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Relativistic generalization of path integral Monte Carlo (PIMC) method has been proposed. The problem of relativistic oscillator has been studied in the framework of this approach. Ultra-relativistic and nonrelativistic limits have been discussed. We show that PIMC method can be effectively used for investigation of relativistic systems.

We investigate local update algorithms for the fully frustrated XY model on a square lattice. In addition to the standard updating procedures like the Metropolis or heat bath algorithm we include overrelaxation sweeps, implemented through single spin updates that preserve the energy of the configuration. The dynamical critical exponent (of order two) stays more or less unchanged. However, the integrated autocorrelation times of the algorithm can be significantly reduced.

A new approach for inverting matrices arising after Raviart-Thomas mixed finite element discretization of second-order elliptic equations is studied. Two Monte Carlo algorithms are considered. The first algorithm is based on a special techniques, which uses different relaxation parameters in the iterative procedure and improves the inverse matrix approximation column by column. The second algorithm derives a good error balancing for these special matrices row by row. All parameters determining the efficiency of the algorithms are controlled automatically by a posteriori criteria based on some essential properties of the residual matrices. Numerical computations for rectangular finite elements are presented. The algorithms under consideration are well parallelized.