Accelerated Metropolis Method

General discussion of the Cambridge quantum Monte Carlo code CASINO; how to install and setup; how to use it; what it does; applications.
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Accelerated Metropolis Method

Post by Vladimir_Konjkov »

Hello all

It is well known that for effective DMC calculation one need, among other things, to choose the optimal time-step. Fort all-electron atomic systems, the linear regime to occur for time steps less than of the order
τ= 1/(3Z^2), where Z is the largest atomic number occurring in the system.
If we have one heavy atom in the molecule optimal time-step have to be very small also for all other atoms.

However, a solution was proposed for this problem by C.J.Urigar ... ett.71.408 - an accelerated Metropolis algorithm, wherein each electron attempts moves that are proportional to its distance from the nearest nucleus.

Further improvement and generalization of this method was proposed in the article

As far as I noticed in CASINO, none of the two approaches described above is implemented.
How difficult to implement one of and whether there are any plans for this?

Best Vladimir.
In Soviet Russia Casino plays you.
Neil Drummond
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Re: Accelerated Metropolis Method

Post by Neil Drummond »

Dear Vladimir,

I played around with "accelerated" Metropolis VMC methods in about 2002 and concluded that one can easily come up with transition probability densities that require fewer Monte Carlo iterations to achieve a given error bar; however, none of these methods was actually more efficient than the current default VMC scheme in CASINO (electron-by-electron moves, Gaussian transition probability density with decorrelation period corper~=3), because the latter only requires wave-function values rather than derivatives, except when energies are calculated, and hence the CPU time per iteration is relatively small. If I recall, the use of a decorrelation period wasn't considered in the papers you mentioned; this is key for making the Gaussian sampling more efficient in practice.

A simple "accelerated Metropolis" VMC scheme that is readily available in CASINO is to use the DMC Green's function as a transition probability density. Perform a "DMC" calculation with branching disabled (set ibran=F in the input file; set use_tmove=F if you are using pseudopotentials; make the "DMC" time step dtdmc as large as you please). If you play with the time step you should find that it is possible to reduce the number of steps required to achieve a given error bar compared to the Gaussian transition probability density; however, the computational expense will be larger (for the energy at least - I didn't look at other expectation values).

Pablo has also looked into this more recently (focusing on backflow) and reached the same conclusions.

Best wishes,

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