Abstract:
We analyze the kinematics of multigrid Monte Carlo algorithms by investigating acceptance rates for nonlocal Metropolis updates. With the help of a simple criterion we can decide whether or not a multigrid algorithm will have a chance to overcome critial 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.

Abstract:
One of the most demanding calculations is to generate random samples from a specified probability distribution (usually with an unknown normalizing prefactor) in a high-dimensional configuration space. One often has to resort to using a Markov chain Monte Carlo method, which converges only in the limit to the prescribed distribution. Such methods typically inch through configuration space step by step, with acceptance of a step based on a Metropolis(-Hastings) criterion. An acceptance rate of 100% is possible in principle by embedding configuration space in a higher-dimensional phase space and using ordinary differential equations. In practice, numerical integrators must be used, lowering the acceptance rate. This is the essence of hybrid Monte Carlo methods. Presented is a general framework for constructing such methods under relaxed conditions: the only geometric property needed is (weakened) reversibility; volume preservation is not needed. The possibilities are illustrated by deriving a couple of explicit hybrid Monte Carlo methods, one based on barrier-lowering variable-metric dynamics and another based on isokinetic dynamics.

Abstract:
An approximation formula is derived for acceptance rates of nonlocal Metropolis updates in simulations of lattice field theories. The predictions of the formula agree quite well with Monte Carlo simulations of 2-dimensional Sine Gordon, XY and phi**4 models. The results are consistent with the following rule: For a critical model with a fundamental Hamiltonian H(phi) sufficiently high acceptance rates for a complete elimination of critical slowing down can only be expected if the expansion of < H(phi+psi) > in terms of the shift psi contains no relevant term (mass term).

Abstract:
We discuss the analytic computation of autocorrelation functions for the generalized Hybrid Monte Carlo algorithm applied to free field theory and compare the results with numerical results for the $O(4)$ spin model in two dimensions. We explain how the dynamical critical exponent $z$ for some operators may be reduced from two to one by tuning the amount of randomness introduced by the updating procedure, and why critical slowing down is not a problem for other operators.

Abstract:
We investigate the properties of the Hybrid Monte-Carlo algorithm (HMC) in high dimensions. HMC develops a Markov chain reversible w.r.t. a given target distribution $\Pi$ by using separable Hamiltonian dynamics with potential $-\log\Pi$. The additional momentum variables are chosen at random from the Boltzmann distribution and the continuous-time Hamiltonian dynamics are then discretised using the leapfrog scheme. The induced bias is removed via a Metropolis-Hastings accept/reject rule. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an $\mathcal{O}(1)$ acceptance probability as the dimension $d$ of the state space tends to $\infty$, the leapfrog step-size $h$ should be scaled as $h= l \times d^{-1/4}$. Therefore, in high dimensions, HMC requires $\mathcal{O}(d^{1/4})$ steps to traverse the state space. We also identify analytically the asymptotically optimal acceptance probability, which turns out to be 0.651 (to three decimal places). This is the choice which optimally balances the cost of generating a proposal, which {\em decreases} as $l$ increases, against the cost related to the average number of proposals required to obtain acceptance, which {\em increases} as $l$ increases.

Abstract:
A new method for sequence optimization in protein models is presented. The approach, which has inherited its basic philosophy from recent work by Deutsch and Kurosky [Phys. Rev. Lett. 76, 323 (1996)] by maximizing conditional probabilities rather than minimizing energy functions, is based upon a novel and very efficient multisequence Monte Carlo scheme. By construction, the method ensures that the designed sequences represent good folders thermodynamically. A bootstrap procedure for the sequence space search is devised making very large chains feasible. The algorithm is successfully explored on the two-dimensional HP model with chain lengths N=16, 18 and 32.

Abstract:
The hermitian Wilson kernel used in the construction of the domain-wall and overlap Dirac operators has exceptionally small eigenvalues that make it expensive to reach high-quality chiral symmetry for domain-wall fermions, or high precision in the case of the overlap operator. An efficient way of suppressing such eigenmodes consists of including a positive power of the determinant of the Wilson kernel in the Boltzmann weight, but doing this also suppresses tunneling between topological sectors. Here we propose a modification of the Hybrid Monte-Carlo algorithm which aims to restore tunneling between topological sectors by excluding the lowest eigenmodes of the Wilson kernel from the molecular-dynamics evolution, and correcting for this at the accept/reject step. We discuss the implications of this modification for the acceptance rate.

Abstract:
We propose a modification of the Hybrid-Monte-Carlo algorithm that allows for a larger step-size of the integration scheme at constant acceptance rate. The key ingredient is that the pseudo-fermion action is split into two parts. We test our proposal at the example of the two-dimensional lattice Schwinger model with two degenerate flavours of Wilson-fermions.

Abstract:
We propose a modification of the Hybrid-Monte-Carlo algorithm that allows for a larger step-size of the integration scheme at constant acceptance rate. The key ingredient is the splitting of the pseudo-fermion action into two parts. We test our proposal at the example of the two-dimensional lattice Schwinger model and four-dimensional lattice QCD with two degenerate flavours of Wilson- fermions.

Abstract:
We introduce a dynamical fermion algorithm which is based on the hybrid Monte Carlo (HMC) algorithm, but without pseudofermions. The molecular dynamics steps in HMC are retained except the derivatives with respect to the gauge fields are calculated with the $Z_2$ noise. The determinant ratios are estimated with the Pa\`{d}e - $Z_2$ method. Finally, we use the Kennedy-Kuti linear accept/reject method for the Monte Carlo step which is shown to respect detailed balance. We comment on the comparison of this algorithm with the pseudofermion algorithm.