[PPT] - MULTILEVEL INTEGRATION FOR FERMIONS II Marco C, Leonardo Giusti, PowerPoint Presentation

SLIDE 1

DOMAIN DECOMPOSITION AND

MULTILEVEL INTEGRATION FOR FERMIONS II

Marco Cè, Leonardo Giusti, Stefan Schaefer Scuola Normale Superiore, Pisa & INFN, Sezione di Pisa

Lattice 2016, July 27th, 2016

SLIDE 2

Case I: disconnected correlation functions

[tr γ5D−1(x, x)][tr γ5D−1(y, y)]
=
x

y

Factorization:

use quark propagator factorization, case I (sink close to source) see talk by Schaefer ⇒ disconnected two point function splits in four contributions

[tr γ5D−1(x, x)][tr γ5D−1(y, y)]
=
[tr γ5D−1

Γ (x, x)][tr γ5D−1 Γ∗ (y, y)]

=
x

y

+
[tr γ5D−1

Γ (x, x)][tr γ5δD−1 Γ∗ (y, y)]

+
x

y

+
[tr γ5δD−1

Γ (x, x)][tr γ5D−1 Γ∗ (y, y)]

+
x

y

+
[tr γ5δD−1

Γ (x, x)][tr γ5δD−1 Γ∗ (y, y)]

+
x

y

1 / 19

SLIDE 3

Numerical test

Wilson plaquette action β = 6 ⇒ a ≈ 0.093 fm Wilson fermions κ = 0.1560 ⇒ mπ ≈ 450 MeV 64 × 243 lattice with open boundary conditions in time Two-level Monte Carlo: n0 = 200 level-0 configurations domain decomposition in two thick time slices n1 = 100 level-1 configurations Estimation of tr γ5D−1: Stochastic volume sources Hopping parameter expansion to reduce UV noise Note: to be computed only

2 / 19

SLIDE 4

Contributions to propagator: factorized

[tr γ5D−1

Γ (x, x)][tr γ5D−1 Γ∗ (y, y)]

+[tr γ5D−1

Γ (x, x)][tr γ5δD−1 Γ∗ (y, y)] + [tr γ5δD−1 Γ (x, x)][tr γ5D−1 Γ∗ (y, y)]

+[tr γ5δD−1

Γ (x, x)][tr γ5δD−1 Γ∗ (y, y)]

Factorized contribution ⇒ independent averaging over level-1 configs

10 20 30 40 50 60 |y0 − x0| 10−6 10−5 10−4 10−3 10−2

Cmlv

Pd

C(f)

Pd

10

20 30 40 50 60 |y0 − x0| 10−6 10−5 10−4 10−3 10−2 δ(f)

Pd

n1 = 1 n1 = 10 n1 = 100

first term carries almost no signal, only noise Note that

tr γ5D−1

Γ (x, x)]

= 0

Multilevel works at full potentiality to reduce the variance

3 / 19

SLIDE 5

Contributions to propagator: correction

[tr γ5D−1

Γ (x, x)][tr γ5D−1 Γ∗ (y, y)]

+[tr γ5D−1

Γ (x, x)][tr γ5δD−1 Γ∗ (y, y)] + [tr γ5δD−1 Γ (x, x)][tr γ5D−1 Γ∗ (y, y)]

+[tr γ5δD−1

Γ (x, x)][tr γ5δD−1 Γ∗ (y, y)]

| − | | − | 10 20 30 40 50 60 |y0 − x0| 10−6 10−5 10−4 10−3 10−2

Cmlv

Pd

C(r1)

Pd

10 20 30 40 50 60 |y0 − x0| 10−6 10−5 10−4 10−3 10−2 δ(r1)

Pd

n1 = 1 n1 = 10 n1 = 100

10 20 30 40 50 60 |y0 − x0| 10−6 10−5 10−4 10−3 10−2

Cmlv

Pd

C(r2)

Pd

10 20 30 40 50 60 |y0 − x0| 10−6 10−5 10−4 10−3 10−2 δ(r2)

Pd

n1 = 1 n1 = 10 n1 = 100

4 / 19

SLIDE 6

Summary: disconnected

10 20 30 40 50 60 |y0 − x0| 10−6 10−5 10−4 10−3 10−2

CPd Cmlv

Pd

10 20 30 40 50 60 |y0 − x0| 10−6 10−5 10−4 10−3 10−2

δPd δmlv

Pd

short-distance: error dominated by the first correction long-distance: error dominated by the factorized contribution ⇒ Factorization and multilevel integration works error decreases exponentially maintain signal for additional 1 fm

