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).

Programming in Math

In a previous post on Solving science problems, I commented on the process of scientific computation. One of the steps was converting the solvable equations into a computer program.

As part of making that step more transparent, I've been working on method for Programming in Mathematical Notation (works best with MathML capable browser).

The program takes a content MathML file as input and outputs a presentation MathML file for nicely rendered display in the browser, and a C++ file to compile and run.

[Edit 2/24/2007 - update link to example]

Fast variance optimization

VMC variance optimization usually uses the reweighted variance.

What is reweighting? Consider a set of particle positions obtained from sampling a wavefunction at a given set of parameters. The energy (and the variance of the energy) explicitly depends on the wavefunction parameters by the local energy formula, and implicitly through the particle positions determined by sampling the wavefunction. When optimizing, one would like to vary the parameters without going through all the trouble and expense of sampling new positions. Computing the energy (or variance) change due to the local energy piece is easy - just put the new parameters in and recalculate. Reweighting is the process of correcting the implicit dependence of the energy (or variance) on the wavefunction parameters.


N. D. Drummond and R. J. Needs describe a VMC optimization method in their paper, A variance-minimization scheme for optimizing Jastrow factors, that uses only the explicit dependence of the variance on the parameters (ie, they use the un-reweighted variance). They restrict their scheme to linear parameters in the Jastrow factor, and this makes the resulting parameter surface a quartic. Also, this scheme only requires a single sum over electron positions (the requisite factors can be accumulated during a single run), making the optimization step very fast.

Even though this scheme uses a less accurate approximation for the variance, it appears to give very good results. In fact, the resulting energies for a neon atom are better using the unreweighted variance for optimization than from using the more traditional reweighted variance.