Chasing Shadows A D Kennedy University of Edinburgh Tuesday, 06 October 2015 QCD & NA, Regensburg.

Chasing Chasing ShadowsShadows

A D KennedyA D Kennedy

University of EdinburghUniversity of Edinburgh

Friday 21 April 2023 QCD & NA, Regensburg

2

Outline

Functional Integrals, Markov Chains, and HMCSymplectic Integrators and Shadow Hamiltonians

Symplectic 2-form and Hamiltonian vector fieldsPoisson bracketsBCH formula

Shadow HamiltoniansShadows for gauge theories

Improved integratorsHessian (force-gradient) integrators

InstabilitiesFree field theoryPseudofermionsMultiple pseudofermionsHasenbusch, RHMC, DD-HMCMultiple shifts and multiple sources


3

Functional Integrals

Friday 21 April 2023

1 Sd eZ

In lattice quantum field theory the goal is to compute the expectation value of some interesting operator

Integral is over all field configurations(At least) one integral for each lattice siteThis become an infinite dimensional integral in the continuum limit

QCD & NA, Regensburg

4

Fermions


Fermion fields are Grassmann-valuedGrassmann integration of quadratic form gives a determinant

detUd d e UM M

We replace these by bosonic integrals over pseudofermion fields with inverse kernel

1

detUd d e UM M


5

1

det UdU U U e

MM 1 1 1

d d dU U e

M M M M1

d d dU e U e

M M

Fermionic Operators


Operators involving Fermion fields can be expressed as non-local bosonic operators Ud d dU e U M

Ud d dU U e

M

Add Fermion sourcesComplete the squareShift Fermion fieldsEvaluate Grassmann integral

Evaluating the inverses of the Fermion kernel is usually the expensive part of measuring operators on dynamical gauge configurations


6

Markov Chains


Evaluate functional integrals by Monte CarloGenerate configurations from Markov chainIterating ergodic Markov step(s) converges to fixed point distributionDetailed balance fixed pointMetropolis accept/reject detailed balance


7

Hybrid Monte Carlo (HMC)


Introduce fictitious momenta p for each q and generate phase space configurations with distribution with the Hamiltonian

,H q pe 212, ( )H q p p S q

Solution of Hamilton’s equations would then generate a new phase space configuration which satisfies detailed balance

Reversible (after a momentum flip)Measure preserving (Liouville’s theorem)Equiprobable (energy conservation)


Momentum & pseudofermion heatbath ergodicity?

8

HMC and Symplectic Integrators


Symmetric symplectic integrators provide discrete approximations to the solution of Hamilton’s equations

Symmetric reversibleSymplectic area preserving

min 1, He

Such integrator only approximately conserve energy, but this may be corrected by Metropolis step with acceptance probability


9

Hamiltonian Dynamics


The basic idea of a Hamiltonian system is that its state at time t is represented by a point in phase space (the cotangent bundle over configuration space )

*T MM

Hamilton’s equations give the time evolution in terms of derivatives of a Hamiltonian which is a function (0-form) on phase spaced dp dq H H

dt dt p dt q q p p q


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More generally the time evolution of the system follows the integral curves of the Hamiltonian vector field , which is related to H through the symplectic 2-form through

Hdq dp

ˆˆ,

HdH X i X H X

ˆ ˆ ˆ ˆp q p q

H Hdp dq dq dp H H H dq H dp

p q p q

ˆ H HH

q p p q


11



Once can define a Hamiltonian vector field for any 0-form A on phase space

The commutator of two such Hamiltonian vector fields is itself a Hamiltonian vector field

where is the Poisson bracket

ˆdA i

ˆ ˆ, ,A B A B

ˆ ˆ, ,A B A B

A B A Bp q q p


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Proofs: I


Definition: Exterior derivative dF of a 0-form F applied to a vector X is dF X XF

ˆˆ ˆ ˆ ˆ ˆ, ,

FAF dF A i A F A A F

Hence for a Hamiltonian vector field

Corollary

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ, , ,A B F AB BA F A B F B A F , , , ,A B F B A F


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Proofs: II


ˆ ˆ ˆ ˆ, , , ,X Y Z X Y Z X Y Z

ˆ ˆ ˆ ˆ ˆ, , , , , , ,X Y Z X Y Z X Y Z Y X Z

Observe that

, , , , ,

, , , , , ,

d X Y Z X Y Z Y Z X Z X Y

X Y Z Y Z X Z X Y

Definition: Exterior derivative of a 2-form field is

ˆ ˆ ˆ, , , , , , , ,d X Y Z X Y Z Y Z X Z X Y Hence for Hamiltonian vector fields


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Proofs: III


This means that 0-forms on a symplectic manifold form a Lie algebra with a Lie bracket that is not a commutator

