saturday, 02 may 2015saturday, 02 may 2015saturday, 02 may 2015saturday, 02 may 2015 2005 taipei...

Tuesday 18 April 2023Tuesday 18 April 20232005 Taipei Summer Institute2005 Taipei Summer Institute

on Strings, Particles and Fields on Strings, Particles and Fields

Simulation Algorithms Simulation Algorithms for Lattice QCDfor Lattice QCD

A D KennedyA D KennedySchool of Physics, The University of EdinburghSchool of Physics, The University of Edinburgh

Tuesday 18 April 2023 A D Kennedy 2

Contents

Monte Carlo methodsFunctional Integrals and QFTCentral Limit TheoremImportance SamplingMarkov Chains

Convergence of Markov ChainsAutocorrelations

Hybrid Monte CarloMDMCPartial Momentum RefreshmentSymplectic IntegratorsDynamical fermions

Grassmann algebras

ReversibilityPHMCRHMC

Chiral FermionsOn-shell chiral symmetryNeuberger’s OperatorInto Five Dimensions

Kernel

Schur ComplementConstraintApproximation

tanhЗолотарев

RepresentationContinued Fraction Partial FractionCayley Transform

Chiral Symmetry BreakingNumerical StudiesConclusions

Approximation TheoryPolynomial approximation

Weierstraß’ theoremБернштейне polynomialsЧебышев’s theoremЧебышев polynomials

Чебышев approximation

Elliptic functionsLiouville’s theoremWeierstraß elliptic functionsExpansion of elliptic functionsAddition formulæJacobian elliptic functions

Modular transformationsЗолотарев’s formula

Arithmetico-Geometric meanConclusionsBibliography

Hasenbusch Acceleration


( )1( )Sd e

Z

The Expectation value of an operator is defined

non-perturbatively by the Functional Integral

The action is S ()

1 1Normalisation constant Z chosen such that

xx

d d d is the appropriate functional measure

Continuum limit: lattice spacing a 0Thermodynamic limit: physical volume V

Defined in Euclidean space-timeLattice regularisation

Functional Integrals and QFT


Monte Carlo methods: I

Monte Carlo integration is based on the identification of probabilities with measures

There are much better methods of carrying out low dimensional quadrature

All other methods become hopelessly expensive for large dimensions

In lattice QFT there is one integration per degree of freedom

We are approximating an infinite dimensional functional integral


Measure the value of on each configuration and compute the average

1

1( )

N

ttN

Monte Carlo methods: II

Generate a sequence of random field configurations chosen from the probability distribution

1 2 t N, , , , ,

( )1( ) tS

t tP d eZ


Central Limit Theorem: I

Distribution of values for a single sample = ()

( )P d P

Central Limit Theorem

where the variance of the distribution of is

2CNO

22C

Law of Large Numbers limN

The Laplace–DeMoivre Central Limit theorem is an asymptotic expansion for the probability distribution of


3

0 3

4 21 4 2

2

2

0

3

C C

C C C

C

The first few cumulants are

Central Limit Theorem: II

Note that this is an asymptotic expansion

Generating function for connected moments

( ) ln ( ) ikW k d P e ln lnik ikd P e e

0 !

n

nn

ikC

n


Central Limit Theorem: III

Connected generating function

lnN

ik Nd P e

1 11

ln expN

N N tt

ikd d P P

N

( ) ln ( ) ikW k d P e

1

1 !

n

nn

n

ik Cn N

ln ik NN e

kNW

N

1 11

1 N

N N tt

P d d P PN

Distribution of the average of N samples


3 4

3 4212 3 3 4

223! 4!2

N

C Cd dik ik CN N dk ikd de e e

Take inverse Fourier transform to obtain distribution P

12

W k ikP dk e e

23 4

23 4 22 3 3 43! 4!

2 2

C Cd d C N

N Nd d

C N

ee

Central Limit Theorem: IV


Central Limit Theorem: V

where and

N

22

2 23 2

32 2

31

6 2

CC C eF

C N C

dP F

d

Re-scale to show convergence to Gaussian distribution


Importance Sampling: I

( ) 1N dx x

Sample from distributionProbability

Normalisation

0 ( ) . .x a e

( )dx f x Integral

( )( )

( )f x

I x dxx

Estimator of integral

Estimator of variance2 2

2( ) ( )( )

( ) ( )f x f x

V x dx I dx Ix x


Importance Sampling: II

2

2

( ) ( )0

( ) ( )V N f y

y y

Minimise varianceConstraint N=1Lagrange multiplier

( )( )

( )opt

f xx

dx f x

Optimal measure

Optimal variance 2 2

opt ( ) ( )V dx f x dx f x


( ) sin( )f x x Example:

12( ) sin( )x x Optimal weight

2 22 2

opt0 0

sin( ) sin( ) 16V dx x dx x

Optimal variance

2

0sindx x

Importance Sampling: III


Importance Sampling: IV2

0sindx x

1 Construct cheap approximation to |sin(x)|

2 Calculate relative area within each rectangle

5 Choose another random number uniformly

4 Select corresponding rectangle

3 Choose a random number uniformly

6 Select corresponding x value within rectangle

7 Compute |sin(x)|


Importance Sampling: V

2 2

0 0

sin( ) sin( ) sin( ) sin( )I dx x x dx x x

But we can do better!

opt 0V For which

With 100 rectangles we have V = 16.02328561

With 100 rectangles we have V = 0.011642808

2

0sindx x


Markov Chains: I

State space (Ergodic) stochastic transitions P’:

Distribution converges to unique fixed point

Q

Deterministic evolution of probability distribution P: Q Q


The sequence Q, PQ, P²Q, P³Q,… is Cauchy

Define a metric on the space of (equivalence classes of) probability distributions

