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

在目前的项目过程中，由Geomathematics Group Siegen构建的两种算法已被进一步开发，用于从脑电图（EEG）和脑磁图（MEG）数据中恢复神经元电流。这些方法，正则化函数匹配追踪（RFMP）和正则化正交函数匹配追踪（ROFMP），迭代地构建一种“最佳基”，以便在该基中以一种近似稳定的方式计算近似解（即仅受数据噪声的轻微影响），并统一不同类型的试验函数的优点。例如，大的全局结构可以用正交多项式表示，而细节结构可以在多尺度结构中通过与局部基函数（如样条和小波）的结合来解决。此外，在先前的项目中，对所涉及的反问题的数学建模也得到了新的结果。除其他外，这些结果为我们提供了关于解决方案中可能的幽灵（人工制品）的新信息。在以前的项目中获得的经验将用于解决地球物理学中一个特别具有挑战性的反问题，即地震走时层析成像。这个问题涉及到根据地震波的传播时间计算地球或地球某一区域的速度模型。这些模型是研究地球内部结构的基础。到目前为止，已经发展了几种数值方法来求解这个反问题。出于这个原因，像RFMP和ROFMP这样的方法适合统一或比较这些方法。这种可能性特别有趣，因为在地震速度模型中识别伪影是非常困难的。RFMP和ROFMP提供了对基系统和其他星座的不同组合运行测试的机会，以便研究解决方案中常见或不同的结构。然而，为了进行这些实验，必须在数值分析和科学计算的背景下取得一些新的发展。例如，与MEG和EEG数据相比，地球物理学中常见的数据集的大小代表了一个新的挑战。此外，对于地震反演问题，目前还没有已知的奇异值分解，而目前的项目已经分别推导出了奇异值分解。其他几个数学细节问题，如在三维空间中沿曲线的特殊函数的有效数值积分，必须解决。该项目的另一个目标是提高方法的适用性和可用性。为此目的，开发的软件将向公众开放。此外，作为另一个应用，将展示一个高分辨率的重力场模型。

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.