Volker Michel

Martin Gutting, Bianca Kretz, Volker Michel et al.

Recently, the regularized functional matching pursuit (RFMP) was introduced as a greedy algorithm for linear ill-posed inverse problems. This algorithm incorporates the Tikhonov-Phillips regularization which implies the necessity of a parameter choice. In this paper, some known parameter choice methods are evaluated with respect to their performance in the RFMP and its enhancement, the regularized orthogonal functional matching pursuit (ROFMP). As an example of a linear inverse problem, the downward continuation of gravitational field data from the satellite orbit to the Earth's surface is chosen, because it is exponentially ill-posed. For the test scenarios, different satellite heights with several noise-to-signal ratios and kinds of noise are combined. The performances of the parameter choice strategies in these scenarios are analyzed. For example, it is shown that a strongly scattered set of data points is an essentially harder challenge for the regularization than a regular grid. The obtained results yield a first orientation which parameter choice methods are feasible for the RFMP and the ROFMP.

Volker Michel, Frederik J. Simons

Slepian functions are orthogonal function systems that live on subdomains (for example, geographical regions on the Earth's surface, or bandlimited portions of the entire spectrum). They have been firmly established as a useful tool for the synthesis and analysis of localized (concentrated or confined) signals, and for the modeling and inversion of noise-contaminated data that are only regionally available or only of regional interest. In this paper, we consider a general abstract setup for inverse problems represented by a linear and compact operator between Hilbert spaces with a known singular-value decomposition (svd). In practice, such an svd is often only given for the case of a global expansion of the data (e.g. on the whole sphere) but not for regional data distributions. We show that, in either case, Slepian functions (associated to an arbitrarily prescribed region and the given compact operator) can be determined and applied to construct a regularization for the ill-posed regional inverse problem. Moreover, we describe an algorithm for constructing the Slepian basis via an algebraic eigenvalue problem. The obtained Slepian functions can be used to derive an svd for the combination of the regionalizing projection and the compact operator. As a result, standard regularization techniques relying on a known svd become applicable also to those inverse problems where the data are regionally given only. In particular, wavelet-based multiscale techniques can be used. An example for the latter case is elaborated theoretically and tested on two synthetic numerical examples.

Volker Michel, Sarah Orzlowski, Naomi Schneider

Due to the uncertainty principle, a function cannot be simultaneously limited in space as well as in frequency. The idea of Slepian functions in general is to find functions that are at least optimally spatio-spectrally localised. Here, we are looking for Slepian functions which are suitable for the representation of real-valued vector fields on a three-dimensional ball. We work with diverse vectorial bases on the ball which all consist of Jacobi polynomials and vector spherical harmonics. Such basis functions occur in the singular value decomposition of some tomographic inverse problems in geophysics and medical imaging, see [V. Michel and S. Orzlowski. On the Null Space of a Class of Fredholm Integral Equations of the First Kind. Journal of Inverse and Ill-posed Problems, 24:687-710, 2016.]. Our aim is to find bandlimited vector fields that are well-localised in a part of a cone whose apex is situated in the origin. Following the original approach towards Slepian functions, the optimisation problem can be transformed into a finite-dimensional algebraic eigenvalue problem. The entries of the corresponding matrix are treated analytically as far as possible. For the remaining integrals, numerical quadrature formulae have to be applied. The eigenvalue problem decouples into a normal and a tangential problem. The number of well-localised vector fields can be estimated by a Shannon number which mainly depends on the maximal radial and angular degree of the basis functions as well as the size of the localisation region. We show numerical examples of vectorial Slepian functions on the ball, which demonstrate the good localisation of these functions and the accurate estimate of the Shannon number.

Simone Gramsch, Max Kontak, Volker Michel

An elementary algorithm is used to simulate the industrial production of a fiber of a 3-dimensional nonwoven fabric. The algorithm simulates the fiber as a polyline where the direction of each segment is stochastically drawn based on a given probability density function (PDF) on the unit sphere. This PDF is obtained from data of directions of fiber fragments which originate from computer tomography scans of a real non-woven fabric. However, the simulation algorithm requires numerous evaluations of the PDF. Since the established technique of a kernel density estimator leads to very high computational costs, a novel greedy algorithm for estimating a sparse representation of the PDF is introduced. Numerical tests for a synthetic and a real example are presented. In a realistic scenario, the introduced sparsity ansatz leads to a reduction of the computation time for 100 fibers from nearly 40 days to 41 minutes.