Thus the closure of the symplectic 2-form implies the Jacobi identity for Poisson brackets , , , , , , 0X Y Z Y Z X Z X Y

0d

ˆ ˆ, , , ,A B F A B F A B F

Using the Jacobi identity we finally get for any F


15

Symplectic Integrators

exp expd dp dqdt dt p dt q

212,H q p T p S q p S q

We are interested in finding the classical trajectory in phase space of a system described by the Hamiltonian

The basic idea of a symplectic integrator is to write the time evolution operator as

ˆexp HH He

q p p q

exp S q T pp q


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PQP Integrator

Define and so that ˆˆ ˆH S T S S qp

T T p

q

Since the kinetic energy T is a function only of p and the potential energy S is a function only of q, it follows that the action of and may be evaluated triviallyTe Se

ˆ

ˆ

: , ,

: , ,

T

S

e f q p f q T p p

e f q p f q p S q

We can thus construct an approximate PQP leapfrog integrator 1 1 1 1

2 2 2 2ˆ ˆ ˆ ˆˆ ˆ;S S S ST T

PQP PQPU e e e U e e e


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Baker-Campbell-Hausdorff formula

1 2

1 2

1 2

221

1 2, , 10

1 ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ, , , ,1 2 ! m

m

m

nm

n n k kk km

k k n

Bc c A B c c A B

n m

More precisely, where and

ˆ ˆ

1

ˆln A Bn

n

e e c 1ˆ ˆc A B

The Bn are Bernoulli numbers

If and belong to any (non-commutative)

algebra then , where is

constructed from commutators of and

ˆ ˆ ˆ ˆ ˆA B A Be e e

A B

A B

Such commutators are in the Free Lie algebra


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BCH formula

Explicitly, the first few terms are

12

1 112 24

1720

ln ,

, , , , , , ,

, , , , 4 , , , ,

6 , , , , 4 , , , ,

2 , , , ,

A Be e A B A B

A A B B A B B A A B

A A A A B B A A A B

A B A A B B B A A B

A B B A

, , , ,B B B B A B

We only include commutators that are not related by antisymmetry or the Jacobi relationThese are chosen from a Hall basis


19

Symmetric Symplectic Integrators

In order to construct reversible integrators we use symmetric symplectic integrators

1 1 32 2

5

ˆ ˆˆ

24

5760

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆln ln , , 2 , ,

ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ7 , , , , 28 , , , ,

ˆ ˆ ˆˆ ˆ ˆ ˆ12 , , , , 32 ,

S STPQPU e e e S T S S T T S T

S S S S T T S S S T

S T S S T T

ˆ ˆ ˆ, , ,

ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ16 , , , , 8 , , , ,

T S S T

S T T S T T T T S T

7O

From the BCH formula the PQP integrator corresponds to the vector field


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Shadow Hamiltonian: I

A symplectic integrator exactly follows the vector field where is constructed from commutators of Hamiltonian vector fields

ˆ ˆ ˆShadowH H

Commutators of Hamiltonian vector fields are themselves Hamiltonian vector fields,

ˆ ˆ, ,A B A B

Therefore is itself the Hamiltonian vector field corresponding to the shadow Hamiltonian obtained by replacing the commutators in with Poisson brackets

ˆShadowH

This only defines the shadow Hamiltonian as a series expansion in ; we will later consider what happens when this series fails to converge


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Shadow Hamiltonian: II

The PQP integrator follows a trajectory that is the exact solution of Hamilton’s equations for the (exactly conserved) shadow Hamiltonian

2

4

24

5760

, , 2 , ,

7 , , , , 28 , , , ,

12 , , , 32 , , , ,

16 , , , 8 , , , ,

PQPH T S S S T T S T

S S S S T T S S S T

S T S S T T T S S T

S T T S T T T T S T

6O


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Scalar Theory

Evaluating the Poisson brackets gives

2 2 2124

44 2 2 2 4 61720

2

6 2 3

PQPH H p S S

p S p SS S S S O

Note that HPQP cannot be written as the sum of a p-dependent kinetic term and a q-dependent potential term

So, sadly, it is not possible to construct an integrator that conserves the Hamiltonian we started with


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How to Tune Integrators

For any (symmetric) symplectic integrator the conserved Hamiltonian is constructed from the same Poisson bracketsThe procedure is therefore

Measure average Poisson brackets during a preliminary runOptimize the integrator (number of pseudofermions, step-sizes, order of integration scheme, etc.) offline using these measured valuesThis can be done because the acceptance rate (and instabilities) are completely determined by

See talk by Paulo Silva for details

shadowH H H


24

12

i i j kjk

jk

d c

iOn a Lie group there are left invariant

forms dual to the generators that

satisfy the Maurer-Cartan equations

Classical Mechanicson Group Manifolds


We first formulate classical mechanics on a Lie group manifold, and then rewrite it in terms of the usual constrained variables (U and P matrices)