1 2 1 2,d Q Q dx Q x Q x

Prove that with > 0, so the Markov process P is a contraction mapping

1 2 1 2, 1 ,d PQ PQ d Q Q

The space of probability distributions is complete, so the sequence converges to a unique fixed pointlim n

nQ P Q

Convergence of Markov Chains: I


1 2 1 2,d PQ PQ dx PQ x PQ x 1 2dx dy P x y Q y dy P x y Q y

dx dy P x y Q y Q y Q y

dx dy P x y Q y

2 mindx dy P x y Q y Q y

dx dy P x y Q y

2min ,a b a b a b

1 2Q y Q y Q y

1y y

Convergence of Markov Chains: II


2 mindy Q y dx dy P x y Q y Q y

2 inf min

ydy Q y dx P x y dy Q y Q y

infy

dy Q y dx P x y dy Q y 1 2(1 ) ,d Q Q

1 2 1 1 0dy Q y dy Q y dy Q y dy Q y Q y dy Q y Q y

1dx P x y

12dy Q y Q y dy Q y

0 infy

dx P x y

1 2,d PQ PQ

Convergence of Markov Chains: III


Markov Chains: II

Use Markov chains to sample from QSuppose we can construct an ergodic Markov process P which has distribution Q as its fixed point

Start with an arbitrary state (“field configuration”)

Iterate the Markov process until it has converged (“thermalized”)

Thereafter, successive configurations will be distributed according to Q

But in general they will be correlated

To construct P we only need relative probabilities of states

Do not know the normalisation of Q

Cannot use Markov chains to compute integrals directly

We can compute ratios of integrals


Metropolis algorithm

min 1,Q x

P x yQ y

Markov Chains: III

How do we construct a Markov process with a specified fixed point ?

Q x dy P x y Q y Integrate w.r.t. y to obtain fixed point conditionSufficient but not necessary for fixed point

P y x Q x P x y Q y Detailed balance

Q xP x y

Q x Q y

Other choices are possible, e.g.,

Consider cases and separately to obtain detailed balance condition

( ) ( )Q x Q y ( ) ( )Q x Q y

Sufficient but not necessary for detailed balance


Markov Chains: IV

Composition of Markov stepsLet P1 and P2 be two Markov steps which have the desired

fixed point distribution

They need not be ergodic

Then the composition of the two steps P2P1 will also have

the desired fixed point

And it may be ergodic

This trivially generalises to any (fixed) number of steps

For the case where P1 is not ergodic but (P1 ) n is the

terminology weakly and strongly ergodic are sometimes used


Markov Chains: V

This result justifies “sweeping” through a lattice performing single site updates

Each individual single site update has the desired fixed point because it satisfies detailed balance

The entire sweep therefore has the desired fixed point, and is ergodic

But the entire sweep does not satisfy detailed balance

Of course it would satisfy detailed balance if the sites were updated in a random order

But this is not necessaryAnd it is undesirable because it puts too much randomness into the system


Markov Chains: VI

Coupling from the PastPropp and Wilson (1996)Use fixed set of random numbersFlypaper principle: If states coalesce they stay together forever

– Eventually, all states coalesce to some state with probability one

– Any state from t = - will coalesce to – is a sample from the fixed point distribution


Autocorrelations: I

This corresponds to the exponential autocorrelation time

expmax

10

lnN

Exponential autocorrelationsThe unique fixed point of an ergodic Markov process

corresponds to the unique eigenvector with eigenvalue

1

All its other eigenvalues must lie within the unit circlemax 1 In particular, the largest subleading eigenvalue is


Autocorrelations: II

2

21 1

1 N N

t tt tN

1

2

21 1 1

12

N N N

t t tt t t tN

Integrated autocorrelationsConsider the autocorrelation of some operator Ω

Without loss of generality we may assume 0

12 2 2

1

1 2 N

N CN N

2

t tC

Define the autocorrelation function


Autocorrelations: III

The autocorrelation function must fall faster that the

exponential autocorrelation exp

maxNC e

2

2 exp1 2 1N

A ON N

2

exp

1

1 2 1N

C ON N

For a sufficiently large number of samples

1

A C

Define the integrated autocorrelation function


In order to carry out Monte Carlo computations including the effects of dynamical fermions we would like to find an algorithm which

Update the fields globally

Because single link updates are not cheap if the action is not local

Take large steps through configuration space

Because small-step methods carry out a random walk which leads to critical slowing down with a dynamical critical exponent z=2

Hybrid Monte Carlo: I

Does not introduce any systematic errors

z relates the autocorrelation to the correlation length of the system,

zA


A useful class of algorithms with these properties is the (Generalised) Hybrid Monte Carlo (HMC) method

Introduce a “fictitious” momentum p corresponding to each dynamical degree of freedom q

Find a Markov chain with fixed point exp[-H(q,p) ] where H is the “fictitious” Hamiltonian ½p2 + S(q)

The action S of the underlying QFT plays the rôle of the potential in the “fictitious” classical mechanical system

This gives the evolution of the system in a fifth dimension, “fictitious” or computer time

This generates the desired distribution exp[-S(q) ] if we ignore the momenta p (i.e., the marginal distribution)

Hybrid Monte Carlo: II


The HMC Markov chain alternates two Markov steps

Molecular Dynamics Monte Carlo (MDMC)(Partial) Momentum Refreshment

Both have the desired fixed pointTogether they are ergodic

Hybrid Monte Carlo: IIIHybrid Monte Carlo: III


MDMC: I

If we could integrate Hamilton’s equations exactly we could follow a trajectory of constant fictitious energy

This corresponds to a set of equiprobable fictitious phase space configurationsLiouville’s theorem tells us that this also preserves the functional integral measure dp dq as required

Any approximate integration scheme which is reversible and area preserving may be used to suggest configurations to a Metropolis accept/reject test

With acceptance probability min[1,exp(-H)]


MDMC: II

Molecular Dynamics (MD), an approximate integrator which is exactly : , ,U q p q p

