The Hybrid Monte Carlo (HMC) algorithm provides a framework for sampling from complex, highdimensional target distributions. In contrast with standard Markov chain Monte Carlo (MCMC) algorithms, it generates nonlocal, nonsymmetric moves in the state space, alleviating random walk type behaviour for the simulated trajectories. However, similarly to algorithms based on random walk or Langevin proposals, the number of steps required to explore the target distribution typically grows with the dimension of the state space. We define a generalized HMC algorithm which overcomes this problem for target measures arising as finite-dimensional approximations of measures π which have density with respect to a Gaussian measure on an infinite-dimensional Hilbert space. The key idea is to construct an MCMC method which is well defined on the Hilbert space itself. We successively address the following issues in the infinite-dimensional setting of a Hilbert space: (i) construction of a probability measure Π in an enlarged phase space having the target π as a marginal, together with a Hamiltonian flow that preserves Π ; (ii) development of a suitable geometric numerical integrator for the Hamiltonian flow; and (iii) derivation of an accept/reject rule to ensure preservation of Π when using the above numerical integrator instead of the actual Hamiltonian flow. Experiments are reported that compare the new algorithm with standard HMC and with a version of the Langevin MCMC method defined on a Hilbert space. c

No abstract

No abstract

No abstract

Currently, the coherent-potential approximation (CPA) implemented via the multiple-scattering theory of Korringa, Kohn, and Rostoker (KKR) gives the best first-principles description of the electronic structure for random substitutional alloys. However, the total energy has an important component of electrostatic energy missing, namely, that arising from the correlation of charges with varying atomic environments. We develop a "charge-correlated" CPA method (cc-CPA) which includes (some) local environmental charge correlations within the KKR-CPA method. We investigate the cc-CPA energetics for several alloys and show that the formation energies are in better agreement with experimental results. These calculations show that the excess charge on a species is almost completely screened by the first-neighbor shell. We then derive a simplified scheme to include the vast majority of the omitted electrostatic energy from charge correlations which requires only a species-dependent shift of the potentials within the original KKR-CPA method. We also discuss the ramifications on the electronic structure.