C04 – Multilevel sparse tensor product approximation for manifolds and for functions and operators on manifolds
High-dimensional problems arise in e.g. data mining, statistical learning theory, financial engineering or stochastics. Without additional knowledge, the curse of dimension dooms any numerical approximation. But fortunately, observed data often stem from a low-dimensional submanifold of the high-dimensional ambient space. In this case, the curse of dimension can be avoided and numerical methods can be applied, provided that they can be restricted to the inherent manifold. To this end, it is necessary to detect the relevant intrinsic dimension and structure of a manifold and to approximate it properly, using a suitable parametrization. This is a crucial step to allow for numerical approximations of functions on manifolds and finally partial differential equations on them. The subject of this project will be fast and efficient multiscale algorithms for these tasks by means of vector-valued adaptive sparse grids for both, function and parametrization representation, and then ultimately for the efficient numerical treatment of PDEs on manifolds.