Disconnected diagrams with multi-level integration Leonardo Giusti, Tim Harris , Alessandro Nada, Stefan Schaefer Lattice 2018, 23.07.18
Outline 1 Motivation 2 Variance reduction for disconnected diagrams 3 Disconnected vector two-point function with multi-level Variance reduction 2 / 14 ■
Motivation Quark-line disconnected diagrams appear when we consider singlet fermion bilinears � flavour singlet currents, e.g. HVP � isosinglet channels in spectroscopy, e.g. f 0 � hadronic matrix elements, e.g. nucleon σ -term � quark condensates These are usually evaluated with a noisy estimator e.g. Hutchinson trace � � � � �� η † ( x ) Γ D − 1 ( x , y ) η ( y ) ... ψ ( x ) Γ ψ ( x ) ... E η (1) = − where η ( x ) are independent random white noise The variance, σ 2 η , of the estimator N tr Γ D − 1 ≈ 1 � η † n ( x ) Γ D − 1 ( x , y ) η n ( y ) (2) N n = 1 is determined by off-diagonal elements η = 1 � � 2 � σ 2 � Γ D − 1 ( x , y ) � � + ... (3) � N x , y x �= y 0Hutchinson, “A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines”; Bernardson, McCarty, and Thron, “Monte Carlo methods for estimating linear combinations of inverse matrix entries in lattice QCD”. Variance reduction 3 / 14 ■
Variance reduction – numerical tests For certain currents, we need to compute a difference, e.g. EM current � � D − 1 light − D − 1 η † Γ η (4) strange whose variance is suppressed due to the covariance between light and strange Investigate using CLS N f = 2 O( a )-improved Wilson fermions E5 ensemble with n 0 = 30 configurations and study the saturation of the variance with N for � pseudoscalar � vector current id a (fm) m PS (MeV) am q κ m MS ( 2GeV ) (MeV) N GCR light 0.0658 450 0.0056 0.13625 32 25 strange 0.0175 0.135808 100 19 Variance reduction 4 / 14 ■
Variance reduction – mass differences As expected variance is reduced in the differ- ence Furthermore. . . pseudoscalar standard light one-end light-strange standard light-strange Similar to the ‘one-end’ trick for TM, we can 10 5 use for light-strange difference D − 1 − D − 1 = D − 1 � D s − D l � D − 1 10 4 s s l l = ( m s − m l ) D − 1 D − 1 , s l variance 10 3 to write a new estimator for the trace tr Γ ( D − 1 − D − 1 ) ≈ ( m s − m l )tr( Γ D − 1 ηη † D − 1 10 2 ). s s l l Note that sample-wise 10 1 10 0 10 1 10 2 10 3 η † Γ ( D − 1 − D − 1 ) η �= ( m s − m l )tr( Γ D − 1 ηη † D − 1 ). s s number of quadratures l l � relevant for HVP � m s − m l smaller than at physical point Variance reduction 5 / 14 ■
Variance reduction – mass differences As expected variance is reduced in the differ- ence Furthermore. . . vector standard light one-end light-strange Similar to the ‘one-end’ trick for TM, we can standard light-strange use for light-strange difference 10 5 D − 1 − D − 1 = D − 1 � D s − D l � D − 1 10 4 s s l l = ( m s − m l ) D − 1 D − 1 10 3 , s l variance 10 2 to write a new estimator for the trace 10 1 tr Γ ( D − 1 − D − 1 ) ≈ ( m s − m l )tr( Γ D − 1 ηη † D − 1 ). s s l l 10 0 Note that sample-wise 10 − 1 η † Γ ( D − 1 − D − 1 ) η �= ( m s − m l )tr( Γ D − 1 ηη † D − 1 10 0 10 1 10 2 10 3 ). s s l l number of quadratures � relevant for HVP � m s − m l smaller than at physical point Variance reduction 5 / 14 ■
Variance reduction – frequency splitting Perform a frequency splitting of the estimator to separate the UV and IR tr Γ D − 1 = tr Γ ( D − 1 − D − 1 1 ) + tr Γ ( D − 1 − D − 1 2 ) + ... + tr Γ D − 1 (5) n 0 0 1 where we compute the differences with noisy estimator for the ‘one-end’ trick tr Γ ( D − 1 − D − 1 ) ≈ ( m j − m i ) η † D − 1 Γ D − 1 (6) η i j j i and use the hopping parameter expansion to order k for the largest mass k − 1 � tr Γ D − 1 B − m H m B − 1 + ξ † Γ B − k H k D − 1 ≈ tr Γ n ξ (7) n m = 0 � �� � � �� � remainder hopping where B = 4 + m + σ · F contains the clover term � use hierarchical probing or spatial dilution to compute hopping contribution exactly with finite quadratures � remainder computed using standard noisy estimator 0Hasenbusch, “Exploiting the hopping parameter expansion in the hybrid Monte Carlo simulation of lattice QCD with two degenerate flavors of Wilson fermions”. 0Stathopoulos, Laeuchli, and Orginos, “Hierarchical probing for estimating the trace of the matrix inverse on toroidal lattices”. Variance reduction 6 / 14 ■
