Random Estimates in QMC

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1 Page 1 Random Estimates in QMC J.R. Trail and R.J. Needs Theory of Condensed Matter Group, Cavendish Laboratory, University of Cambridge, UK February 2007

2 Page 2 Numerical Integration Unkown function, r sample points 1 1 f(x) 0.5 f(x) x x Evenly sampled grid: 1 0 f(x)dx a n f(x n ), ǫ p ( 1 r) p/d Points sampled randomly, with PDF P(x): 1 0 f(x)dx 1 r f(xn )/P(x n ), e 1 r

3 Page 3 Sample 3N dimensional space with PDF P(R) Quantum Monte Carlo (VMC) Est[E tot ] = ψ 2 E L /P ψ2 /P = ψ Ĥ ψ + Y ψ ψ + X where E L = ψ 1 Ĥψ. Simplest case is Standard Sampling : Choose P(R) = Aψ(R) 2, then Est[E tot ] = 1 r EL = ψ Ĥ ψ ψ ψ + W W is the random error in a sum of random variables, so what is its distribution? IF the CLT is valid then it is Gaussian with mean 0, and variance σ/r 1/2.

4 Page 4 3N 1 dimension Why?: Easier to deal with the general case analytically A change of the random variable from spatial to energy: E tot = = V ψ 2 E L d 3N R/ P ψ 2(E)EdE V ψ 2 d 3N R with P ψ 2(E) = E=E L P(R) R E L d3n 1 R A histogram of E L approximates the seed PDF P ψ 2 R E L results from curvilinear co-ordinates and change of variables. Useless numerically, but useful analytically.

5 Page 5 What can we say about P ψ 2? Singularities in the local energy: E L (R) = Rψ ψ + i<j 1 r ij i Z r i E L (R) = E tot if the trial wavefunction, ψ, is exact Enforce Kato cusp conditions no Coulomb singularities Nodal surface is ψ = 0, and is 3N 1 dimensional Kinetic energy part gives singularity on a 3N 1 dimensional surface Singularities provide information about P ψ 2 for large E

6 Page 6 What can we say about P ψ 2? P ψ 2(E) = E=E L P(R) R E L d3n 1 R S ˆn nodal surface ψ = a 1 S +... E L = b 1 S P(R)/ E L = c 4 S P ψ 2(E) = d 4 E or more completely P ψ 2(E) = (E E 0 ) 4 (e 0 + ) e 1 (E E 0 ) +... E E 0

7 Page 7 Jastrow + 48 determinants + backflow: Example: All-electron isolated Carbon atom ψ = e J(R) m a m D m (R )D m (R ) with R = R (R) 10 0 σ P ψ 2(EL) % correlation energy at VMC level (E L µ)/σ Estimated seed probability density function Also shown is 2 π σ 3 σ 4 +(E µ) 4, and a Normal distribution

8 Page 8 Random error in total energy estimate Est[E tot ] = 1 r (E E r ) Product of probability of r samples energies that add up to re tot convolution integrals P r=2 (2E tot ) = P r (re tot ) = P ψ 2 P ψ 2... P ψ 2 Take Fourier transform of P ψ 2(E) Take the r th power P ψ 2(E 1 )P ψ 2(E 2 )δ(e 1 + E 2 2E tot )de 1 de 2 = P ψ 2 P ψ 2 Take the inverse Fourier transform Rescale some variables to get the PDF of averages instead of sum P r (y) = [ η d 3 ( ) ] [ 1 λ 2π r dy + O e y2 /2 + 3 r 3π 1 d 3 ( ) ( y 1 r dy 3D + O 2 r) ] y = (E tot µ)/σ

9 Page 9 PDF of estimate of Total energy 10 0 σ/ r Pr(Etot) (E tot µ) 5 10 r/σ Approximate PDF from 10 4 estimates of total energy, with r = 10 3 For small E, PDF is dominated by 1 2π e x2 /2 For large E, PDF is dominated by CLT is true in its weakest form 2 π λ r 1/x 4 (λ 1 for Carbon trial wavefunction)

10 Page 10 PDF of estimate of the Residual Variance, v Optimisation of wavefunctions using the residual variance, v (Ĥ Etot ) ψ = δ = (EL E tot )ψ v = δ 2 dr 0, and zero for exact ψ To optimise the wavefunction v is usually minimised Analyse effect of tails, as before: P r (v) = [ ] 3 1 v σ 2 2 [ exp v σ 2 π 2γ 2γ 2γ [ sgn ] v σ 2 K 1/3 v σ 2 2γ with the width of the PDF decided by the magnitude of the tails γ = [ 6λ 2 πr ] K 2/3 v σ 2 2γ 3 ] 1/3 σ 2 (1)