5 / 19

SLIDE 7

Case II: connected correlation functions

meson propagator

baryon propagator
Factorization less obvious. Two steps:

1

factorization of the quark propagator, case II (sink far from source) see talk by Schaefer

x y ≈ − x y +O

eMπ∆

D−1(x, y) −D−1

¯ Γ (y, ·)D∂Γ∗ D−1 Γ (·, x)

2

factorization of the hadron two-point function With the factorized propagator:

3

multilevel integration

6 / 19

SLIDE 8

Numerical test

Wilson plaquette action β = 6 ⇒ a ≈ 0.093 fm Wilson fermions κ = 0.1560 ⇒ mπ ≈ 450 MeV 64 × 243 lattice with open boundary conditions in time Test of quark propagator factorization: stochastic sources on time slice x0 = 4a first cut at x0 = 24a ∆ = 8a, 12a and 16a ⇒ second cut at 24a − ∆ Contraction of two or three quark lines: pseudoscalar propagator

x =24 Δ

nucleon propagator

x =24 Δ

7 / 19

SLIDE 9

Meson

Pseudoscalar two-point function

10 20 30 40 50 60 y0 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2

CPc C(0)

Pc

C(1)

Pc − C(0) Pc

C(2)

Pc − C(1) Pc

C(rest)

Pc

x =24 Δ

CPc = C(0)

Pc +

C(1)

Pc − C(0) Pc

+
C(2)

Pc − C(1) Pc

+ C(rest)

Pc

C(0): ∆ = 8a C(1): ∆ = 12a C(2): ∆ = 16a

8 / 19

SLIDE 10

Baryon

Nucleon two-point function

10 20 30 40 50 60 y0 10−16 10−15 10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−7 10−6

CN C(0)

N

C(1)

N

− C(0)

N

C(2)

N

− C(1)

N

C(rest)

N

x =24 Δ

CN = C(0)

N +

C(1)

N − C(0) N

+
C(2)

N − C(1) N

+ C(rest)

N

C(0): ∆ = 8a C(1): ∆ = 12a C(2): ∆ = 16a

9 / 19

SLIDE 11

Factorized propagator

Factorization of propagator in principle works small ∆ already gives excellent approximation can be improved by hierarchy of ∆i ⇒ potential for multi-level Problem: Natural building blocks have two or three propagator on surface (6V3)2 or (6V3)3 complex numbers ⇒ too much to be saved to disk

10 / 19

SLIDE 12

Projection of the propagator

x y

Cut the fermion line with P: S(y, x) = −D−1

¯ Γ D∂Γ∗·P·D−1 Γ (x, y)

P projects on lower dimensional space P =

N

i=1

ψi ψ†

i

Reduce memory of building block (6V3)2 → N2 Several possibilities: stochastic sources ⇒ does not seem to work, large N required local deflation subspace eigenmodes of block Dirac operator

11 / 19

SLIDE 13

Projections: deflation subspace

Pseudoscalar two-point function with P: deflation subspace at x0/a = 24

10 20 30 40 50 60 y0 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2

CPc ˜ C(0)

Pc

˜ C(1)

Pc − ˜

C(0)

Pc

˜ C(2)

Pc − ˜

C(1)

Pc

˜ C(rest)

Pc

10 20 30 40 50 60 y0 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2

CPc C(0)

Pc

C(1)

Pc − C(0) Pc

C(2)

Pc − C(1) Pc

C(rest)

Pc

Deflation subspace: Ns = 60 block modes, 44 blocks Lüscher ‘07 Factorization only ∆/a = 8, 12, 16 ⇒ Virtually no difference visible

12 / 19

SLIDE 14

Projections: block eigenvectors subspace

Pseudoscalar two-point function with P: eigenvectors of Dirac operator restricted to x0 ∈ [24 − ∆, 24 + ∆]

10 20 30 40 50 60 y0 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2

CPc ˜ C(0)

Pc

˜ C(1)

Pc − ˜

C(0)

Pc

˜ C(2)

Pc − ˜

C(1)

Pc

˜ C(rest)

Pc

10 20 30 40 50 60 y0 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2

CPc C(0)

Pc

C(1)

Pc − C(0) Pc

C(2)

