Effiziente Quasi Monte Carlo Methoden und ihre Anwendung in der Quantenfeldtheorie

• Antragstellerinnen / Antragsteller: Privatdozent Dr. Karl Jansen; Professorin Dr. Caren Tischendorf, seit 5/2016
• Fachliche Zuordnung: Kern- und Elementarteilchenphysik, Quantenmechanik, Relativitätstheorie, Felder ; Mathematik
• Förderung: Förderung von 2013 bis 2018
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 248869916

The project aims at the exploration of quasi-Monte Carlo (QMC) methods for applications in Euclidean quantum field theory and statistical mechanics. We plan to explore and construct efficient effective--dimension reduction techniques based on derivative information of the underlying integrands using new techniques of algorithmic differentiation. These techniques will be used for Monte Carlo simulations of models in Euclidean lattice field theory and for the computation of so-called dis-connected diagrams. An efficient effective--dimension reduction renders integration problems in Euclidean quantum field theory and statistical mechanics amenable to QMC methods. Since the very high-dimensional problems considered are usually smooth, we aim at reaching an order of convergence O(1/Nr), with r ≥=1 and N the number of samples, whereas by classical Monte Carlo (MC) methods only O(1/N½ ) convergence is achievable. The application of new algorithmic differentiation techniques in this project pursues two different goals: First, we will explore dimension reduction techniques based on derivative information of the target integrands which will represent systems of lattice field theory with compact integration variables as needed for gauge theories and the stochastic evaluation of the discretized quark propagator. To this end, we investigate the effective--dimensions of a problem via Sobol' derivative-based sensitivities. It has been recently shown in a paper by the applicants that efficient algorithmic differentiation techniques can drastically speed-up the required calculations for Sobol' derivative-based sensitivities. Second, we aim at using derivative information for an effective construction of QMC rules tailored to the target integrands. In another publication, it was shown by some of the applicants that new algorithmic differentiation techniques can be used to construct efficient lattice rules for weighted spaces, one of the mayor techniques in QMC methods that we aim to apply to the Euclidean quantum field problems. We will also apply and test the efficiency of Sobol' sequences with additional uniformity properties that provide an additional guarantee of uniformity for high-dimensional problems even at a small number of sampled points. We plan to extent the efficient construction and application of QMC methods and (effective) dimension reduction techniques to other classes of problems. We intend to look at problems where the relevant degrees of freedom are in a compact group. Another problem to be addressed are so-called dis-connected contributions which are very difficult to tackle with conventional MC techniques. Here, a successful application of the QMC methods would provide a breakthrough for lattice quantum chromodynamics (QCD) calculations.

• DFG-Verfahren: Sachbeihilfen
• Ehemaliger Antragsteller: Professor Dr. Andreas Griewank, bis 5/2016 (†)