Optimal multigrid algorithms for the massive Gaussian model and path integrals

Multigrid algorithms are presented which, in addition to eliminating the critical slowing down, can also eliminate the volume factor. The elimination of the volume factor removes the need to produce many independent fine-grid configurations for averaging out their statistical deviations, by averaging over the many samples produced on coarse grids during the multigrid cycle. Thermodynamic limits of observables can be calculated to relative accuracy _r in justO(_r^-2) computer operations, where _r is the error relative to the standard deviation of the observable. In this paper, we describe in detail the calculation of the susceptibility in the one-dimensional massive Gaussian model, which is also a simple example of path integrals. Numerical experiments show that the susceptibility can be calculated to relative accuracy _r in about 8_r^-2 random number generations, independent of the mass size.