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

• Applicant: Professor Dr. Daniel Potts
• Subject Area: Mathematics
• Term: from 2017 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 380648269

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.

• DFG Programme: Research Grants