VMC sampling efficiency

BackgroundPractical tests

Summary and recommendations

Pablo Lopez Rıos

TCM group. Cavendish Laboratory. University of Cambridge.

July 25, 2010

Pablo Lopez Rıos VMC sampling efficiency

VMC samplingCommon modifications

The VMC algorithm

In VMC we sample configurations {R1, . . . ,RM} distributedaccording to |Ψ(R)|2

We evaluate the variational energy as EVMC = 1M ∑

Mm=1 EL(Rm)

This energy has an uncertainty given by ∆ = σ√M/ncorr

σ2 is the variance of the sample of local energies, whichdepends on Ψncorr is the (integrated) correlation length of the sample oflocal energies, which depends on how we sample configurations

A VMC calculation is more efficient the less time it takes toachieve a target errorbar: E =

(∆2MTiter

)−1=

(σ2ncorrTiter

It is inefficient to attempt to maximize this directly withrespect to any parameter due to the multiple evaluations ofncorr that this would require

The VMC algorithm

Mm=1 EL(Rm)

(∆2MTiter

)−1=(σ2ncorrTiter

The VMC algorithm

Mm=1 EL(Rm)

(∆2MTiter

The VMC algorithm

Mm=1 EL(Rm)

(∆2MTiter

The VMC algorithm

Mm=1 EL(Rm)

(∆2MTiter

VMC sampling

{Rm}m=1,...,M are generated using the Metropolis algorithm:

Propose move from Rm to R′m with probability T(R′m← Rm)

Compute A(R′m← Rm) = min(

1, T(Rm←R′m)T(R′m←Rm)

|Ψ(R′m)|2|Ψ(Rm)|2

)Draw random number 0 < ζ < 1 from a uniform distribution,and

If ζ < A(R′m← Rm), make Rm+1 = R′m (accept move)Otherwise, set Rm+1 = Rm (reject move)

To achieve reasonable acceptance ratios, proposedconfigurations are the original plus a normally-distributedrandom displacement of variance τ

This causes serial correlation (ncorr > 1)

VMC sampling

|Ψ(R′m)|2|Ψ(Rm)|2

VMC sampling

|Ψ(R′m)|2|Ψ(Rm)|2

VMC sampling

|Ψ(R′m)|2|Ψ(Rm)|2

VMC sampling

|Ψ(R′m)|2|Ψ(Rm)|2

VMC sampling

|Ψ(R′m)|2|Ψ(Rm)|2

Electron-by-electron sampling

It is possible to use a variation of the Metropolis algorithmwhere one proposes single-electron moves and accepts orrejects them individually

Advantage: larger steps can be taken with high acceptanceratios, thus reducing ncorr

Disadvantage: the evaluation of N single-electronwave-function ratios is more expensive than that of oneall-electron wave function ratio, and especially for complicatedfunctional forms (e.g., Slater determinants with backflowtransformations), which increases Titer

Decorrelation loops

One can perform p > 1 Metropolis steps between evaluationsof the local energy

Advantage: ncorr decreases

Disadvantage: the extra moves increase Titer

Decorrelation loops

Averaging successive local energies

The mth local energy can be replaced by the average[1−A(R′m← Rm)]EL(Rm) + A(R′m← Rm)EL(R′m)

Advantage: more statistics, especially important at lowacceptance ratios, potentially reducing ncorr

Disadvantage: needs more energy evaluations, increasing Titer

This has proved inefficient in electron-by-electron sampling, sowill only test in configuration-by-configuration sampling

MethodologyTest resultsFunctional form of E (p)

Things to look into

Optimal value of τ?

Electron-by-electron versus configuration-by-configuration -which to use when?

Decorrelation loops - optimal length?

Is averaging energies over proposed configurations useful?

