### Project Part 06 (P. Kritzer): Approximation of Integrals and Functions by New Types of Quasi-Monte Carlo Algorithms

(1.SFB funding period)

In this project part, we will consider recent developments in QMC theory. One topic is hybrid point sets, obtained by concatenating the components of conventional QMC points, applications of which are, e.g., to be found in computer graphics and high-dimensional simulation. By using hybrid point sets, one has increased flexibility in integrating functions with different properties regarding different components.
There has been much work on the analysis of hybrid point sets as such, but only few results on the functions that we can efficiently integrate by QMC algorithms based on these. Therefore, we will study "hybrid'' function spaces and the corresponding algorithms. As pointed out in many papers, one can employ QMC methods also for function approximation, so studying hybrid point sets also for approximating functions is near at hand. Another new approach is obtaining exponential error convergence for QMC algorithms when dealing with analytic functions. In this vein, multivariate algorithms have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. Considerable progress has been made on these questions during the last years, and we will continue research in this context, with a particular focus on function approximation. The previous analysis of algorithms for analytic functions gave rise to new definitions of tractability, and we will analyze our new methods with respect to these concepts and compare the findings to those for classical tractability notions.

(2.SFB funding period)

Our project part will continue exploring the use of novel QMC techniques for multivariate problems that reflect recent developments  in the literature on QMC-methods and their applications. This will involve the consideration of different (or more general) function spaces  than usual, particular kinds of coordinate weights, and/or algorithms that were obtained by different methods than usually outlined in the standard literature.

Our research will partly be motivated by recent QMC applications to partial  differential equations with random coefficients, in particular to elliptic boundary-value problems. In this context, it is crucial to adapt QMC-methods to the $\mathbb{R}^s$, and try to obtain higher-order error convergence. The topic of truncation algorithms shall be continued in the second phase, with some additional viewpoints (as, e.g., links to singular value decompositions of matrices, and applications to Hermite spaces) to take care of. We are also going to include an approach to integration problems based on a randomization over integration rules  with varying numbers of integration nodes. In this context, we study the so-called randomized error setting, and we shall be including tractability results.

Our main cooperation partners are expected to be A. Hinrichs (Project Part 13), G. Leobacher (Project Part 08), F. Pillichshammer (Project Part 09), F.Y. Kuo (Sydney), D. Nuyens (Leuven), and G.W. Wasilkowski (Kentucky).