We build the MDMC step out of three parts

A Metropolis accept/reject step

Area preserving, *

,det det 1

,

q pU

q p

Reversible, 1F U F U

A momentum flip :F p p

1H Hq qF U e y y e

p p

The composition of these gives

with y being a uniformly distributed random number in [0,1]


The Gaussian distribution of p is invariant under F

The extra momentum flip F ensures that for small the momenta are reversed after a rejection rather than after an acceptance

For = / 2 all momentum flips are irrelevant

This mixes the Gaussian distributed momenta p with Gaussian noise

cos sin

sin cos

p pF

Partial Momentum Refreshment


Special casesThe usual Hybrid Monte Carlo (HMC) algorithm is the special case where = / 2

= 0 corresponds to an exact version of the Molecular Dynamics (MD) or Microcanonical algorithm (which is in general non-ergodic)

the L2MC algorithm of Horowitz corresponds to choosing arbitrary but MDMC trajectories of a single leapfrog integration step ( = ). This method is also called Kramers algorithm.

Hybrid Monte Carlo: IV


Hybrid Monte Carlo: V

Further special casesThe Langevin Monte Carlo algorithm corresponds to choosing = / 2 and MDMC trajectories of a single leapfrog integration step ( = ).

The Hybrid and Langevin algorithms are approximations where the Metropolis step is omitted

The Local Hybrid Monte Carlo (LHMC) or Overrelaxation algorithm corresponds to updating a subset of the degrees of freedom (typically those living on one site or link) at a time

Hybrid Monte Carlo: V


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,B })

A B A Be e e

Symplectic Integrators: I

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


Symplectic Integrators: IIExplicitly, 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


Symplectic Integrators: III

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 such a symplectic integrator is to write the time evolution operator as

ˆexp HH He

q p p q

exp S q T pp q


Symplectic Integrators: IV

Define and so that̂H P Q P S qp

Q 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 triviallyPe Qe

: , ,

: , ,

Q

P

e f q p f q T p p

e f q p f q p S q


Symplectic Integrators: V

From the BCH formula we find that the PQP symmetric symplectic integrator is given by

1 12 2

//

0( ) P PQU e e e

3 5124exp , , 2 , ,P Q P P Q Q P Q O

2 4124exp , , 2 , ,P Q P P Q Q P Q O

ˆ 2P QHe e O

In addition to conserving energy to O (² ) such symmetric symplectic integrators are manifestly area preserving and reversible


Symplectic Integrators: VI

For each symplectic integrator there exists a Hamiltonian H’ which is exactly conserved

For the PQP integrator we haveˆ H HH

p q q p

2124

4 615760

, , 2 , ,

32 , , , , 16 , , , ,

28 , , , , 12 , , , ,

8 , , , , 7 , , , ,

Q P P P Q Q P Q

Q Q P P Q P Q Q P Q

Q P P P Q P Q P P Q O

Q Q Q P Q P P P P Q


Symplectic Integrators: VII

Substituting in the exact forms for the operations P and Q we obtain the vector field

ˆ H HH

p q q p

2 2112 2p S pS SS p S

q p q p

43 2

544 61

7204 2

2

4 6 3

30 6

3 2

p S S SS pq

p S OS S SS p

pS S SS


Symplectic Integrators: VIII

Solving these differential equations for H’ we find

2 2 2124

44 2 2 2 4 61720

2

6 2 3

H H p S S

p S p SS S S S O

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


Fermion fields are Grassmann valued

Required to satisfy the spin-statistics theorem

Even “classical” Fermion fields obey anticommutation relationsGrassmann algebras behave like negative dimensional manifolds

Dynamical fermions: I


Grassmann algebras: I

Grassmann algebrasLinear space spanned by generators {1,2,3,…} with

coefficients a, b,… in some field (usually the real or complex numbers)

Algebra structure defined by nilpotency condition for elements of the linear space ² = 0

There are many elements of the algebra which are not in the linear space (e.g., 12)

Nilpotency implies anticommutativity + = 00 = ² = ² = ( + )² = ² + + + ² = + = 0

Anticommutativity implies nilpotency, 2² = 0 Unless the coefficient field has characteristic 2, i.e., 2 = 0


A natural antiderivation is defined on a Grassmann algebra

Linearity: d(a + b) = a d + b d

Anti-Leibniz rule: d() = (d) + P()(d)

Grassmann algebras have a natural grading corresponding to the number of generators in a given product

deg(1) = 0, deg(i ) = 1, deg(ij ) =

2, ...

All elements of the algebra can be decomposed into a sum of terms of definite grading

Grassmann algebras: II

The parity transform of is

deg1P


Grassmann algebras: III

Gaussians over Grassmann manifoldsLike all functions, Gaussians are polynomials over Grassmann algebras

1i ij j

ij i ij j

aa

i ij jij ij

e e a

There is no reason for this function to be positive even for real coefficients

Definite integration on a Grassmann algebra is defined to be the same as derivation

Hence change of variables leads to the inverse Jacobian

1 1 det in n

j

d d d d


Grassmann algebras: IV

Where Pf(a) is the Pfaffian

Pf(a)² = det(a)

11 deti ij j

ij

a

nn ijd d d d e a

If we separate the Grassmann variables into two “conjugate” sets then we find the more familiar result

Despite the notation, this is a purely algebraic identityIt does not require the matrix a > 0, unlike its bosonic analogue

Gaussian integrals over Grassmann manifolds

1 Pfi ij j

ija

n ijd d e a


Dynamical fermions: II

PseudofermionsDirect simulation of Grassmann fields is not feasible

The problem is not that of manipulating anticommuting values in a computer

We therefore integrate out the fermion fields to obtain the fermion determinant detMd d e M

