High-dimensional efficient approximation based on sampling along rank-1 structures with applications

基于对多元三角多项式进行秩1格空间离散化重构的最新成果，我们将在空间域上使用多个秩1格的并集作为空间离散化。这些采样集似乎结合了稀疏网格和秩1格的优点。高维傅里叶变换的快速计算算法的开发以及新采样算子近似性质的估计是基本的研究重点。此外，我们将采用已经开发的概念来识别稀疏近似，新的采样策略将果断地改进该方法。此外，我们将使用不同的周期化技术来建立非周期高维问题的快速算法。在此背景下，我们计划与一家商业企业合作。带随机系数的偏微分方程是所开发算法的一个重要应用。特别是对于依赖于大量随机参数的数学问题，新算法可以有效地处理。一个关键的优势是重要参数的自动确定和不同参数的相互作用。

Based on the latest results on reconstructing rank-1 lattices as spatial discretizations for multivariate trigonometric polynomials, we will use the union of multiple rank-1 lattices as spatial discretizations in spatial domain. These sampling sets seem to combine the advantages of both, sparse grids and rank-1 lattices. The development of fast algorithms for the computation of the high-dimensional Fourier transform as well as the estimate of the approximation properties of the new sampling operators are fundamentalresearch focuses.Furthermore, we will pursue the already developed concept for the identification of sparse approximations and the new sampling strategy will decisively improve the approach. In addition, we will use different periodization techniques in order to establish fast algorithms for non-periodic high dimensional problems. In this context, we plan a cooperation with a commercial enterprise.Partial differential equations with random coefficients are one important application of the developed algorithms.In particular, mathematical problems that depend on a big number of random parameters can be efficiently treated by thenew algorithms. One crucial advantage is the automated determination of significant parameters and the interaction of different parameters.