Variance reduction – frequency splitting id am q κ m MS ( 2GeV ) (MeV) N GCR m 0 0.0056 0.13625 32 25 m 1 0.0876 0.133272 500 11 m 2 0.34 0.125 1914 5 pseudoscalar, k = 4 standard light doublet hopping Contributions depend on the intermediate exact hopping heavy doublet remainder masses, order of the hopping parameter ex- 10 5 pansion and channel 10 4 Can be tuned with just a few noise samples 10 3 variance Hierarchical probing computes the exact es- 10 2 timator for the hopping H k − 1 after k 4 / 8 10 1 quadratures 10 0 Spatial dilution with even-odd blocks does 10 3 10 4 10 5 10 6 the same, also for k �= 2 m cost (core-s) Variance reduction 7 / 14 ■
Variance reduction – frequency splitting id am q κ m MS ( 2GeV ) (MeV) N GCR m 0 0.0056 0.13625 32 25 m 1 0.0876 0.133272 500 11 m 2 0.34 0.125 1914 5 pseudoscalar, k = 8 standard light doublet hopping Contributions depend on the intermediate exact hopping heavy doublet remainder masses, order of the hopping parameter ex- 10 5 pansion and channel 10 4 Can be tuned with just a few noise samples 10 3 variance Hierarchical probing computes the exact es- 10 2 timator for the hopping H k − 1 after k 4 / 8 10 1 quadratures 10 0 Spatial dilution with even-odd blocks does 10 3 10 4 10 5 10 6 the same, also for k �= 2 m cost (core-s) Variance reduction 7 / 14 ■
Variance reduction – frequency splitting id am q κ m MS ( 2GeV ) (MeV) N GCR m 0 0.0056 0.13625 32 25 m 1 0.0876 0.133272 500 11 m 2 0.34 0.125 1914 5 vector, k = 4 Contributions depend on the intermediate standard light doublet hopping exact hopping heavy doublet remainder masses, order of the hopping parameter ex- 10 5 pansion and channel 10 4 Can be tuned with just a few noise samples 10 3 variance Hierarchical probing computes the exact es- 10 2 timator for the hopping H k − 1 after k 4 / 8 10 1 quadratures 10 0 Spatial dilution with even-odd blocks does 10 − 1 the same, also for k �= 2 m 10 3 10 4 10 5 10 6 cost (core-s) Variance reduction 7 / 14 ■
Variance reduction – frequency splitting id am q κ m MS ( 2GeV ) (MeV) N GCR m 0 0.0056 0.13625 32 25 m 1 0.0876 0.133272 500 11 m 2 0.34 0.125 1914 5 vector, k = 8 Contributions depend on the intermediate standard light doublet hopping exact hopping heavy doublet remainder masses, order of the hopping parameter ex- 10 5 pansion and channel 10 4 Can be tuned with just a few noise samples 10 3 variance Hierarchical probing computes the exact es- 10 2 timator for the hopping H k − 1 after k 4 / 8 10 1 quadratures 10 0 Spatial dilution with even-odd blocks does 10 − 1 the same, also for k �= 2 m 10 3 10 4 10 5 10 6 cost (core-s) Variance reduction 7 / 14 ■
Variance reduction – conclusions 10 5 pseudoscalar, k = 4 standard hierarchical probing splitting estimator 10 4 variance Gain of ∼ 4 in the cost for the pseudoscalar, 10 3 vector Similar to hierarchical probing for pseu- 10 2 doscalar, faster for vector 10 2 10 3 10 4 10 5 10 6 10 7 cost (core-s) Parameters can be further optimized 10 5 pseudoscalar, k = 8 Expect efficient estimator at smaller quark standard mass hierarchical probing splitting estimator 10 4 Scalar and axial vector similar to pseu- variance doscalar and vector, respectively 10 3 10 2 10 2 10 3 10 4 10 5 10 6 10 7 cost (core-s) Variance reduction 8 / 14 ■
Variance reduction – conclusions 10 5 vector, k = 4 standard 10 4 hierarchical probing splitting estimator 10 3 variance 10 2 Gain of ∼ 4 in the cost for the pseudoscalar, vector 10 1 Similar to hierarchical probing for pseu- 10 0 doscalar, faster for vector 10 2 10 3 10 4 10 5 10 6 10 7 cost (core-s) Parameters can be further optimized 10 5 vector, k = 8 Expect efficient estimator at smaller quark standard 10 4 mass hierarchical probing splitting estimator Scalar and axial vector similar to pseu- 10 3 variance doscalar and vector, respectively 10 2 10 1 10 0 10 2 10 3 10 4 10 5 10 6 10 7 cost (core-s) Variance reduction 8 / 14 ■
Recap domain decomposition Ω ∗ Ω ∗ 1 As seen in talk by A. Nada, there exists a representa- 0 tion �� � � � � O 0 ( x ) O 1 ( x ) 1 W N 0 N O 1 ( x ) O 2 ( y ) 〈 O 0 ( x ) O 1 ( y ) 〉 QCD = � W N � N where � [ dU Λ 0 ] e − S G [ U Λ 0 ] det Q Ω ∗ Λ 0 Λ 1 Λ 2 � � O 0 ( x ) 0 = 0 O 0 ( x ) U ∈ Λ 0 A factorizable multi-level estimator has a variance n 1 level-1 1 σ 2 ∝ n 0 n 2 1 To investigate multi-level scaling, our estimator’s vari- ance should be dominated by the gauge noise n 0 level-0 0Cè, Giusti, and Schaefer, “A local factorization of the fermion determinant in lattice QCD”. Variance reduction 9 / 14 ■
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