It is that is not positive, and thus we get poorimportance sampling

FS Me e

and always occur quadratically

The overall sign of the exponent is unimportant


Dynamical fermions: III

0

, , Me

Any operator can be expressed solely in terms of the bosonic fields

1( , ) ( , )G x y x y M x y

E.g., the fermion propagator is


Dynamical fermions: IV

Represent the fermion determinant as a bosonic Gaussian integral with a non-local kernel 1

det MM d d e

The fermion kernel must be positive definite (all its eigenvalues must have positive real parts) otherwise the bosonic integral will not convergeThe new bosonic fields are called pseudofermions

The determinant is extensive in the lattice volume, thus again we get poor importance sampling

Including the determinant as part of the observable to

be measured is not feasible

det

detB

B

S

S

M

M


Dynamical fermions: V

It is usually convenient to introduce two flavours of fermion and to write 1†2 †det det

M MM M M d d e

The evaluation of the pseudofermion action and the corresponding force then requires us to find the solution of a (large) set of linear equations 1†M M

This not only guarantees positivity, but also allows us to generate the pseudofermions from a global heatbath by applying to a random Gaussian distributed field

†M

The equations for motion for the boson (gauge) fields are

†1 1 1† † † † †B BS S

M M M M M M M M


Dynamical fermions: VI

It is not necessary to carry out the inversions required for the equations of motion exactly

There is a trade-off between the cost of computing the force and the acceptance rate of the Metropolis MDMC step

The inversions required to compute the pseudofermion action for the accept/reject step does need to be computed exactly, however

We usually take “exactly” to by synonymous with “to machine precision”


Reversibility: I

Are HMC trajectories reversible and area preserving in practice?

The only fundamental source of irreversibility is the rounding error caused by using finite precision floating point arithmetic

For fermionic systems we can also introduce irreversibility by choosing the starting vector for the iterative linear equation solver time-asymmetrically

We do this if we to use a Chronological Inverter, which takes (some extrapolation of) the previous solution as the starting vector

Floating point arithmetic is not associativeIt is more natural to store compact variables as scaled integers (fixed point)

Saves memoryDoes not solve the precision problem


Reversibility: II

Data for SU(3) gauge theory and QCD with heavy quarks show that rounding errors are amplified exponentially

The underlying continuous time equations of motion are chaotic

Ляпунов exponents characterise the divergence of nearby trajectories

The instability in the integrator occurs when H » 1

Zero acceptance rate anyhow


Reversibility: III

In QCD the Ляпунов exponents appear to scale with as the system approaches the continuum limit

= constant

This can be interpreted as saying that the Ляпунов exponent characterises the chaotic nature of the continuum classical equations of motion, and is not a lattice artefact

Therefore we should not have to worry about reversibility breaking down as we approach the continuum limit

Caveat: data is only for small lattices, and is not conclusive


Data for QCD with lighter dynamical quarks

Instability occurs close to region in where acceptance rate is near one

May be explained as a few “modes” becoming unstable because of large fermionic force

Integrator goes unstable if too poor an approximation to the fermionic force is used

Reversibility: IV


PHMC

Polynomial Hybrid Monte Carlo algorithm: instead of using Чебышев polynomials in the multiboson algorithm, Frezzotti & Jansen and deForcrand suggested using them directly in HMC

Polynomial approximation to 1/x are typically of order 40 to 100 at presentNumerical stability problems with high order polynomial evaluationPolynomials must be factoredCorrect ordering of the roots is importantFrezzotti & Jansen claim there are advantages from using reweighting


RHMC

Rational Hybrid Monte Carlo algorithm: the idea is similar to PHMC, but uses rational approximations instead of polynomial ones

Much lower orders required for a given accuracy

Evaluation simplified by using partial fraction expansion and multiple mass linear equation solvers

1/x is already a rational function, so RHMC reduces to HMC in this case

Can be made exact using noisy accept/reject step


Chiral Fermions

Conventions

We work in Euclidean spaceWe work in Euclidean space

γγ matrices are Hermitianmatrices are Hermitian

We writeWe write

We assume all Dirac We assume all Dirac

operators are operators are γγ55 HermitianHermitian

†5 5D D

D D


It is possible to have chiral symmetry on the lattice without doublers if we only insist that the symmetry holds on shell

On-shell chiral symmetry: I

Such a transformation should be of the form

(Lüscher)

is an independent field from

has the same Spin(4) transformation properties as

does not have the same chiral transformation

properties

as in Euclidean space (even in the continuum)

y†y

y

551 1;i aD i aDe e

y

†y

†y†y


On-shell chiral symmetry: II

For it to be a symmetry the Dirac

operator must be invariant 5 51 1i aD i aDD e De D

5 51 1 0aD D D aD For an infinitesimal transformation this implies that

5 5 52D D aD D

Which is the Ginsparg-Wilson relation


5 5 5†sgnˆ W

w

W W

D MD M

D M D M

Both of these conditions are satisfied if we define

(Neuberger)

Neuberger’s Operator: I

We can find a solution of the Ginsparg-Wilson relation as follows

† †15 5 5 5 5 52 1 ;ˆ ˆ ˆaD aD aD

Let the lattice Dirac operator to be of the form

This satisfies the GW relation iff 25 1ˆ

It must also have the correct continuum limit

Where we have defined where W WD Z O a 2WZM aZ

2 25 5 52 1 1ˆ WD

D Z aZ O a O aM


Into Five Dimensions

H Neuberger hep-lat/9806025A Boriçi hep-lat/9909057, hep-lat/9912040, hep-lat/0402035A Boriçi, A D Kennedy, B Pendleton, U Wenger hep-lat/0110070R Edwards & U Heller hep-lat/0005002趙挺偉 (T-W Chiu) hep-lat/0209153, hep-lat/0211032, hep-lat/0303008R C Brower, H Neff, K Orginos hep-lat/0409118 Hernandez, Jansen, Lüscher hep-lat/9808010


