Numerical investigation of dictionary-based regularization for inverse problems and approximation problems on spheres and balls - with applications to seismic tomography and high-dimensional geophysical modelling

点击翻译按钮获取中文摘要

In the current course of the project, two algorithms, which were constructed by the Geomathematics Group Siegen, have been further developed for the recovery of neuronal currents from electroencephalography (EEG) and magnetoencephalography (MEG) data. These methods, the Regularized Functional Matching Pursuit (RFMP) and the Regularized Orthogonal Functional Matching Pursuit (ROFMP), iteratively construct a kind of a 'best basis' in order to compute an approximate solution in this basis in a way such that the approximation is stable (i.e. it is only slightly affected by noise on the data) and it unifies the advantages of different types of trial functions. For instance, large global structures can be represented by orthogonal polynomials, whereas detail structures can be resolved in a multi-scale structure due to a combination with localized basis functions such as splines and wavelets.Furthermore, novel results for the mathematical modelling of the involved inverse problems have been derived in the previous project. Amongst others, these results provide us with new information on possible phantoms (artefacts) in the solution.The experience which has been gained in the previous project will be used to solve a particularly challenging inverse problem from geophysics, the seismic traveltime tomography. This problem is concerned with the computation of a velocity model for the Earth or for a region of the Earth from traveltimes of seismic waves. Such models are fundamental for the investigation of structures in the Earth's interior. So far, several numerical methods have been developed for solving this inverse problem. For this reason, a method like the RFMP and the ROFMP is suitable to unify or compare such approaches. This possibility is particularly interesting, because the identification of artefacts in seismic velocity models is very difficult. RFMP and ROFMP yield the opportunity to run tests for different unions of basis systems and other constellations in order to investigate common or differing structures in the solution.However, for conducting these experiments, several new developments in the context of Numerical Analysis and Scientific Computing have to be made. For example, the size of the data sets which are common in geophysics represents a new challenge, in contrast to MEG and EEG data. Furthermore, no singular value decomposition is known for the seismic inverse problem, whereas such representations are available respectively have been derived in the current project. Several other mathematical detail problems, such as an efficient numerical integration of special functions along curves in 3D space, have to be addressed.Another objective of the project is to enhance the applicability and the usability of the methods. For this purpose, the developed software will be made available to the public. Moreover, as another application, a high-resolution gravitational field modelling will be demonstrated.