Multi-grid Monte Carlo Research Articles

Anisotropic seismic inversion using a multigrid Monte Carlo approach

We propose a new approach for the inversion of anisotropic P-wave data based on Monte Carlo methods combined with a multigrid approach. Simulated annealing facilitates objective minimization of the functional characterizing the misfit between observed and predicted traveltimes, as controlled by the Thomsen anisotropy parameters (ɛ, δ). Cycling between finer and coarser grids enhances the computational efficiency of the inversion process, thus accelerating the convergence of the solution while acting as a regularization technique of the inverse problem. Multigrid perturbation samples the probability density function without the requirements for the user to adjust tuning parameters. This increases the probability that the preferred global, rather than a poor local, minimum is attained. Undertaking multigrid refinement and Monte Carlo search in parallel produces more robust convergence than does the initially more intuitive approach of completing them sequentially. We demonstrate the usefulness of the new multigrid Monte Carlo (MGMC) scheme by applying it to (a) synthetic, noise-contaminated data reflecting an isotropic subsurface of constant slowness, horizontally layered geologic media and discrete subsurface anomalies; and (b) a crosshole seismic data set acquired by previous authors at the Reskajeage test site in Cornwall, UK. Inverted distributions of slowness (s) and the Thomson anisotropy parameters (ɛ, δ) compare favourably with those obtained previously using a popular matrix-based method. Reconstruction of the Thomsen ɛ parameter is particularly robust compared to that of slowness and the Thomsen δ parameter, even in the face of complex subsurface anomalies. The Thomsen ɛ and δ parameters have enhanced sensitivities to bulk-fabric and fracture-based anisotropies in the TI medium at Reskajeage. Because reconstruction of slowness (s) is intimately linked to that ɛ and δ in the MGMC scheme, inverted images of phase velocity reflect the integrated effects of these two modes of anisotropy. The new MGMC technique thus promises to facilitate rapid inversion of crosshole P-wave data for seismic slownesses and the Thomsen anisotropy parameters, with minimal user input in the inversion process.

Multigrid Monte Carlo simulation viaXYembedding. II. Two-dimensional SU(3) principal chiral model

We carry out a high-precision simulation of the two-dimensional SU(3) principal chiral model at correlation lengths \ensuremath{\xi} up to $\ensuremath{\sim}4\ifmmode\times\else\texttimes\fi{}{10}^{5}$, using a multigrid Monte Carlo (MGMC) algorithm and approximately one year of Cray C-90 CPU time. We extrapolate the finite-volume Monte Carlo data to infinite volume using finite-size-scaling theory, and we discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is \ensuremath{\approx}12% at \ensuremath{\xi}\ensuremath{\sim}25, decreases to \ensuremath{\approx}2% at $\ensuremath{\xi}\ensuremath{\sim}4\ifmmode\times\else\texttimes\fi{}{10}^{5}$. We also analyze the dynamic critical behavior of the MGMC algorithm using lattices up to 256\ifmmode\times\else\texttimes\fi{}256, finding the dynamic critical exponent ${z}_{\mathrm{int},{\mathcal{M}}^{2}}\ensuremath{\approx}0.45\ifmmode\pm\else\textpm\fi{}0.02$ (subjective 68% confidence interval). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated.

Multi-grid Monte Carlo via XY embedding. General theory and two-dimensional O( N)-symmetric non-linear σ-models

We introduce a variant of the multi-grid Monte Carlo (MGMC) method, based on the embedding of an XY model into the target model, and we study its mathematical properties for a variety of non-linear σ-models. We then apply the method to the two-dimensional O( N)-symmetric nonlinear σ-models (also called N-vector models) with N = 3,4,8 and study its dynamic critical behavior. Using lattices up to 256 × 256, we find dynamic critical exponents Z int, M 2 0.70 ± 0.08, 0.60 ± 0.07, 0.52 ± 0.10 for N = 3, 4, 8, respectively (subjective 68% confidence intervals). Thus, for these asymptotically free models, critical slowing down is greatly reduced compared to local algorithms, but not completely eliminated; and the dynamic critical exponent does apparently vary with N. We also analyze the static data for N = 8 using a finite-size scaling extrapolation method. The correlation length ξ agrees with the four-loop asymptotic-freedom prediction to within 1 % over the interval 12 ⪅ ξ ⪅ 650.

Dynamic critical behavior of multi-grid Monte Carlo for two-dimensional nonlinear σ-models

We introduce a new and very convenient approach to multi-grid Monte Carlo (MGMC) algorithms for general nonlinear $\sigma$-models: it is based on embedding an $XY$ model into the given $\sigma$-model, and then updating the induced $XY$ model using a standard $XY$-model MGMC code. We study the dynamic critical behavior of this algorithm for the two-dimensional $O(N)$ $\sigma$-models with $N = 3,4,8$ and for the $SU(3)$ principal chiral model. We find that the dynamic critical exponent $z$ varies systematically between these different asymptotically free models: it is approximately 0.70 for $O(3)$, 0.60 for $O(4)$, 0.50 for $O(8)$, and 0.45 for $SU(3)$. It goes without saying that we have no theoretical explanation of this behavior.

Multi-grid Monte Carlo (III). Two-dimensional O(4)-symmetric non-linear σ-model

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm with piecewise-constant interpolation applied to the two-dimensional O(4)-symmetric nonlinear $\sigma$-model [= SU(2) principal chiral model], on lattices up to $256 \times 256$. We find a dynamic critical exponent $z_{int,{\cal M}^2} = 0.60 \pm 0.07$ for the W-cycle and $z_{int,{\cal M}^2} = 1.13 \pm 0.11$ for the V-cycle, compared to $z_{int,{\cal M}^2} = 2.0 \pm 0.15$ for the single-site heat-bath algorithm (subjective 68% confidence intervals). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated. For a $256 \times 256$ lattice, W-cycle MGMC is about 35 times as efficient as a single-site heat-bath algorithm.

Multi-grid Monte Carlo (II). Two-dimensional XY model

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm applied to the two-dimensional XY model, on lattices up to 128 × 128. In the low-temperature (spin-wave) phase, we find that critical slowing-down is completely eliminated ( z = 0). On the high-temperature side of critically, we find that the dynamic critical exponent is z = 1.4±0.3, compared to z = 2.1±0.3 for the heat-bath algorithm. We attribute the remaining critical slowing-down on the high-temperature side of criticality to the limited effectiveness of the MGMC updates in creating and destroying widely separated vortex-antivortex pairs.

Multigrid Monte Carlo method. Conceptual foundations.

We present details of a stochastic generalization of the multigrid method, called multigrid Monte Carlo (MGMC), that reduces critical slowing down in Monte Carlo computations of lattice field theories. For Gaussian (free) fields, critical slowing down is completely eliminated. For a ${\ensuremath{\varphi}}^{4}$ model, numerical experiments show a factor of \ensuremath{\approx} 10 reduction, over a standard heat-bath algorithm, in the CPU time needed to achieve a given accuracy. For the two-dimensional $\mathrm{XY}$ model, experiments show a factor of \ensuremath{\approx} 10 reduction on the high-temperature side of criticality, growing to an unbounded reduction in the low-temperature regime. The algorithm is also applicable to nonlinear $\ensuremath{\sigma}$ models, and to lattice gauge theories with or without bosonic matter fields.