Is DN local?It is not ultralocal (Hernandez, Jansen, Lüscher)

It is local iff DW has a gap

DW has a gap if the gauge fields are smooth enough

It seems reasonable that good approximations to DN

will be local if DN is local and vice versa

Otherwise DWF with n5 → ∞ may not be local

Neuberger’s Operator: II

1, 1 1 sgn

52D H H

N 10 μ


Four dimensional space of algorithms

Neuberger’s Operator: III

Representation (CF, PF, CT=DWF)

Constraint (5D, 4D)

,

( )sgn( ) ( )

( )n

n mm

P HH H

Q HApproximation

5 WH D MKernel


Kernel

Shamir kernel

55

5

;2

WT T T

W

a D MH D aD

a D M

Möbius kernel

5 5

55 5

;2

WM M M

W

b c D MH D aD

b c D M

5W WH D M

Wilson (Boriçi) kernel


1 0 0 111 11 0 0 1

A A B

CA D CA B

1 0 0 1

1 0 0 1

1 0

1 11 0

A B

CA D CA B

Schur Complement

It may be block diagonalised by an LDU factorisation (Gaussian elimination)

1det det

A BAD ACA B

C DIn particular

A B

C D

Consider the block matrixEquivalently a matrix over a skew field = division ring

The bottom right block is the Schur complement


1

2

2

1

n

n

D

00

00

1

2

52

1

n

n

D

Constraint: I

1

2

2

1

n

n

DU

So, what can we do with the Neuberger operator represented as a Schur complement?Consider the five-dimensional system of linear

equations00

00

1L 1LL

The bottom four-dimensional component is, Nn n DD


Constraint: II

Alternatively, introduce a five-dimensional pseudofermion field 1 12 n

Then the pseudofermion functional integral is

† 15†

5 ,1

det det det detn

Dj j

j

d d e D LDU D D

So we also introduce n-1 Pauli-Villars fields

†,

11 1

†,

1 1

detj j jj

n nD

j j j jj j

d d e D

and we are left with just det Dn,n = det DN


Approximation: tanh

Pandey, Kenney, & Laub; Higham; NeubergerFor even n (analogous formulæ for odd n)

11 1

1,11

1tanh tanh

1

nxx

n n nxx

x n x

2

2 2

2

2 2

1

tan

1

12tan

1

n

n

kx

nk

kx

nk

xn

2

2 2 21 12 2cos sin1

2 1n

x k kk n n

xn

ωj


Approximation: Золотарев

2

2/ 2

21

212

sn ;1

sn / ; sn 2 / ;1sn ; sn ;

1sn 2 / ;

n

m

z k

z M iKm n k

z k M z k

iK m n k

sn(z/M,λ)

sn(z,k)

ωj


Approximation: Errors

The fermion sgn problem

Approximation over 10-2 < |x| < 1Rational functions of degree (7,8)

ε(x) – sgn(x)

log10 x

0.01

0.005

-0.01

-0.005

-2 1.5 -1 -0.5 0.50

Золотарев tanh(8 tanh-1x)


Representation: Continued Fraction I

0

1

2

3

1 0 0

1 1 0

0 1 1

0 0 1

A

A

A

A

Consider a five-dimensional matrix of the form

10

01 10 11

1 121 2

1 32

1 0 0 0 1 0 00 0 01 0 0 0 1 00 0 0

0 1 0 0 0 0 0 0 10 0 0

0 0 1 0 0 0 1

SSS SS

S S SS

S

Compute its LDU decomposition

0 01

1; n n

n

S A S AS

where

3 3 3 32

2 21

10

1 1 11 1

1

S A A AS A A

S AA

then the Schur complement of the matrix is the

continued fraction


Representation: Continued Fraction II

We may use this representation to linearise our rational approximations to the sgn function

1

1 21

1

1 0

1

22

, ( )

n

n

n

n

n

n

cc

H HH

HH

cc

c cc

c

0

01

0 0

0 02

01

21

21 1

22 1 2

21 2 1

0

1

10

c c cn n n

c c cn n n

c c c

c c c c

Hn

Hn

H

H

Hc

as the Schur complement of the five-dimensional matrix


Representation: Partial Fraction I

2

22

1

1

1

2

2

0 0

0 0 0

0 0

0 0 0

0 0

1 1

1 1

1

1

11

x

p

p x

x

p

q

p

x

q

R

Consider a five-dimensional matrix of

the form (Neuberger & Narayanan)


Compute its LDU decomposition

So its Schur complement is

12 2

1

22 2

2

p x

x q

p xR

x q

2

2 2 2

2 2

2

1

1 1 1

2

1 2

2

1

1 0 0 0 0

1 0 0 0

0 0 1 0 0

0 0 1 0

1

p

x

p q x q

x x q x q

p

xp q x q

x x q x q

12 2

1 1

2

1

1

22 2

2 2

2

2

2

2 2

2

2 2

1

(

( )

)