11 Page 11 PDF of estimate of the Residual Variance, v γ Pr(v) (v σ 2 )/(2γ) Approximate PDF from 10 4 estimates of residual variance, with r = 10 3 Small v, e x3. Large v 1/x 5/2 PDF has no variance, γ has no vigorous statistical estimate and is r 1/3

12 Page 12 CLT is valid in its weakest form for the total energy CLT not valid for residual variance CLT is likely to be invalid for estimates of other physical quantites Because: ψ 2 samples E L rarely where it is largest, at the nodal surface Can the CLT be reinstated?

13 Page 13 Residual sampling Instead of sampling with P = Aψ 2, sample with P = Aψ 2 /w, then and the residual variance, Est[E tot ] = wel w = ψ Ĥ ψ + Y ψ ψ + X Est [ ] δ 2 dr = w(el E tot ) 2 w = ψ 2 (E L E tot ) 2 dr + Y ψ2 dr + X Choose the weighting function w(e L ) = ǫ 2 (E L E 0 ) 2 + ǫ 2 to interpolate beween sampling the numerator and denominator perfectly. No singularities, and no power law tails Quotient of two correlated random variables, each a sum of random variables

14 Page 14 Fieller s Theorem 5 µ µ 1 (µ 2, µ 1 ) that give Est = µ 2 /µ 1

15 Page 15 Fieller s Theorem 5 µ µ 1 (µ 2, µ 1 ) that give Est = µ 2 /µ 1 Ellipse containing 39% of probability from covariance matrix and bivariate CLT

16 Page 16 Fieller s Theorem 5 µ µ 1 (µ 2, µ 1 ) that give Est = µ 2 /µ 1 Ellipse containing 39% of probability from covariance matrix Wedge that contains 68.3% of probability m 1 < µ 2 /µ 1 < m 2 with confidence 68.3%

17 Page 17 Estimate of total energy Est[E tot ] Histogram of 10 3 total energy estimates, each total energy estimate from 10 3 configurations. Residual sampling (filled) and standard sampling (unfilled) are not significantly different Residual sampling reduces error by 30% For other systems standard sampling may give power law outliers (depending on λ) For all systems residual sampling does not give power law outliers

18 Page 18 Estimate of residual variance Est[v] Histogram of 10 3 residual variance estimates, each estimated from 10 3 configurations. Residual sampling and standard sampling are very different Standard sampling shows the v 5/2 tails and outliers expected Residual sampling gives well defined confidence limits for estimate via the bivariate CLT Standard sampling does not

19 Page 19 Optimisation 1) Take samples using wavefunction with parameter α 0, {R} r 2) These define random sample from a distribution of Optimisation functions, O(α) 3) Find the minimum of O(α), at α = α min 4) set α 0 = α min, and return to 1) What is the distribution of O(α)? O(α) = a 0 + a 1 (α α 0 ) +... b 0 + b 1 (α α 0 ) +... For each type of sampling/optimisation this expansion gives statistics of random error For Standard Sampling error is not normal unless the nodes are suppressed by introducing a weight function into O(α) solely for this purpose (eg g(e L )δ 2 d 3N R). For Residual sampling the error is always normal

20 Page 20 Optimisation r = 10 5 configurations. Total energy Residual variance Etot/(a.u.) Std. Res. Var δ Std. Res. E exp no. cycle no. cycle. Std. - standard sampling with nodal surface suppressed (93% E corr ) Res. - residual sampling of residual variance (95% E corr ) New sampling provides lower total energy and a lower residual variance than standard sampling with nodal surface suppressed

21 Page 21 Conclusions We cannot assume the CLT is true for estimates in standard sampling QMC r large enough must be shown to be true for each estimate in standard sampling QMC The CLT can be reinstated by using an alternative sampling strategy Random functions whose minimum gives optimum wavefunctions are not generally normally distributed. Some do not converge as r The residual sampling strategy can guarantee that the CLT is valid for estimates and optimisation functions, as long as they exist With residual sampling optimisation functions can be chosen on physical grounds - to give a good wavefunction at the nodal surface and small fixed node error in DMC Acknowledgements Financial support was provided by the Engineering and Physical Sciences Research Council (EPSRC), UK, and computational resources were provided by the Cambridge-Cranfield High Performance Computing Facility.