March meeting

I'll be at the APS March Meeting.

I'm giving a talk in session V27 (on Thursday).

If anyone else going to the meeting wants to meet and talk about QMC, MC, computational physics, or other topics, feel free to email me.

Unbiased exponential estimators

The penalty method allows a Metropolis random walk to be correct even with the presence of noise in the probability function being sampled.

It eventually dawned on me that this same method can be used to correct any exponential estimator.

Imagine we wish to compute the estimator, exp(x), where x is some sample point that is contaminated with noise. Suppose the variance of the noise is sigma, and we know the noise is gaussian. Then the estimator exp(x-sigma^2/2) will be unbiased.

Free Energy Links

Radford Neal has written a paper on computing free energy differences. The paper presents the Linked Importance Sampling method, but describes other methods as well. The paper contains references to both the physics literature and the statistics literature, and uses physics language.

If you're really interested in free energy differences, check out Arjun Acharya's thesis, Free Energy differences: Representations, estimators, and sampling strategies (also on arxiv). The review chapter covers several methods, but most of the thesis deals with the Phase Mapping technique, which increases the overlap between phases. It does not eliminate the problem entirely, however, and many of the other methods can be used to deal with the remaining issues.

[Edit 2/27/06: updated links from comments, and revised the entry]

Linear scaling and noisy forces

This paper by Krajewski and Parrinello,
Linear scaling electronic structure calculations and accurate sampling with noisy forces, has two intriguing ideas.

They compute electronic structure using a new linear scaling method that involves sampling, and thus results in noisy forces. Those forces are used to drive a molecular dynamics simulation of the nuclei (note that one of the authors is Parrinello).

For the electronic structure part of the problem, they replace a determinant in the independent particle grand potential with an integral over some fields. These fields can be sampled using a Langevin equation. This Langevin equation apparently involves only matrix-vector multiplication with a sparse matrix, which delivers the scaling.

The second part (starting in the paragraph before equation 11) is how to deal with the noisy forces in molecular dynamics. Their solution for this is to again use a Langevin equation, and to choose the size of the damping terms to cancel out the noise on average. In this way they obtain correct sampling of the Boltzmann distribution.

Polarization

Calculating observable properties of systems has been a weak point of QMC methods. The paper, Dielectric Response of Periodic Systems from Quantum Monte Carlo Calculations by Umari, Williamson, Galli, and Marzari remedies some of that deficiency by demonstrating a calculation of polarization using DMC.

"Programming in Math" updated

The paper on Programming in Mathematical Notation has been updated. (see previous blog entry about Programming in Math)

I've added a page about integration by random sampling. This example adds Greek characters, and a foreign function interface for the random number generator. Also, the translation of several separate sums (defining average, variance, etc) into C++ requires the sums to be merged into a single loop.

[Edit 2/24/2007 - update link]

Cluster algorithm

In the paper, Cluster algorithm for pairwise-interacting particles, Stephen Whitelam and Phillip Geissler describe an algorithm for creating a cluster of particles to move as a group.

The cluster starts with a randomly chosen particle and grows by adding particles with some probability (based on the Boltzmann factor). Particles that strongly interact with cluster are more likely to be part of the cluster. Then a normal trial move is made of the cluster as a whole.

Importantly, the potential and temperature chosen for this operation are not the same as the potential and temperature of the system. In fact, the cluster size can be controlled by adjusting the temperature (at infinite temperature each cluster is only a single particle). This cluster temperature is varied during the simulation run to sample moves with different cluster sizes.

See the paper for more details about controlling the range of energies involved in the cluster (and the cluster interface energies).