Applicability of Quasi-Monte Carlo for lattice systems

Transcription

1 Applicability of Quasi-Monte Carlo for lattice systems Andreas Ammon 1,2, Tobias Hartung 1,2,Karl Jansen 2, Hernan Leovey 3, Andreas Griewank 3, Michael Müller-Preussker 1 1 Humboldt-University Berlin, Physics Department, 2 NIC, DESY, Zeuthen, 3 Humboldt-University Berlin, Mathematics Department July st International Symposium on Lattice Field Theory. corresponding publication: arxiv: , (in referee process with CPC)

2 Outline Motivation The (An)Harmonic Oscillator on the Lattice Results Outlook/Conclusions

3 typical lattice problem: Z = Dx e S[x] ; x = (x 1,..., x d ) (1) O = Z 1 Dx e S[x] O[x] (2) stochastic approximation through Markov chain Monte Carlo methods: Metropolis algorithm, HMC,... finite Markov chain: x 1,..., x N N samples of O: O 1,..., O N O i random variables with variance σo 2 estimate O = 1 N N i=1 O i has standard error O = σ O N need 100 times more statistics to get additional digit of precision past improvements: reduce σ O and auto-correlation Improved error scaling would be highly desirable!

4 improved error scaling? quasi-monte Carlo (QMC) is an approach to improve the asymptotic error behaviour see for example F. Kuo, Ch. Schwab and I. Sloan, 2012[KSS12] construction of deterministic low-discrepancy point-sets in arbitrary many dimensions low-discrepancy more uniform (see below) promises N 1 asymptotic error behaviour for integrands with certain properties (e.g. Gaussian) two times more digits with the same number of samples!! applied successfully to financial problems (see bibliography)

5 QMC point sets are more uniform How does an actual uniform sampling in two dimensions look like? Example: 512 two-dimensional pseudo-random points pseudo random 2d point set histogram of counts count frequency sample 512 points introduce grid of 8 8 equal squares count number of points in each square count occurrence of 1, 2,... points in a square (histogram of histogram) Poisson distribution with λ = n = 8 uneven sampling larger stochastic error

6 QMC point set (2d Sobol samples) : quasi Monte Carlo point set (Sobol) histogram of counts count frequency Figure: 512 uniform 2d Sobol points each square contains same number of points delta distribution even coverage less stochastic fluctuations simulate effect of higher statistics with much less samples in this sense QMC is exactly what we want randomisation possible (RQMC) w/o changing properties practical error estimation

7 problem description lattice action (see Creutz and Freedman [CF81]): d ( ) M0 (x i+1 x i ) 2 S = a + µ2 2 a 2 2 x i 2 + λxi 4 i=1 M 0... particle mass µ 2 = M 0ω 2... frequency/spring constant a... lattice spacing d... number of lattice sites T = da... time extent ; x d+1 = x 1 (p.b.) λ = 0 harmonic oscillator λ > 0 anharmonic oscillator, µ 2 < 0 double well potential Anharmonic potential µ 2 > 0 Anharmonic potential µ 2 < 0 Figure: two cases for the anharmonic potential

8 observables primary observables derived quantities x 2 = 1 xi 2 (3) d i x 4 = 1 xi 4 (4) d i x k x k+j = 1 x i x i+j... correlator (5) d i E 0 = 3λ x 4 + µ 2 x 2 + µ4 16 (6) E 1 E 0 = energy gap from correlator fit (7) theoretically known for a 0, T = da (iterative method) Blankenbecler, DeGrand and Sugar 1980 [BDS80]

9 experiment I: harmonic oscillator (λ = 0, µ 2 > 0 ) partition function can be written as multivariate Gaussian integral ( Z = Dx exp 1 ) 2 x t C 1 x C 1 = 2M0 a ((1 + a2 µ 2 )δ ij 1 ) 2M 0 2 (δ ij+1 + δ ij 1 ) covariance matrix: C = SDS t D = diag(β 1,..., β d ) β i R + ( x = Sw Z Dw exp ) 1 wi 2 2β i i (8) (9) (10) sampling algorithm generate uniform z [0, 1] d (pseudo random / QMC ) w i = β i Φ 1 (z i ), Φ 1... inverse standard normal CDF ordering of eigenvalues β 1 > β 2 >... > β d when using QMC like ordering of importance z 1 > z 2 >... > z d x i = S ij w j ( Hartley transformation, involutive: S = S 1 = S t )