0 0 0 0

0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0

x

p

p x q

xq

p x

x q

x

p

p x q

R

xqp x

x q

1

2

1

1 1

2 2

1 1

2

2 2

2 2

2 2

1 0 0

0 1 0 0

0 0 1

0 0 0 1

0 0 0 0 1

p

p

x

x

p

xq x q

x q x qp

xq x q

x q x q

1 11

11 01

1

1 12

21 02

1 0

0 0

0 0

0 0

0 0

0

x

pp x

q

x

R

x

pp

q

Representation: Partial Fraction II


Representation: Partial Fraction III

This allows us to represent the partial

fraction expansion of our rational function as

the Schur complement of a five-dimensional

linear system

1, 2 2

1

( )n

jn n

j j

pH H

H q


1

2 1

3 2 1

4 3 2 1

1 0 0 0 1 0 0

0 1 0 0 0 1 0

0 0 1 0 0 0 1

0 0 0 0 0 0 1

A

A A

A A A

C A A A A

1

2 1

3 2 1

4 3 2 1

1 0 0

0 1 0

0 0 1

0 0 0

A

A A

A A A

C A A A A

2

3

4

1 0 0 0

1 0 0

0 1 0

0 0 1

A

A

A

Representation: Cayley Transform I

1

2

3

4

1 0 0

1 0 0

0 1 0

0 0

A

A

A

A C

Consider a five-dimensional matrix of the

formCompute its LDU decomposition

CT 1 2 1n nS C A A A A So its Schur complement is

Neither L nor U depend on C

1CT 2

1 112 51

11 1

1P P T

TS

T

If where , and , then

1 nT T T 1s sA T

1C P P


Representation: Cayley Transform II

The Neuberger operator is

CTN

CT

,1

P P SD H

S

1 1

( ) ;1 1

T x xx T x

T x x

T(x) is the Euclidean Cayley transform of, ( ) sgn( )n m x x

1

x x T xT x

0 0 0 1T jj

j

xT x

x

For an odd function we have

In Minkowski space a Cayley transform maps between Hermitian (Hamiltonian) and unitary (transfer) matrices


(1) (1) (1)

(2) (2) (2)

(3) (3) (3)

(4) (4) (4)

00

00

D D P D PD P D D P

D P D D PD P D P D

The Neuberger operator with a general Möbius kernel is related to the Schur complement of

D5 (μ) (1) (1) (1) (1)

(2) (2) (2) (2)

(3) (3) (3) (3)

(4) (4) (4) (4) (4)

0 0 0 00 0 0 0

0 0 0 00 0 0

D D D DD D D D

P PD D D D

D D D D D

Representation: Cayley Transform III

with and( ) ( )

5

( ) ( )5

( ) 1

( ) 1

s ss W

s ss W

D b D M

D c D M

152 1P

( ) ( ) ( ) ( )5 55 5 5 5 5 5;s s s s

s

b cb c b c b c

P-μP-

P+ μP+


(1) (1) (1) (1)

(2) (2) (2) (2)

5 (3) (3) (3) (3)

(4) (4) (4) (4)

0 0 0 0

0 0 0 0( )

0 0 0 0

0 0 0 0

D D D D

D D D DD P P

D D D D

D D D D

P

1 0 0 0 0 0 0 1

0 1 0 0 1 0 1 0

0 0 1 0 0 1 0 0

0 0 0 1 0 0 1 0

P P

P

Cyclically shift the columns of the right-handed part where

5 5D D P

Representation: Cayley Transform IV

P+P- μP+μP-


( ) ( ) ( )s s sQ D P D P

Representation: Cayley Transform V

1 0 0 0

0 1 0 0

0 0 1 0

10 0 0 P P

C

(1)

(2)

(3)

(4)

0 0 0

0 0 0

0 0 0

0 0 0

Q

Q

Q

Q

Q1( ) ( )s s s M

ss M

HT Q Q

H

With some simple rescaling

11

12

13

14

1 0 0

1 0 01

0 1 05

10 0

( )

T

T

T

T P P

D

Q PCx

The domain wall operator reduces to the form introduced before


Representation: Cayley Transform VI

It therefore appears to have exact off-shell chiral symmetryBut this violates the Nielsen-Ninomiya theorem

q.v., Pelissetto for non-local versionRenormalisation induces unwanted ghost doublers, so we cannot use DDW for dynamical (“internal”) propagatorsWe must use DN in the quantum action instead

We can us DDW for valence (“external”) propagators, and thus use off-shell (continuum) chiral symmetry to manipulate matrix elements

5 5( ) (1)D D We solve the equation

Note that satisfies1 1DW ND D a

1 1 1 15 5 5 5 5, , 2 , 2 0DW N N N ND D a D D D a


Chiral Symmetry Breaking

Ginsparg-Wilson defect5 5 5 52 LD D aD D Using the approximate Neuberger operator 1

52 1aD H

L measures chiral symmetry breaking 212 1La H

The quantity is essentially the usual

domain wall residual mass (Brower et al.)

†

res †

tr

trLG G

mG G

mres is just one moment of L

G is the quark propagator


Numerical Studies

Used 15 configurations from the RBRC dynamical DWF dataset

316 32 12 2

0.8 1.8 0.02

V n L ns fM

Matched π mass for Wilson and Möbius kernelsAll operators are even-odd preconditionedDid not project eigenvectors of HW


mres per Configuration

mres is not sensitive to this small eigenvalue

But mres is sensitive to this one

ε


Cost versus mres


Conclusions: I

Relatively goodZolotarev Continued FractionRescaled Shamir DWF via Möbius (tanh)

Relatively poor (so far…)Standard Shamir DWFZolotarev DWF (Chiu)

Can its condition number be improved?

Still to doProjection of small eigenvalues

HMC5 dimensional versus 4 dimensional dynamicsHasenbusch acceleration

5 dimensional multishift?Possible advantage of 4 dimensional nested Krylov solvers

Tunnelling between different topological sectorsAlgorithmic or physical problem (at μ=0)Reflection/refraction


Approximation Theory

We review the theory of optimal polynomial and rational Чебышев approximations, and the theory of elliptic functions leading to Золотарев’s formula for the sign function over the range R={z : ε ≤|z|≤1}. We show how Gauß’ arithmetico-geometric mean allows us to compute the Золотарев coefficients on the fly as a function of ε. This allows us to calculate sgn(H) quickly and accurately for a Hermitian matrix H whose spectrum lies in R.


What is the best polynomial approximation p(x) to a continuous function f(x) for x in [0,1] ?

