Interesting notes

While searching for Monte Carlo information on the web, I ran across Jonathon Goodman's Monte-Carlo Course Materials. One interesting part was Alan Sokal's introductory notes near the bottom (see also this CiteSeer link). I liked these notes because they covered a practical method for computing the autocorrelation time.

Optimization

Recently there were two papers in the Journal of Chemical Physics related to QMC optimization. The first, Geometry optimization in quantum Monte Carlo with solution mapping: Application to formaldehyde, is concerned with optimization of nuclear positions using only energy information, and not gradient (force) information, since it is noisier and harder to obtain. They use statistical experiment design techniques to pick coordinates at which to perform the energy computations, and fit that data to a quadratic form. The the geometry was then optimized using that fitted form.

The second paper is concerned with optimizing the one-body part of the wavefunction: Transcorrelated method for electronic systems coupled with variational Monte Carlo calculation. The authors use a transcorrelated Hamiltonian (some of the correlation has been moved into the Hamilitonian), and compute the SCF solution for the one-body orbitals. They use regular VMC to optimize the Jastrow factors. Note that the two calcuations are coupled, and to reach a solution the procedure is to alternately optimize the one-body and Jastrow factors.

Monte Carlo on the Graphics Card

A recent short paper on The Graphics Card as Streaming Computer (see related presentation slides) got me wondering if Monte Carlo could be programmed onto a graphics card.

The proposed Protein Explorer from Japan, using a GRAPE type architecture, follows a similar programming model.

One limitation is the precision of the calculations. Single precision at best, some cards only have 8-16 bits. The penalty method or related methods may be useful for correcting errors due to finite precision. (See also Ball, Fink, and Bowler)

Another limitation is the size and complexity of the code that can execute on the graphics card. This might be more useful for simple classical Monte Carlo simluations that QMC.

I suspect this falls into the category of "cool ideas that would be fun to play with, but spending the time researching better algorithms is probably more effective".

Of Wavefunctions and Wikis

and mailing lists (but that didn't alliterate nicely in the title).

There's a new QMC Mailing list in town. Be the first on your block to sign up.

Also there's a QMC Wiki (A wiki is an easily editable set of community web pages) (No link yet - you'll have to sign up for the mailing list and look at the Sept. archives to get the address)

Both of these are brought to you by Alan Aspuru-Guzik.

Linear charged particle simulations

Today's paper is by Joerg Rottler and A.C. Maggs, entitled "A Continuum,O(N) Monte-Carlo algorithm for charged particles". There are some earlier papers in the references where this method was developed for lattice systems.

As near as I can tell, the algorithm works by writing the total energy in terms of the electric field rather than the electric potential. The electric field is put on a grid. Particle moves then only require local updates to the electric field grid.

The electrostatic interaction is generated as a result of minimizing the energy functional (functional of the electric field). The algorithm they give is supposed to recover the proper interaction by a Monte Carlo means. This is the part I don't quite understand yet.

It seems like a good example of exploiting Monte Carlo to reduce the complexity of a problem by solving it on average.

Linear scaling QMC

"Linear scaling of the local energy in quantum Monte Carlo" JCP 119, 1307 by Manten and Luchow.

Cutting off the orbitals and using sparse matrix computations for the determinant, and cutting of the Jastrow factor are the time consuming parts.

One interesting point - they use a topological cutoff for the orbitals rather than a strictly spatial cutoff.

This is the scaling of the local energy with the number of electrons - I'm curious how the correlation time varies with system size.

50th Anniversary of the Metropolis Algorithm

Last week I attended a conference on the 50th Anniversary of the Metropolis Algorithm at Los Alamos. To give proper credit to all the authors, it should be the Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (MRRTT) method, but most people just call it the "Metropolis method" (or algorithm).

I enjoyed meeting people at the conference, learning more about various techniques (look for future posts), and visiting New Mexico (my first time there).

One of the sessions was a history of the Metropolis method, and Marshall Rosenbluth gave a talk. It was good to hear from one of the original authors.