Things to look into

Methodology

Choose 6 relevant systems of different sizes

Run short (but significant) VMC calculations spanning 16values of τ and 10 values of p

Run electron-by-electron and configuration-by-configurationversions of the above, the latter with and without averagingover successive energies

Use Slater-Jastrow and Slater-Jastrow-backflow wave functionforms

Total: 5760 runs

Use the data to locate maximum efficiency for each case,compare, analyze, etc

Methodology

Total: 5760 runs

Methodology

Total: 5760 runs

Methodology

Total: 5760 runs

Methodology

Total: 5760 runs

Methodology

Total: 5760 runs

Pseudo Nitrogen atom, Slater-Jastrow, EBES vs CBCS

0.001 0.01 0.1 1 10 100Timestep

0.001 0.01 0.1 1 10 1000

0.010.020.030.040.050.06

0.001 0.01 0.1 1 10 1000

r. tim

0.001 0.01 0.1 1 10 1000

1.5e+05

Assorted quantities vs VMC timestepSystem: nitrogen_pp. Sampling method: ebe.

0.0001 0.001 0.01 0.1 1Timestep

0.0001 0.01 10

0.0001 0.001 0.01 0.1 10

r. tim

0.0001 0.01 10

Assorted quantities vs VMC timestepSystem: nitrogen_pp. Sampling method: cbc.

HEG, Slater-Jastrow, EBES vs CBCS

0.001 0.01 0.1 1 10 100Timestep

0.001 0.01 0.1 1 10 1000

0.050.1

0.150.2

0.001 0.01 0.1 1 10 1000

r. tim

0.001 0.01 0.1 1 10 1000

50010001500200025003000

Assorted quantities vs VMC timestepSystem: heg. Sampling method: ebe.

0.0001 0.001 0.01 0.1 1Timestep

0.0001 0.001 0.01 0.1 10

0.0005

0.0015

0.0001 0.001 0.01 0.1 10

r. tim

0.0001 0.001 0.01 0.1 10

100200300400500600

Assorted quantities vs VMC timestepSystem: heg. Sampling method: cbc.

Pseudo NiO molecule, backflow, EBES vs CBCS

0.001 0.01 0.1 1 10 100Timestep

0.001 0.01 0.1 1 10 1000

0.010.020.030.040.050.06

0.001 0.01 0.1 1 10 1000

r. tim

0.001 0.01 0.1 1 10 1000

50100150200250300

Assorted quantities vs VMC timestepSystem: nio_pp. Sampling method: ebe. Wfn: bf.

0.0001 0.001 0.01 0.1 1Timestep

0.0001 0.001 0.01 0.1 10

0.0005

0.0015

0.0001 0.001 0.01 0.1 10

r. tim

0.0001 0.001 0.01 0.1 10

Assorted quantities vs VMC timestepSystem: nio_pp. Sampling method: cbc. Wfn: bf.

All-electron N2H4, backflow, CBCS vs CBCS2

0.0001 0.001 0.01Timestep

0.0001 0.001 0.010

0.00010.00020.00030.00040.00050.00060.0007

0.0001 0.001 0.010

r. tim

0.0001 0.001 0.010

50100150200250300

Assorted quantities vs VMC timestepSystem: n2h4. Sampling method: cbc. Wfn: bf.

0.0001 0.001 0.01 0.1Timestep

0.0001 0.001 0.01 0.10

0.00010.00020.00030.00040.00050.00060.0007

0.0001 0.001 0.01 0.10

r. tim

0.0001 0.001 0.01 0.10

Assorted quantities vs VMC timestepSystem: n2h4. Sampling method: cbc. Wfn: bf.

Functional form of E (p)

Cost of one energy evaluation: Titer(p) = pTmove + Tenergy

Assuming M→ ∞, and that the autocorrelation of the localenergies is dominated by a single exponential,

ncorr(p) = 1 + 2 (ncorr−1)p

(ncorr+1)p−(ncorr−1)p

One can minimize Titer(p)ncorr(p) numerically if ncorr andTenergy/Tmove are know.

Use electron-by-electron sampling

Optimize τ so as to achieve a 50% acceptance ratio

Set p to 3-5, or compute ncorr from a short run and maximizeE numerically

Do not average over successive energies

We’ve been doing it right all along!

Use electron-by-electron sampling

Optimize τ so as to achieve a 50% acceptance ratio

Set p to 3-5, or compute ncorr from a short run and maximizeE numerically

Do not average over successive energies

We’ve been doing it right all along!

VMC sampling efficiency

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