Haar measure is 1 2 NdU

ie


25

Fundamental 2-form

1

2

i i i i i i

i i

i i i i j kjk

i

d p dp p d

dp p c

We may therefore define a closed symplectic fundamental 2-form

This specifies the Poisson bracket

ˆ ˆ, ( , )A B A B


26

Hamiltonian Vector Field


We may now follow the usual procedure to find the equations of motion

Introduce a Hamiltonian function (0-form) H on the cotangent bundle (phase space) over the group manifoldDefine a vector field such that

HdH i H

ˆ k ki ji ii j i

i jk

H HH e c p e H

p p p


27

Poisson Brackets


For we have vector fields

ˆ k ki jii j i

i jk

T TT e c p

p p p

ˆi i

i

S e Sp

2

if 2

i k k ji ji i

i jk

pp e c p p T p

p

,H p q T p S q


28

and the Hamiltonian vector corresponding to it

, ,, ,k k

i ji ii j ii jk

S T S TS T e c p e S T

p p p

k k ji i ji j i j ie S e c p e S p e e S

p

More Poisson Brackets


We thus compute the lowest-order Poisson bracket

12

ˆ ˆˆ ˆ, , ,i i i i j k ijk iS T S T dp p c S T p e S


29

Even More Poisson Brackets


, , ( ) ( )

, , ( )

, , , , 2 ( ) ( ) ( ) ( )

, , , , 3 ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

, , , , 0

, , , , 2

i i

i ji j

i ji j k k i k j k

i ijk j k k

i jk k i j k i i k k j

S S T e S e S

T S T p p e e S

T T S S T p p e e e S e S e e S e e S

S T T S T c p p e S e e S e e S

p p e S e e e S e e S e e S e e S

T S S S T

S T S S T e

( ) ( ) ( )

, , , , ( )

, , , , 0

i j i j

i j ki j k

S e S e e S

T T T S T p p p p e e e e S

S S S S T


30

Integrators


2/ 2 / 2

22 2

2 26 3 62 2

3 3 3

36 2 2

Integrator Update Steps Shadow Hamiltonian

, , 2 , ,24

2 , , , ,24

Omelyan , ,72

Omelyan

S T S

T TS

S S ST T

S ST T

PQP e e e T S S S T T S T

QPQ e e e T S S S T T S T

SST e e e e e T S S S T

TST e e e e e

3

26 3 2T, S,T

24

S

T S


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Hessian Integrators


We may therefore evaluate the integrator explicitly

We may therefore evaluate the integrator explicitly

3, ,S S Te

An interesting observation is that the Poisson bracket depends only of q

An interesting observation is that the Poisson bracket depends only of q

, ,S S T

The force for this integrator involves second derivatives of the action

Using this type of step we can construct efficient Hessian (Force-Gradient) integrators

The force for this integrator involves second derivatives of the action

Using this type of step we can construct efficient Hessian (Force-Gradient) integrators


32

Higher-Order Integrators

We can eliminate all the leading order Poisson brackets in the shadow Hamiltonian leaving errors ofThe coefficients of the higher-order Poisson brackets are much smaller than those from the Campostrini integrator


4O


33

Campostrini Integrator


3 33 3

3 33 3 3 3

3 3 3 3

3 3 3

3

4 2 2 44 2 2 4612

4 2 2 14 2 2 2 4 2 2 2312 12

4 2 2 4 4 2 2 46 12

40 4+40 2+48 , , , , + 20 2+32 , , , ,

60 4+

Integrator Campostrini

Update Steps

Shadow Hamiltonian

ST

ST T

S T

T S

S S S S T T T S S T

e e

e e e

e e

3 3

3 3 3

80 2+104 , , , , 20 4+8 , , , ,

+ 180 4+240 2+312 , , , , 5 2+8 , , , ,4

34560

S T T S T T S S S T

S T S S T T T T S T


34

Hessian Integrators


3

36 8 3

48 , ,

192

33 8 6

2259 , , , , 4224 , , , ,

768 , , , , 5616 , , , ,

3024 , , , , 896 , , , ,

Integrator Update Steps Shadow Hamiltonian

Force-Gradient 1

T S T

S S S T

T S T

T S

S S S S T T T S S T

S T T S T T S S S T

S T S S T T T T S T

e e e

e

e e e

3

6 2

48 , ,

72

62

46635520

41 , , , , +126 , , , ,

+72 , , , , +84 , , , ,

+36 , , , , +54 , , , ,4

155520

Force-Gradient 2

S T

S S S T

ST

T S

S S S S T T T S S T

S T T S T T S S S T

S T S S T T T T S T

e e

e

e e


35

Multiple Timescale Integrators

Use different integration step sizes for different contributions to the action (Sexton—Weingarten)

