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

基于多元三角多项式空间离散化的最新结果，我们将多个秩1格的并集作为空间域的空间离散化。这些采样集似乎结合了稀疏网格和秩1晶格的优点。发展高维傅里叶变换的快速算法以及估计新采样算子的逼近性质是最基本的研究重点，此外，我们将继续追求已经发展的稀疏近似识别的概念，新的采样策略将决定性地改进该方法。此外，我们将使用不同的周期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。带有随机系数的偏微分方程组是所开发算法的一个重要应用。特别是，新算法可以有效地处理依赖于大量随机参数的数学问题。一个关键的优点是自动确定重要参数以及不同参数之间的相互作用。

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.