  In lattice field theory, the interactions of elementary particles can be
computed via high-dimensional integrals. Markov-chain Monte Carlo (MCMC)
methods based on importance sampling are normally efficient to solve most of
these integrals. But these methods give large errors for oscillatory
integrands, exhibiting the so-called sign-problem. We developed new quadrature
rules using the symmetry of the considered systems to avoid the sign-problem in
physical one-dimensional models for the resulting high-dimensional integrals.
This article gives a short introduction to integrals used in lattice QCD where
the interactions of gluon and quark elementary particles are investigated,
explains the alternative integration methods we developed and shows results of
applying them to models with one physical dimension. The new quadrature rules
avoid the sign-problem and can therefore be used to perform simulations at
until now not reachable regions in parameter space, where the MCMC errors are
too big for affordable sample sizes. However, it is still a challenge to
develop these techniques further for applications with physical
higher-dimensional systems.
