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

基于字典的正则化球体反演问题和近似问题的数值研究 - 及其在地震层析成像和高维地球物理建模中的应用

基本信息

批准号：
226407518
负责人：
Professor Dr. Volker Michel
金额：
--
依托单位：
Arbeitsgruppe Geomathematik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2012
资助国家：
德国
起止时间：
2011-12-31 至 2021-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/226407518?language=en
关键词：
Numerical investigation dictionary based regularization

项目摘要

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.

在当前的项目过程中，由锡根地理数学小组构建的两种算法得到了进一步开发，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“最佳基础”，以便在此基础上计算近似解，使得近似为稳定（即它仅受数据噪声的轻微影响）并且它统一了不同类型的试验函数的优点。例如，大型全局结构可以用正交多项式表示，而细节结构可以通过与样条和小波等局部基函数的组合在多尺度结构中解析。此外，所涉及的数学建模的新颖结果逆问题在之前的项目中已经导出。除此之外，这些结果为我们提供了有关解决方案中可能出现的幻像（伪影）的新信息。在之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。该问题涉及根据地震波的传播时间计算地球或地球某个区域的速度模型。这些模型是研究地球内部结构的基础。到目前为止，已经开发了几种数值方法来解决这个反问题。因此，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，因为识别地震速度模型中的伪影非常困难。 RFMP 和 ROFMP 提供了对基础系统和其他星座的不同联合运行测试的机会，以便研究解决方案中的共同或不同结构。然而，为了进行这些实验，数值分析和科学计算背景下的一些新发展已经出现待制作。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，对于地震反问题，尚无已知的奇异值分解，而当前项目中已经分别导出了此类表示。必须解决其他几个数学细节问题，例如沿 3D 空间中曲线的特殊函数的有效数值积分。该项目的另一个目标是增强这些方法的适用性和可用性。为此，开发的软件将向公众开放。此外，作为另一个应用，将演示高分辨率引力场建模。