Multilevel sparse tensor product approximation for manifolds and for functions and operators on manifolds (C04)

在许多高维应用中，观测数据来自高维环境空间的低维子流形。在这个项目中，我们设计了新的方法来检测这个流形的内在结构，以规避维数灾难，这注定了任何经典的数值逼近。我们联合收割机几个非线性近似步骤，以确定从高维输入数据出现的过程。这将允许以具有成本效益的方式处理流形上的回归问题和偏微分方程。

In many high-dimensional applications observed data stem from a low-dimensional submanifold of the high-dimensional ambient space. In this project, we devise novel methods to detect the intrinsic structure of this manifold in order to circumvent the curse of dimensionality, which dooms any classical numerical approximation. We combine several nonlinear approximation steps to identify processes which emerge from the high-dimensional input data. This will allow the treatment of regression problems and PDEs on manifolds in a cost-efficient way.