Post on 05-Feb-2016
description
transcript
Saturday, April 22, 2023Saturday, April 22, 2023
Speeding up HMC with
better integrators
Speeding up HMC with
better integrators
A D Kennedy and M A ClarkA D Kennedy and M A ClarkSchool of Physics & SUPA, The University of EdinburghSchool of Physics & SUPA, The University of Edinburgh
Boston UniversityBoston University
A D Kennedy and M A ClarkA D Kennedy and M A ClarkSchool of Physics & SUPA, The University of EdinburghSchool of Physics & SUPA, The University of Edinburgh
Boston UniversityBoston University
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 33
OutlineOutlineSymmetric symplectic integrators in HMCShadow Hamiltonians and Poisson bracketsTuning integrators using Poisson bracketsHessian or Force-Gradient integratorsSymplectic integrators and Poisson brackets on Lie groupsResults for single-link updates
Symmetric symplectic integrators in HMCShadow Hamiltonians and Poisson bracketsTuning integrators using Poisson bracketsHessian or Force-Gradient integratorsSymplectic integrators and Poisson brackets on Lie groupsResults for single-link updates
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 44
Symplectic IntegratorsSymplectic Integrators
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
exp expd dp dqdt dt p dt q
The idea of a symplectic integrator is to write the time evolution operator (Lie derivative) as
ˆexp HH He
q p p q
exp S q T pp q
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 55
Symplectic IntegratorsSymplectic Integrators
Define and so
that H T S
S S qp
T T pq
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 trivially
Se
Te
: , ,
: , ,
T
S
e f q p f q T p p
e f q p f q p S q
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 66
If A and B belong to any (non-commutative)
algebra then , where constructed from commutators of A and B (i.e.,
is in the Free Lie Algebra generated by A and
B )
A B A Be e e
Symplectic IntegratorsSymplectic Integrators
1 2
1 2
1 2
221
1 2, , 10
1, , , ,
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
1
ln A Bn
n
e e c
1c A B
Baker-Campbell-Hausdorff (BCH) formula
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 77
Symplectic IntegratorsSymplectic Integrators
Explicitly, the first few terms are
1 1 12 12 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
In order to construct reversible integrators we use symmetric symplectic integrators
2 2 124
15760
ln , , 2 , ,
7 , , , , 28 , , , ,
12 , , , , 32 , , , ,
16 , , , , 8 , , , ,
A B Ae e e A B A A B B 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
The following identity follows directly from the BCH formula
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 88
Symplectic IntegratorsSymplectic Integrators
From the BCH formula we find that the PQP symmetric symplectic integrator is given by
1 12 2
//
0( ) S STU e e e
3 5124
exp , , 2 , ,T S S S T T S T O
ˆ 2S THe e O
In addition to conserving energy to O (² ) such symmetric symplectic integrators are manifestly area preserving and reversible
2 4124exp , , 2 , ,T S S S T T S T O
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 99
Shadow HamiltoniansShadow Hamiltonians
This may be obtained by replacing the
commutators in the BCH expansion
of
with the Poisson bracket
,S T
ln S Te e
,S T
For each symplectic integrator there
exists a Hamiltonian H’ which is exactly
conserved
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 1010
Conserved HamiltonianConserved Hamiltonian
For the PQP integrator we have
124
15760
' , , 2 , ,
7 , , , , 28 , , , ,
12 , , , 32 , , , ,
16 , , , 8 , , , ,
H 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
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 1111
Tuning HMCTuning HMCFor any (symmetric) symplectic integrator the conserved Hamiltonian is constructed from the same Poisson bracketsA procedure for tuning such integrators is
Measure the Poisson brackets during an HMC 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 δH = H’ - H
For any (symmetric) symplectic integrator the conserved Hamiltonian is constructed from the same Poisson bracketsA procedure for tuning such integrators is
Measure the Poisson brackets during an HMC 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 δH = H’ - H
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 1212
Simple Example (Omelyan)Simple Example (Omelyan)
1 12 21 2Q QPP Pe e e e e Consider the PQPQP integrator
The conserved Hamiltonian is thus
2
3 56 6 1 1 6' , , , ,
12 24H H S S T T S T O
Measure the “operators” and minimize the cost by adjusting the parameter α
2
3 5, , , ,6 6 1 1 6
12 24SH OS T T S T
, ,12 ,
14 ,
T S T
S S T
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 1313
Hessian IntegratorsHessian 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 actionUsing this type of step we can construct very efficient Force-Gradient integrators
The force for this integrator involves second derivatives of the actionUsing this type of step we can construct very efficient Force-Gradient integrators
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 1414
Higher-Order IntegratorsHigher-Order Integrators
We can eliminate all the leading order Poisson brackets in the shadow Hamiltonian leaving errors of O (δτ2)The coefficients of the higher-order Poisson brackets are much smaller than those from the Campostrini integrator
We can eliminate all the leading order Poisson brackets in the shadow Hamiltonian leaving errors of O (δτ2)The coefficients of the higher-order Poisson brackets are much smaller than those from the Campostrini integrator
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 1515
Beyond Scalar Field TheoryBeyond Scalar Field Theory
We need to extend the formalism beyond a scalar field theoryFermions are easy
† 1 1 †TrFS U U U M M1
1 1
U U
M M
M M
How do we extend all this fancy differential geometry formalism to gauge fields?
