Multigrid Monte Carlo Algorithm Research Articles

Multigrid Monte Carlo method. IV. One-dimensional O(4)-symmetric nonlinear sigma model.

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm with piecewise-constant interpolation and a W-cycle, applied to the one-dimensional $O(4)$-symmetric nonlinear $\sigma$-model [= $SU(2)$ principal chiral model], on lattices from $L=128$ to $L=16384$. Our data for the integrated autocorrelation time $\tau_{int,{\cal M}^2}$ are well fit by a logarithmic growth. We have no idea why the critical slowing-down is not completely eliminated.

Multigrid Monte Carlo algorithm with higher cycles in the sine-Gordon model.

We study the dynamical critical behavior of multigrid Monte Carlo for the two dimensional Sine Gordon model on lattices up to 128 x 128. Using piecewise constant interpolation, we perform a W-cycle (gamma=2). We examine whether one can reduce critical slowing down caused by decreasing acceptance rates on large blocks by doing more work on coarser lattices. To this end, we choose a higher cycle with gamma = 4. The results clearly demonstrate that critical slowing down is not reduced in either case.

Theoretical analysis of acceptance rates in multigrid Monte Carlo

We analyze the kinematics of multigrid Monte Carlo algorithms by investigating acceptance rates for nonclonal Metropolis updates. With the help of a simple criterion we can decide whether or not a multigrid algorithms will have a chance to overcome critical slowing down for a given model. Our method is introduced in the context of spin models. A multigrid Monte Carlo procedure for nonabelian lattice gauge theory is described, and its kinematics is analyzed in detail.

Does the topological charge scale in the lattice CP 3-model?

Using a multigrid Monte Carlo algorithm with piecewise-linear interpolation operator for the two dimensional lattice CP 3 model with the standard quadratic action, we compare our results for the correlation length and the topological susceptibility with recent overrelaxation data. We do not confirm the scaling for the universal topological susceptibility found for smaller correlation length in the overrelaxation simulation. We find clear evidence that a single state saturates the 2-point correlation function at large distances.

Kinematics of multigrid Monte Carlo

We study the kinematics of multigrid Monte Carlo algorithms by means of acceptance rates for nonlocal Metropolis update proposals. An approximation formula for acceptance rates is derived. We present a comparison of different coarse-to-fine interpolation schemes in free field theory, where the formula is exact. The predictions of the approximation formula for several interacting models are well confirmed by Monte Carlo simulations. The following rule is found: For a critical model with fundamental Hamiltonian H(phi), absence of critical slowing down can only be expected if the expansion of <H(phi + psi)> in terms of the shift psi contains no relevant (mass) term. We also introduce a multigrid update procedure for nonabelian lattice gauge theory and study the acceptance rates for gauge group SU(2) in four dimensions.

Testing accelerated algorithms in the latticeCP3model

We present a comparison of the multigrid Monte Carlo algorithm with piecewise-linear interpolation operator with a microcanonical overrelaxation algorithm for the two-dimensional lattice ${\mathrm{CP}}^{3}$ model. For the overrelaxation algorithm we obtain a dynamical critical experiment ${z}_{or}=1.22\ifmmode\pm\else\textpm\fi{}0.05$ as compared with ${z}_{\mathrm{MGMC}}=0.20\ifmmode\pm\else\textpm\fi{}0.04$ for our multigrid implementation. Asymptotic scaling with the bare coupling does not appear to set in for the standard action in the range of parameters studied.

Multigrid acceleration for asymptotically free theories.

We present a multigrid Monte Carlo algorithm with piecewise-linear interpolation operator for nonlinear \ensuremath{\sigma} models with global chiral SU(N)\ifmmode\times\else\texttimes\fi{}SU(N) symmetry or ${\mathrm{CP}}^{\mathit{N}\mathrm{\ensuremath{-}}1}$ manifold. We demonstrate its efficiency in the two-dimensional SU(3)\ifmmode\times\else\texttimes\fi{}SU(3) chiral model and the ${\mathrm{CP}}^{3}$ nonlinear \ensuremath{\sigma} model for correlation length up to 77 in lattice units. On lattices up to size 512\ifmmode\times\else\texttimes\fi{}512 we achieve almost complete elimination of critical slowing down in the ${\mathrm{CP}}^{3}$ case, with a dynamical critical exponent ${\mathit{z}}_{\mathrm{MGMC}}$=0.2\ifmmode\pm\else\textpm\fi{}0.1. Asymptotic scaling with the bare coupling does not appear to set in for the standard action of both models in the range of parameters studied.

Multigrid acceleration for asymptotically free theories

We present a multigrid Monte Carlo algorithm with piecewise-linear interpolation operator for nonlinear σ models with global chiral SU ( N ) × SU ( N ) symmetry or CP N −1 manifold. We demonstrate its efficiency in the two dimensional SU (3) × SU (3) chiral model and the CP 3 nonlinear σ model for correlation length up to 77 in lattice units. On lattices up to size 512 × 512 we achieve almost complete elimination of critical slowing down in the CP 3 case, with a dynamical critical exponent z MGMC = 0.2 ± 0.1. Asymptotic scaling with the bare coupling does not appear to set in for the standard action of both models in the range of parameters studied.

Noncritical multigrid Monte Carlo: O(3) non-linear σ model

We describe a multigrid Monte Carlo technique for asymptotically free lattice field theories like the O( n) non-linear σ model and the SU( N) × SU( N) chiral model in two dimensions. We study this multigrid Monte Carlo algorithm with a smooth interpolation operator in the two-dimensional O(3) σ-model. On lattices up to 256 × 256 we achieve almost complete elimination of critical slowing down. With piecewise-constant interpolation we find a remaining critical slowing down with a dynamic critical exponent of z ≈ 1.

Multigrid Monte Carlo for a bose field in an external gauge field

We describe and test a multigrid Monte Carlo algorithm for a scalar field in two dimensions interacting with an external U(1) gauge field. The algorithm leads to autocorrelation times scaling like τ ≈ 0.14 ξ 0.4.

Multigrid Monte Carlo in an external U(1) gauge field

We test a Multigrid Monte Carlo algorithm on a two dimensional scalar field in an external U(1) gauge field.

Gauge-invariant coarse-to-fine transformations for multigrid Monte Carlo algorithms in pure lattice gauge theories

A class of gauge-invariant coarse lattice to fine lattice transformations is constructed for pure lattice gauge theories. Coarse-to-fine transformations will play an essential role in multigrid Monte Carlo updating algorithms which are believed to cope with the problem of critical slowing down. Efficiency estimates for multigrid Monte Carlo updating algorithms are given and preliminary results from the implementation of such an algorithm for four-dimensional U(1) lattice gauge theory are reported.