Polynomial approximation

Best with respect to the appropriate norm

where n 1

1/1

0

( ) ( )n

np f dx p x f xn


Weierstraß’ theorem

Weierstraß: Any continuous function can be arbitrarily well approximated by a polynomial

0 1minmax ( ) ( )

p xp f p x f x

Taking n →∞ this is the Taking n →∞ this is the minimax norm norm


Бернштейне polynomials

0

(1 )n

n n kn

k

nkf x x

n kp x

The explicit solution is provided by Бернштейне polynomials


Чебышев’s theorem

The error |p(x) - f(x)| reaches its maximum at exactly d+2 points on the unit interval

0 1max

xp f p x f x

ЧебышевЧебышев: There is always : There is always a unique polynomial of any a unique polynomial of any degree d which minimises degree d which minimises


Чебышев’s theorem: NecessitySuppose p-f has less than d+2 extrema of equal magnitudeThen at most d+1 maxima exceed some magnitude

And whose magnitude is smaller than the “gap”The polynomial p+q is then a better approximation than p to f

This defines a “gap”We can construct a polynomial q of degree d which has the opposite sign to p-f at each of these maxima (Lagrange interpolation)Lagrange

was born in Turin!


Чебышев’s theorem: Sufficiency

Then at each of the d+2 extrema i i i ip x f x p x f x

Thus p’ – p = 0 as it is a polynomial of degree d

Therefore p’ - p must have d+1 zeros on the unit interval

Suppose there is a polynomialp f p f


The notation is an old transliteration of Чебышев !

Чебышев polynomials

Convergence is often exponential in d The best approximation of degree d-1 over [-

1,1] to xd is 1

112

dd

d dp x x T x

1cos cosdT x d x

Where the Чебышев polynomials are

1 ln21 2

2

dd

d ddx p x T x e

The error is


Чебышев rational functions

Чебышев’s theorem is easily extended to rational approximations

Rational functions with nearly equal degree numerator and denominator are usually best

Convergence is still often exponential

Rational functions usually give a much better approximation

A simple (but somewhat slow) numerical algorithm for finding the optimal Чебышев rational approximation was given by Ремез


Чебышев rationals: ExampleA realistic example of a rational approximation is

x 2.3475661045 x 0.1048344600 x 0.00730638141

0.3904603901x 0.4105999719 x 0.0286165446 x 0.0012779193x

Using a partial fraction expansion of such rational functions allows us to use a multishift linear equation solver, thus reducing the cost significantly.

1 0.0511093775 0.1408286237 0.59648450330.3904603901

x 0.0012779193 x 0.0286165446 x 0.4105999719x

The partial fraction expansion of the rational function above is

This is accurate to within almost 0.1% over the range [0.003,1]

This appears to be numerically stable.


Polynomials versus rationals

Optimal L2 approximation with weight is

2

1

1 x2 1

0

( ) 4( )

(2 1)

jn

jj

T xj

Optimal L∞ approximation cannot be too much better (or it would lead to a better L2 approximation)

lnn

e Золотарев’s formula has L∞ error

This has L2 error of O(1/n)


Elliptic functions

Elliptic function are doubly periodic analytic functionsJacobi showed that if f is not a constant it can have at most two primitive periods


Liouville’s theorem

( )P

dzf z

0

0

( )dzf z

( )dzf z

( )dzf z

0

( )dzf z

0

( ) ( )dz f z f z

0

( ) ( )dz f z f z

ω

ω’ ω+ω’

0

Res ( ) 0f


Liouville’s theorem: Corollary I

( )( ) ln ( )

( )f z

g z f zf z

Apply this to the logarithmic derivative

1( ) ( ) ( )j jr rj jf z c z O z

Laurent expansion about each pole ζj

( ) 2 0jjP

dzg z i r

Hence number of poles must equal the number of zeros (with multiplicity)


Liouville’s theorem: Corollary II

It follows that there are no non-constant holomorphic elliptic functions

If f is an analytic elliptic function with no poles then f(z)-a could have no zeros either

The more common form of

Liouville’s theorem was

actually due to Cauchy


Liouville’s theorem: Corollary III

( ) 2P

dz h z i n n

Apply Liouville’s theorem to h(z) = z g(z)

Where n and n’ and the number of times f(z) winds around the origin as z is taken along a straight line from 0 to ω or ω’ respectively

1

m

k kk

n n

If αk and βk are the locations of the poles and zeros of f, then the sum of the locations of the poles minus the location of the zeros vanishes modulo the periods


Weierstraß elliptic functions: I

A simple way to construct a doubly periodic analytic function is as the convergent sum

3,

1( ) 2

m m

Q zz m m

Note that Q is an odd function


Weierstraß elliptic functions: II

The derivative of an elliptic function is clearly also an elliptic function, but in general its integral is not

Q z z c

1 12 2

But since Q is odd its integral is even, and thus

2 30

1 2zz d Q

z

So the Weierstraß function is elliptic

The only singularity is a double pole at the origin and its periodic images

The name of the function is , and the function is called the Weierstraß

function, but what is its name called?

(with apologies to the White Knight)


Weierstraß elliptic functions: III

satisfies the differential equation

where

2 3

2 34z z g z g

3

6,

0

1140 m m

m m

g

m m

24

,

0

160 m m

m m

g

m m

Thus we may express the functional inverse of as the elliptic integral

32 34Z

dwz

w g w g


Weierstraß elliptic functions: IV

The integral of - is not an elliptic function

20

1 1zz du u

z u

1 12 2z z

But it is an odd function, and so satisfies

2 i And Liouville’s theorem gives the useful identity

Simple pole at the origin with unit residue


Weierstraß elliptic functions: V

That was so much fun we’ll do it again.