Evaluate cheap forces that give a large contribution to the shadow Hamiltonian more frequentlyOriginal application failed because largest contribution to shadow Hamiltonian was also the most expensive


36

Instabilities in Free Field Theory


For free field theory the Hamiltonian may be diagonalized to for some set of frequencies

For free field theory the Hamiltonian may be diagonalized to for some set of frequencies

2 2 212 i i ij

H p q i

The shadow Hamiltonian is exactlyThe shadow Hamiltonian is exactly

2

2

2i i

Shadow i ii i

pH q

21 12cos 1PQP

i i 2141PQP

i i i

This has a singularity at This has a singularity at max 2

37

Integrator Instabilities

When the step size exceeds some critical value the BCH expansion for the shadow Hamiltonian divergesNo real conserved shadow HamiltonianSymplectic integrators become exponentially unstable in trajectory length


maxexp 2 2PQPH

38

Symptoms


Acceptance rate falls (exponentially) with trajectory length for

Floating point rounding errors can also cause this

For light fermions maximum frequency corresponds to inverse of smallest eigenvalue of fermion kernel

This can grow as gauge fields equilibrate

Often erroneously ascribed to exceptional configurations

Real problem is exponential amplification due to instabilities

39

Higher Order Integrator Instabilities


The values of at which instabilities occur depends on the choice of integrator

max

Integrator Criticalvalue

PQP, QPQ 2

P2MN2 2.22

P24FG 2.61

P5MN4 3.14

Q5MN4FG 3.10

Campostrini 1.57

Mike Clark

Omelyan, Mryglod, Folk Computer Physics Communcations 151 (2003) 272-314

Joo et al Phys Rev D62 (2000) 114501 (hep-lat/0005023)

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Pseudofermions


1†

1 † 1 1

1 1 1 † 1 1 1 1 1

2 2

2 2 1

tr ln

tr

tr

F PF

F PF

F PF

n nF PFPQP PQPn n

S U S U

S S

S S

H H

M M

M M M M M

M M M M M M 2M M M M M M M M

O O

We must distinguish the intrinsic fermion part from that due to pseudofermionic noise

The pseudofermion contribution to the shadow Hamiltonian has one more power of the inverse fermion kernel than the intrinsic fermion contributionThe pseudofermions are fixed during a trajectory, so probably do not suppress this extra inverse power


41

This does not change the intrinsic fermion part at all

Instability is driven by most singular kernelMust balance cost of applying kernels with reduction in contribution to shadow Hamiltonian

No improvement with more than two or three pseudofermions so far

Acceptance rate no longer limited by instability but by bulk contributions?Opportunity to use improved/higher-order integratorsRôle of finite density of fermion spectrum near zero?Eventually limited by intrinsic fermion part

tr ln tr ln tr lnF FS S1 2 1 2M M M M

Multiple Pseudofermions


The effect of the extra inverse power can be reduced by introducing multiple pseudofermions with less singular kernels

† 1 † 11 2 21

1 2det det det d d e 1 2M M

1 2M M M


42

Multiple Pseudofermion Techniques


1H HM M M M

1 3/ 2

s ssM M M M M

3/ 2sM

Hasenbusch:Heavy pseudofermion cheap to invertParticularly cost-effective if we make use of existing pseudofermion (e.g., s quark):

RHMCNo parameter tuning necessaryCan implement fractional flavours (e.g., for above)

Lüscher DD-HMCDominated by Schur complement part towards chiral limit?

Evidence of this from PAC-CS data?Other benefits of DD?QCD & NA, Regensburg

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Multi-Everything Solvers

For RHMC we evaluate fractional powers of the fermion kernel using optimal (Chebyshev) partial fraction rational approximationsWe thus need to find solutions for this kernel (the Dirac operator or its square) for

Multiple shifts: shifts are poles of rational approximationMultiple right hand sides: for multiple pseudofermions

We already use a multishift solver, butHow to do so for multiple right hand sides?Can we use an initial guess extrapolated from the past?Can we restart the solver to avoid stagnation?


45

Outlook

Large-scale computations with light dynamical fermions require

Avoiding integrator instabilities as far as possibleUse multiple pseudofermion fields

Reducing volume contributions to HShadow

Tune integrators by measuring Poisson bracketsUse good higher order integrators, such as Hessian/Force Gradient

Using longer trajectoriesReduces autocorrelations as physical correlation lengths growAmortizes cost of pseudofermion heatbath and Metropolis energy calculation

Improving solver technology


Chasing Shadows A D Kennedy University of Edinburgh Tuesday, 06 October 2015 QCD & NA, Regensburg.

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Transcript of Chasing Shadows A D Kennedy University of Edinburgh Tuesday, 06 October 2015 QCD & NA, Regensburg.