Multiscale Approximation Methods for Scattered Scalar- and Manifold-Valued Data

在应用数学中，从离散样本逼近未知函数是非常重要的，它在许多科学领域都有应用。通常，信息是以任意分散的数据站点上的大量数据值的形式给出的。这些信息可以表示物理量的测量样本，也可以表示物理或生物过程的数值模拟。随着现代医学成像等数据收集设备的进步，对更复杂的数据处理技术的需求越来越大。因此，经典的线性方法常常被更灵活的非线性模型所取代。最近几年流行的一种这样的数学模型是基于多值函数的。然而，即使是标量值函数的逼近也是一个具有挑战性的任务。在这个项目中，将开发一种新的数值方法来逼近标量和流形数据，并对其进行数学分析。该方法应高度适用于涉及海量数据集的问题，并应适用于进一步的数据处理步骤，如去噪、检测、超分辨率和压缩。我们的近似方案将基于使用径向和其他正定核的准内插过程，并结合多尺度技术。以多尺度方式部署准内插的一个主要动机是为标量和多值数据形成一种简单而有效的工具。我们将分三个主要步骤实现这一目标。在第一步中，我们打算研究标量值函数的多尺度拟内插。当在多尺度格式中使用径向核时，我们渴望确定它们的条件以改善它们的收敛。在第二步中，我们旨在推广多尺度方法来逼近流形函数。适应过程，其中我们调整我们的方法以流形数据，是基于流形上的内在平均。特别是，我们将定义新的技巧，研究它们的性质，并获得类似于我们将在标量值情况下证明的理论收敛结果。在最后一步，我们将研究我们的多尺度方法在各种应用中的适用性。我们的目标是证明，导出的效率和理论保证也将导致真实世界数据的理想数值行为。所提出的方法有可能成为未来处理海量数据集的技术的基础。这项研究将为许多实际应用中出现的复杂近似问题提供一个数学框架，从大规模科学模拟到成像设备的数据处理。此外，由于该建议涉及基本的数学方法，它将影响应用数学和其他科学学科的进一步研究领域。

The approximation of an unknown function from its discrete samples is essential in applied mathematics with applications in many science areas. Often, the information is given as a large number of data values at arbitrarily scattered data sites. The information could represent measured samples of a physical quantity or the numerical simulation of a physical or biological process. With advances in data-collecting devices, such as modern medical imaging, there is a growing need for more sophisticated data processing techniques. Consequently, classical linear methods have often been replaced by nonlinear, more flexible models. One such mathematical model, which became popular in recent years, is based on manifold-valued functions. However, even the approximation of scalar-valued functions is still a challenging task. In this project, a new numerical method for the approximation of scalar- and manifold-valued data will be developed and mathematically analyzed. The method shall be highly suitable for problems that involve massively large datasets and shall be applicable to further data processing steps such as denoising, detection, super- resolution, and compression. Our approximation scheme will be based on quasi-interpolation processes using radial and other positive definite kernels in combination with multiscale techniques. A chief motivation for deploying quasi-interpolation in a multiscale way is to form a simple yet efficient tool for both scalar- and manifold-valued data. We will pursue this goal in three main steps. In the first step, we intend to investigate multiscale quasi-interpolation for scalar-valued functions. We aspire to determine conditions on radial kernels for improving their convergence when using them in a multiscale scheme. In the second step, we aim at extending the multiscale method for the approximation of manifold-valued functions. The adaptation process, where we adjust our methods to manifold data, is based upon intrinsic averaging over the manifold. In particular, we will define the new techniques, investigate their properties and obtain theoretical convergence results analogous to the ones we will prove in the scalar-valued case. In the final step, we will study the applicability of our multiscale method for various applications. Our goal is to show that the derived efficiency and theoretical guarantees will also lead to a desirable numerical behavior for real-world data. The proposed methods have the potential of becoming a basis for future techniques for the processing of massive data sets. This study will provide a mathematical framework for complex approximation problems that arise in many real-world applications, from large-scale scientific simulations to processing data of imaging devices. Furthermore, since the proposal deals with basic mathematical methods, it will impact further research areas in applied mathematics and other scientific disciplines.