Speaker: Julia Volmer (NIC, DESY Zeuthen)
Title: Improving Monte Carlo Integration by Symmetrization
Abstract: The error scaling for Markov Chain - Monte Carlo techniques with N samples behaves like $1/\sqrt{N}$. This scaling makes it often very time intensive to reduce the error of calculated observables, in particular for applications in lattice QCD. It is therefore highly desirable to have alternative methods at hand which show an improved error scaling. One candidate for such an alternative integration technique we tried is based on a new class of polynomially exact integration rules on U(N) and SU(N) which are derived from polynomially exact rules on spheres. In the talk we examine these quadrature rules and their efficiency at the example of a 0+1 dimensional QCD for a non-zero quark mass and chemical potential. In particular, here we have to integrate highly oscillatory functions which manifest itself in form of the sign problem. We demonstrate this problem by the failure of Monte Carlo methods in such applications and show that we can obtain arbitrary precision results using the new polynomially exact integration rules. Additionally we show first attempts to apply the method to the 1+1 dimensional pure U(1) gauge model. Because of the large number of degrees of freedom in this case it is necessary to develop new methods on how to combine the integration rules on spheres for all degrees of freedom efficiently.