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

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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 Consequently, classical linear methods have often been replaced by nonlinear, more flexible models.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.