation by means of post-smoothing, which provides results, which are indistinguishable from results obtained by direct discretization using cubic splines.

Abhishake Rastogi, Peter Mathé

We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of convergence for the regularized solution under the prior assumptions and a certain link condition. We express the error in terms of certain distance functions. For regression functions with smoothness given in terms of source conditions the error bound can then be explicitly established.

Abhishake Rastogi, Gilles Blanchard, Peter Mathé

We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization, MOR) approach to reconstruct the estimator of the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the ansatz of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.

Bernd Hofmann, Peter Mathé

We study Tikhonov regularization for certain classes of non-linear ill-posed operator equations in Hilbert space. Emphasis is on the case where the solution smoothness fails to have a finite penalty value, as in the preceding study 'Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales'. Inverse Problems 34(1), 2018, by the same authors. Optimal order convergence rates are established for the specific a priori parameter choice, as used for the corresponding linear equations.

Shuai Lu, Peter Mathé, Sergiy Pereverzyev

This paper studies a Nyström type subsampling approach to large kernel learning methods in the misspecified case, where the target function is not assumed to belong to the reproducing kernel Hilbert space generated by the underlying kernel. This case is less understood, in spite of its practical importance. To model such a case, the smoothness of target functions is described in terms of general source conditions. It is surprising that almost for the whole range of the source conditions, describing the misspecified case, the corresponding learning rate bounds can be achieved with just one value of the regularization parameter. This observation allows a formulation of mild conditions under which the plain Nyström subsampling can be realized with subquadratic cost maintaining the guaranteed learning rates.