DMC timestep bias asymptotic

[Neil Drummond]

In these cases, instead of fitting a+b.tau^(1/2)+c.tau... to the DMC energy, it might be better to fit (a+b.tau+c.tau^2)/(1+D.tau) to the energy against time step tau. This describes a crossover between two different linear bias regimes.

Best wishes,

Neil.

[Vladimir_Konjkov]

In these cases, instead of fitting a+b.tau^(1/2)+c.tau... to the DMC energy, it might be better to fit (a+b.tau+c.tau^2)/(1+D.tau) to the energy against time step tau. This describes a crossover between two different linear bias regimes.

Best wishes,

Neil.

woohoo excellent!

timestep bias

be2.png (7.78 KiB) Viewed 101065 times

but this is very bad news for me, because the correlation length is where the red line, and 1/(3*Z^2) is where the black.

[Neil Drummond]

Thanks for sending the graph. I think I'll stick to plane waves and pseudopotentials...

Best wishes,

Neil.

[Vladimir_Konjkov]

Thanks for sending the graph. I think I'll stick to plane waves and pseudopotentials...

Best wishes,

Neil.

Hello Neil

I want to make one very important comment for that fitting

E = (a+b.tau+c.tau^2)/(1+D.tau)

Where:
a is E at (tau->0)
b is slope at (tau->0)
c/d is slope at (tau->inf)

I interpolated all my data with this equation and got better or comparable to other eqs (within the error) results, but when the number of points at small tau is insufficient to determine the b value because error of b is compatible with its abs value, we can effectively set b=0 and then reinterpolate the data with simple equation. In this case we can nevertheless determine a with a small enough error and find out a parabolic-like dependence between E and tau at small tau.
This also explains many of the previously published E(tau) dependencies.

Best Vladimir.

[Neil Drummond]

Dear Vladimir,

Thanks very much for investigating this.

The slope at tau=0 is b-D.a, so setting b=0 doesn't get rid of the slope at tau=0 (but I can nevertheless believe that setting b=0 gives a reasonable fitting function for the situation you describe).

Best wishes,

Neil.

[Vladimir_Konjkov]

If you look at Fig. 13 of Umrigar, Nightingale, Runge, J. Chem. Phys. 1993, you will see that it is nearly quadratic near tau=0.
I do not know of a theoretical argument for the behaviour of the time-step error.
What is the x-scale on your plots?

Cyrus

Hello Cyrus and everybody usnig QMC methods. In the process of implementing DMC in my program I would like to clarify way to calculate effective time step.
As described in the above link UNR (24) and also in Casino Manual 13.5.1 The effective time step (60):
Teff = averages are over all attempted moves of the electrons i in configuration α at time step m with weights of acceptance probability of the electron move. Then Teff used in UNR (23) to calculate branching factor of particular configuration.
What formula is required to calculate Teff in the case configuration-by-configuration sampling CBCS? In my guess Teff simply equals T in this case.

Also Teff using in timedependent normalization constant - Et, UNR (11) or Casino manual (58) whether it is necessary in this case to use Teff and by what formula to calculate it in case of CBCS?

In my calculations I do not use the weighted DMC so after branching each configuration has a weight = 1. In this way I replace averaging with weights by averaging after branching.

Another question about configuration that crossed the nodal surface and it`s move is rejected. Does acceptance probability UNR (22) for this configuration be set to 0 in UNR (24)?

Best Vladimir.

[Neil Drummond]

Dear Vladimir,

For configuration-by-configuration DMC (in CASINO at least), the effective time step for any particular configuration that has just moved is tau_eff=p.tau, where p is the configuration-move acceptance probability and tau is the real time step.

The mean effective time step tau_EFF=<tau_eff>, used when setting the reference energy E_T, is evaluated like any other expectation value.

CASINO does not use weighted DMC by default. It can be switched on by setting lwdmc to T.

Even in unweighted DMC, one still has to calculate move-acceptance probabilities.

The move acceptance probability for a node-crossing move is zero if complex_wf=F. In practice very few moves are rejected due to node-crossing attempts, and it doesn't really seem to matter what one does with node-crossing attempts. If complex_wf=T then we don't bother with this at all.

Hope this helps. Best wishes,

Neil.

[Vladimir_Konjkov]

thanks a lot Neil for your answer.