10 harmonic oscillator results parameters: µ 2 = 2.0, M 0 = 0.5, a = 0.5 & d = 100 Error of <x 2 > for the Harmonic Oscillator Fit QMC, α = Error of <x 2 > 1e 07 1e 05 1e 03 Mc MC Rand. QMC 10 i N 10 i N i = 5 : 5 <X 2 > χ 2 = ; dof = Number of samples N log(n) Figure: left: asymptotic error behaviour of MC/QMC, right: fit of QMC error N α QMC at work trivial, but successful application to physical problem

11 experiment II: anharmonic oscillator (λ = 1, µ 2 < 0 ) direct sampling not possible because of anharmonic part of the potential do reweighting ( Z = Dx exp 1 2 xt C 1 x aλ ) xi 4 i (11) ( C 1 indefinite (µ 2 < 0) define C 1 sim = 2M 0 (1 + µ a 2 sim with µ 2 sim > 0, arbitrary insert the productive 0 Z = = Dx exp 1 2 xtc 1 sim x }{{} sampling 1 2 xt (C 1 C 1 sim )x aλ i a 2 )δ 2M 0 ij 1 ( ) ) δij+1 + δ 2 ij 1 xi 4 } {{ } reweighting (12) Dx e 1 2 xt C 1 sim x W (x) (13) sampling like harmonic oscillator but with C C sim observable estimation from samples (x j ) j=1,...,n : j O W (xj )O(x j ) j W W (x) = e (xj )

12 numerical results/anharmonic oscillator parameters: M 0 = 0.5, a = 0.015, µ 2 = 16 fit: O CN α O α log C χ 2 /dof d = 100 X (8) 2.0(1) 7.9 / 6 X (8) 4.0(1) 13.2 / 6 E (9) 4.0(1) 8.3 / 6 d = 1000 X (14) 2.0(2) 5.0 / 4 X (14) 4.0(2) 5.7 / 4 E (13) 4.0(2) 4.0 / 4 Error of <X 2 >, d = 1000 Error of <X 4 >, d = 1000 Error of E 0, d = 1000 <X 2 > 5e 04 2e <X 4 > E N N N ( arxiv: , K. Jansen, H. Leovey, A. Ammon, A. Griewank, M. Müller-Preußker, 2013[JLA + 13])

13 energy gap asymptotic behaviour of correlator non-trivial observable not possible to detect on present parameter setup (T too small) changed µ 2 = 16 µ 2 = 4, energy gap: result obtained for d = 100, N = 2 5, 2 8, 2 11, 2 14 and 400 Sobol sequences each: α = 0.735(13) (Tobias Hartung, 2013, personal communication)

14 outlook & conclusions harmonic oscillator: QMC works perfectly (as expected) anharmonic oscillator: significantly improved error scaling N 3 4 remaining questions: Why do we observe this N 3 4 behaviour?? further improvements by generalised choice of C sim? other, possibly non-gaussian, sampling methods next step: one-dimensional spin model in cosine discretisation S[φ] = Ia i 1 a 2 cos(φ i+1 φ i ) (14) study χ Q (topological susceptibility) and E = E 1 E 0 (energy gap)

15 R. Blankenbecler, T. A. DeGrand and R. Sugar, Moment Method for Eigenvalues and Expectation Values, Phys.Rev. D21, 1055 (1980). M. Creutz and B. Freedman, A STATISTICAL APPROACH TO QUANTUM MECHANICS, Annals Phys. 132, 427 (1981). P. Glasserman, Monte Carlo methods in financial engineering, volume 53 of Applications of Mathematics (New York), Springer-Verlag, New York, 2004, Stochastic Modelling and Applied Probability. K. Jansen, H. Leovey, A. Ammon, A. Griewank and M. Müller-Preußker, Quasi-Monte Carlo methods for lattice systems: a first look, (2013), F. Kuo, C. Schwab and I. Sloan, Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond, ANZIAM Journal 53(0) (2012).