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 1616
Hamiltonian MechanicsHamiltonian Mechanics
,A B A B
A Bp q q p
: 0d dq dp
Flat Manifold General
Symplectic 2-form
Hamiltonian vector field
Equations of motion
Poisson bracket
ˆ H HH
p q q p
H
dH i
,H H
q pp q
ˆdH
dt
ˆ ˆ, ( , )A B A B
Darboux theorem:
All manifolds are locally flat
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 1717
Maurer-Cartan EquationsMaurer-Cartan Equations
12
i i j kjk
jk
d c
The left invariant forms dual to the
generators of a Lie algebra satisfy the
Maurer-Cartan equations
i
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 1818
We can invent any Classical Mechanics we want…So we may therefore define a closed symplectic 2-form which globally defines an invariant Poisson bracket by
i i
i
d p
Fundamental 2-formFundamental 2-form
12
i i i i j kjk
i
dp p c
i i i i
i
dp p d
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 1919
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 manifold
Hamiltonian Vector FieldHamiltonian Vector Field
Define a vector field such thatH
dH i H
k ki ji ii j i
i jk
H HH e c p e H
p p p
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 2020
Integral CurvesIntegral CurvesThe classical trajectories are then the integral curves of h:
,t t tQ P t th
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 2121
Poisson BracketsPoisson Brackets
ˆ k ki ji ii j i
i jk
H HH e c p e H
p p p
Recall our Hamiltonian vector field
For H(q,p) = T(p) + S(q) 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
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 2222
More Poisson BracketsMore Poisson Brackets
12
ˆ ˆˆ ˆ, , ,i i i i j kjkS T S T dp p c S T
We thus compute the lowest-order Poisson bracket
Retrii
Sp e S PU
U
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
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 2323
Even More Poisson BracketsEven 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 T 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
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 2424
IntegratorsIntegrators
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
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 2525
Campostrini IntegratorCampostrini 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
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 2626
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
Hessian IntegratorsHessian Integrators
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 2727
One-Link ResultsOne-Link Results
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 1 2 3 4 5 6 7 8 9 10
t
x 1
07
dH
dH Shadow
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 2828
Scaling BehaviourScaling Behaviour
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
-1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3
-log10 Step Size
log
10 E
rror
PQPOmelyan {S,{S,T}}CampostriniForce-Gradient
Saturday, April 22, 2023Saturday, April 22, 2023
A D KennedyA D Kennedy 2929
ConclusionsConclusionsWe hope that very significant performance improvements can be obtained using Force-Gradient integratorsFor fermions one extra inversion of the Dirac operator is requiredPure gauge force terms and Poisson brackets get quite complicated to programReal-life speed-up factors will be measured really soon…
We hope that very significant performance improvements can be obtained using Force-Gradient integratorsFor fermions one extra inversion of the Dirac operator is requiredPure gauge force terms and Poisson brackets get quite complicated to programReal-life speed-up factors will be measured really soon…