Pc − C(1) Pc

C(rest)

Pc

Block eigenvectors subspace: Nev = 120 vectors Factorization only ∆/a = 8, 12, 16 ⇒ Virtually no difference visible

13 / 19

SLIDE 15

Projections: deflation subspace

Nucleon two-point function with P: deflation subspace at x0/a = 24

10 20 30 40 50 60 y0 10−16 10−15 10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−7 10−6

CN ˜ C(0)

N

˜ C(1)

N

− ˜ C(0)

N

˜ C(2)

N

− ˜ C(1)

N

˜ C(rest)

N

10 20 30 40 50 60 y0 10−16 10−15 10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−7 10−6

CN C(0)

N

C(1)

N

− C(0)

N

C(2)

N

− C(1)

N

C(rest)

N

Deflation subspace: Ns = 60 block modes, 44 blocks Lüscher ‘07 Factorization only ∆/a = 8, 12, 16

14 / 19

SLIDE 16

Projections: block eigenvectors subspace

Nucleon two-point function with P: eigenvectors of Dirac operator restricted to x0 ∈ [24 − ∆, 24 + ∆]

10 20 30 40 50 60 y0 10−16 10−15 10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−7 10−6

CN ˜ C(0)

N

˜ C(1)

N

− ˜ C(0)

N

˜ C(2)

N

− ˜ C(1)

N

˜ C(rest)

N

10 20 30 40 50 60 y0 10−16 10−15 10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−7 10−6

CN C(0)

N

C(1)

N

− C(0)

N

C(2)

N

− C(1)

N

C(rest)

N

Block eigenvectors subspace: Nev = 120 vectors Factorization only ∆/a = 8, 12, 16 ⇒ Surprise: also the nucleon can be saturated by low-modes

15 / 19

SLIDE 17

Summary of factorization

Factorization of pion and nucleon two-point functions in principle possible. Two approximations involved: approximate factorization of propagator projection to low-mode subspace Surprise: the nucleon propagator is saturated by low modes! ⇒ Multilevel in quenched is possible

16 / 19

SLIDE 18

Multilevel

10 20 30 40 50 60 y0 10−16 10−15 10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2

CN Cmlv

N

C(f)

N

10 20 30 40 50 60 y0 10−2 10−1 100

δmlv

N /δN

δ(f),mlv

N

/δN δ(1)−(0)

N

/δN δ(2)−(1)

N

/δN δ(rest)

N

/δN

38 40 42 44 46 48 50 10−15 10−14 10−13

Two-level Monte Carlo integration: level-0 n0 = 50 configurations level-1 n1 = 20 updates Factorized contribution with two-level algorithm Correction term with source at x0 = 4a only on level-0

17 / 19

SLIDE 19

Detailed improvement

Total Source average

10 20 30 40 50 60 y0 10 20 30 40 50 60

δ(f)

std

δ(f)

n1=20,mlv

10 20 30 40 50 60 y0 1 2 3 4 5 6 7 8

δ(f)

std

δ(f)

n1=1

10 20 30 40 50 60 y0 1 2 3 4 5 6 7 8 √ 5 √ 10 √ 20 δ(f)

n1=1

δ(f)

n1

n1 = 5 n1 = 10 n1 = 20

10 20 30 40 50 60 y0 1 2 3 4 5 6 7 8 √ 5 √ 10 √ 20 δ(f)

n1

δ(f)

n1,mlv

n1 = 5 n1 = 10 n1 = 20

Ord. n1 avg.

Two-level

18 / 19

SLIDE 20

Conclusions & Outlook

Two-level methods work: quark-line disconnected meson correlation functions

tr
Γ 1

D(x, x)

tr
Γ 1

D(y, y)

also gluonic correlation functions ¯

q(x0)¯ q(y0) Garcia-Vera, Schaefer ‘16 For quark line connected there is hope: correlation functions shown to be factorizable. two-level is only partial solution signal-to-noise require √ N ∝ emx0 − → N ∝ emx0 ⇒ Number of configurations to reach target statistics ∼ square root of the standard case

19 / 19

SLIDE 21

Conclusions & Outlook

Two-level methods work: quark-line disconnected meson correlation functions

tr
Γ 1

D(x, x)

tr
Γ 1

D(y, y)

also gluonic correlation functions ¯

q(x0)¯ q(y0) Garcia-Vera, Schaefer ‘16 For quark line connected there is hope: correlation functions shown to be factorizable. two-level is only partial solution signal-to-noise require √ N ∝ emx0 − → N ∝ emx0 ⇒ Number of configurations to reach target statistics ∼ square root of the standard case Outlook: generalize quark line factorization to multilevel factorization of the fermion determinant

19 / 19

SLIDE 22

Thanks for your attention!

