Markus Hegland

Markus Hegland, Bernd Hofmann

Based on the variable Hilbert scale interpolation inequality bounds for the error of regularisation methods are derived under range inclusions. In this context, new formulae for the modulus of continuity of the inverse of bounded operators with non-closed range are given. Even if one can show the equivalence of this approach to the version used previously in the literature, the new formulae and corresponding conditions are simpler than the former ones. Several examples from image processing and spectral enhancement illustrate how the new error bounds can be applied.

Markus Hegland, Robert S. Anderssen

Operationally, index functions of variable Hilbert scales can be viewed as generators for families of spaces and norms. Using a one parameter family of index functions based on the dilations of a given index function, a new class of scales (dilational Hilbert scales (DHS)) is derived which generates new interpolatory inequalities (dilational interpolatory inequalities (DII)) which have the ordinary Hilbert scales (OHS) interpolatory inequalities as special cases. They therefore represent a one-parameter family generalization of OHS, and are a precise and concise subset of VHS approriate for deriving error estimates for deconvolution. The role of the Hilbert scales in deriving error estimates for the approximate solution of inverse problems is discussed along with an application of DHS to deconvolution sharpening.

Markus Hegland, Greg W. Wasilkowski

We consider approximation problems for a special space of d variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show that, depending on the norm for measuring the error, the problems are strongly polynomially or quasi-polynomially tractable even in the model of computation where functional evaluations have the cost exponential in the number of active variables.

Bishnu P. Lamichhane, Markus Hegland

The thin plate spline is a popular tool for the interpolation and smoothing of scattered data. In this paper we propose a novel stabilized mixed finite element method for the discretization of thin plate splines. The mixed formulation is obtained by introducing the gradient of the smoother as an additional unknown. Working with a pair of bases for the gradient of the smoother and the Lagrange multiplier which forms a biorthogonal system, we can easily eliminate these two variables (gradient of the smoother and Lagrange multiplier) leading to a positive definite formulation. The optimal a priori estimate is proved by using a superconvergence property of a gradient recovery operator.

Markus Hegland

Spectral enhancement -- which aims to undo spectral broadening -- leads to integral equations which are ill-posed and require special regularisation techniques for their solution. Even when an optimal regularisation technique is used, however, the errors in the solution -- which originate in data approximation errors -- can be substantial and it is important to have good bounds for these errors in order to select appropriate enhancement methods. A discussion of the causes and nature of broadening provides regularity or source conditions which are required to obtain bounds for the regularised solution of the spectral enhancement problem. The source conditions do only in special cases satisfy the requirements of the standard convergence theory for ill-posed problems. Instead we have to use variable Hilbert scales and their interpolation inequalities to get error bounds. The error bounds in this case turn out to be of the form $O(ε^{1-η(ε)})$ where $ε$ is the data error and $η(ε)$ is a function which tends to zero when $ε$ tends to zero. The approach is demonstrated with the Eddington correction formula and applied to a new spectral reconstruction technique for Voigt spectra. In this case $η(ε)=O(1/\sqrt{|\logε|})$ is found.