SLIDE 23

Backup

SLIDE 24

Quark propagator factorization, case I

Two regions: Γ ∪ Γ∗. The Dirac operator is D = DΓ D∂Γ D∂Γ∗ DΓ∗

For the Wilson Dirac operator, D∂Γ and D∂Γ∗ act on the boundaries only.

The propagator is D−1 =

S−1

Γ

−S−1

Γ D∂ΓD−1 Γ∗

−D−1

Γ∗ D∂Γ∗S−1 Γ

S−1

Γ∗

with the Schur complements

SΓ = DΓ − D∂ΓD−1

Γ∗ D∂Γ∗

SΓ∗ = DΓ∗ − D∂Γ∗D−1

Γ D∂Γ

Dirichlet boundary conditions: D∂Γ = D∂Γ∗ = 0 D−1 ≈ D−1

Γ

D−1

Γ∗

Correction term:

δD−1

Γ

= 1 DΓ − D∂ΓD−1

Γ∗ D∂Γ∗ − 1

DΓ = 1 DΓ − D∂ΓD−1

Γ∗ D∂Γ∗ D∂ΓD−1 Γ∗ D∂Γ∗ 1

DΓ = −D−1D∂Γ∗D−1

Γ 1 / 5

SLIDE 25

Quark propagator factorization, case I

Graphically: see talk by Schaefer = − D−1 = D−1

Γ

− D−1D∂Γ∗D−1

Γ

r

x x x

+ =

Correction term δD−1

Γ

has propagator to and from boundary Tr γ5δD−1

Γ

∝ e−mπ∆

x y

Dir.b.c Dir.b.c

2 / 5

SLIDE 26

Quark propagator factorization, case II

see talk by Schaefer D−1 =

S−1

Γ

−S−1

Γ D∂ΓD−1 Γ∗

−SΓ∗D∂Γ∗D−1

Γ

S−1

Γ∗

Now take x ∈ Γ and y ∈ Γ∗, in different domains:

x y

= − x

y

D−1(x, y) = −D−1(y, ·)D∂Γ∗ D−1

Γ (·, x)

x y

= −

x y

3 / 5

SLIDE 27

Quark propagator factorization, case II

see talk by Schaefer D−1 =

S−1

Γ

−S−1

Γ D∂ΓD−1 Γ∗

−SΓ∗D∂Γ∗D−1

Γ

S−1

Γ∗

Now take x ∈ Γ and y ∈ Γ∗, in different domains:

x y

≈ − x

y

D−1(x, y) ≈ −D−1

¯ Γ (y, ·)D∂Γ∗ D−1 Γ (·, x)

x y

≈ −

Δ

Approximation with a second Dirichlet b.c. cut ⇒ correction ≈ e Mπ∆/2

3 / 5

SLIDE 28

Some error analysis

Garcia-Vera, Schaefer ‘16

O(x) O'(y) L B R

ˆ A = 1 N0

N0

i=1

1 N2

1 N1

j=1

N1

k=1

OijO′ik Specialize to O = O′ = 0; connected contribution. σ2

A =

1 N0N2

1

VarL(O)VarR
O′

B + 1

N0

[O]2

L

O′2

R

B − ¯

A2 + 1 N0N1

VarL(O)
O′2

R + VarR

O′

[O ]2

L

B

VarL(O) =

O2

L − [O]2 L 4 / 5

SLIDE 29

Some error analysis

σ2

A =

1 N0N2

1

VarL(O)VarR

O′

B → const + 1 N0N1 VarL(O)

O′2

R + VarR

O′

[O ]2

LB

→ e−2m∆ + 1 N0

[O]2

L

O′2

RB − ¯

A2 → e−2m∆ Contributions which do not profit from multilevel inevitable. These decay exponentially with the distance from the boundary ∆. Difference between [O(x)]L(UB) and O = [O(x)]LB ([O]L − O)2B = ([Ot]R − Ot)([O]L − O)B ∝ e−m|x0−y0| ...assuming time-slice boundary.

5 / 5