0

1exp

zz z u

u

Integrating gives the odd function ln

12z

z e z

With the periodic behaviourNo poles, and simple zero at the origin


Expansion of elliptic functions: I

Let f be an elliptic function with periods ω and ω’ whose poles and zeros are at αj and βj respectively

Recall that ; w.l.g. set this to 0 1

m

k kk

n n

1 2

1 2

n

n

z b z b z bg z

z a z a z a

The function g then has the same poles and zeros as f

And is periodic, and hence elliptic

1

expn

j jj

g z g z g z


Expansion of elliptic functions: II

By Liouville’s theorem f/g is an elliptic function with no poles, and is thus a constant C

1 2

1 2

n

n

z b z b z bf z C

z a z a z a

Every elliptic function may thus be written in the “rational” form


Addition formulæ: I

Similarly, any elliptic function f can be expanded in “partial fraction” form in terms of Weierstraß ζ functions.

12

u vu v u v

u v

By considering this allows us to show

that the ζ function satisfies the addition formula

uf u

u v


Addition formulæ: II

2

14

u vu v u v

u v

And by differentiation we obtain an addition formula for too

The principal conclusion is that any elliptic function f can be represented in terms of rational functions of Weierstraß functions of the same argument with the same periods

e of z R z R z z


Jacobian elliptic functions: I

Although Weierstraß elliptic functions are most elegant for theoretical studies, in practice it is useful to follow Jacobi and introduce the function sn(z;k) through the inversion of the elliptic integral

sn

2 2 20 1 1

z dtz

t k t


Jacobian elliptic functions: II

12

2 3

sn ;z

z kz e z e

Of course, this cannot be anything new – it must be expressible in terms of the Weierstraß functions with the same periods

Where the “roots” of the cubic form in Weierstraß differential equation are

2 2 2

1 2 3

1 6 1 6 1, ,

6 12 12k k k k k

e e e


Jacobian elliptic functions: III

We are hiding the fact that the Weierstraß functions depend not only on z but also on periods ω and ω’

1

2 2 20

( )1 1

dtK k

t k t

It is not too hard to show that the periods of the Jacobian elliptic functions are where and the complete elliptic integral is

4 ( ), 2 ( )K k iK k 2 2 1k k


Jacobian elliptic functions: IV

2

213

1sn ;

1z k

z k

For later purposes we note the relation between the even Jacobi and Weierstraß elliptic functions

A corollary of this is that any even elliptic function with periods may be expressed as a rational function of sn(z;k)2

4 ( ), 2 ( )K k iK k


Modular transformations: I

We observed earlier that there are many choices of periods ω and ω’ which generate the same period lattice: if we choose

with this will be the case.

,

det 1

Since the period lattices are the same, elliptic

functions with these periods are rationally expressible in terms of each other

This is called a first degree transformation


Modular transformations: II

We can also choose periods whose lattice has the original one as a sublattice, e.g., , /n

Elliptic functions with these periods must also be rationally expressible in terms of ones with the original periods, although the inverse may not be true

This is called a transformation of degree n


Let us construct an elliptic function with periods

4 ( ), 2 ( ) /K k iK k n

Modular transformations: III

This is easily done by taking sn(z;k) and scaling z by some factor 1/M and choosing a new parameter λ

We thus consider sn(z/M; λ) with periods for z/M of

4 4 , 2 2L K iL iK

The periods for z are thus 4LM and 2iL’M, so we must have LM=K and L’M=K’/n


Modular transformations: IV

The even elliptic function must therefore

be expressible as a rational function of

sn / ;

sn ;

z M

z k

2sn ;z k

Matching the poles and zeros, and fixing the overall normalisation from the behaviour near z=0 we find

2

2/ 2

21

212

sn ;1

sn / ; sn 2 / ;1sn ; sn ;

1sn 2 / ;

n

m

z k

z M iKm n k

z k M z k

iK m n k


Modular transformations: V

The quantity M is determined by evaluating this identity at the half period K where sn ; sn / ; sn ; 1K k K M L

Likewise, the parameter λ is found by evaluating the identity at z = K + iK’/n

/ 2 2

21

1sn ;

1

nm

m m

czM M c

It is useful to write the identity in parametric form


Im z

Re z0 2K

K-2K

-K

iK’

-iK’

Золотарев’s formula: I

Real period 4K, K=K(k)Imaginary period 2iK’, K’=K(k’) , k2+k’2=1

Fundamental region

∞1/k10-1

-1/k


Золотарев’s formula: IIsn(z/M,λ)

sn(z,k)

Im z

Re z0 2K

K-2K

-K

iK’

-iK’


Arithmetico-Geometric mean: I

Long before the work of Jacobi and Weierstraß

Gauß considered the following sequence 1

1 12 ,n n n n n na a b b a b

Since these are means

1 1;n n n n n na a b a b b

Furthermore 22 2 11 1 4 0n n n na b a b

1 1n n n na a b b

So it converges to the AG mean

a b


Gauß transformation

2

1 sn ;2sn 1 ;

1 1 sn ;

k u kkk u

k k u k

Gauß discovered the second degree modular transformation (but not in this notation, of course)

Let 1

2, sn 1 ; , sn ;

1n n

n nn n

a b kk s k u s u k

a b k

The the Gauß transformation tells us that

1 1

2 21 1

1 21

n n nn

n n n n n n

k s a ss

ks a b a b s

This is just a special case of the degree n modular transformation we used

before


Arithmetico-Geometric mean: II

On the other hand

1

2 2 21 1 42 20 0 1 121

11

n ns s

dt dt

t k t kt t

k

uk

So the quantity

is a constant (the RHS is independent of n)

1

2 2 2 2 2 2 21 1

2 21 1 1 11 0 0

n n

n n n n

s s

dt dt

t a t b t t a t b tn

ua


Arithmetico-Geometric mean: III

Therefore it must have the value

2 2 2 2

1

21 11 0 0

sin1

21

s s

t a t b tn

su dtdta a at

We may thus compute

if we start with , then also

21 sn ; ,1, 1ns u k f u k

2

sin if

, , 2with , , if

2

au a b

f u a b a a bf u ab a b

a b a b

2

K ka

0 0

11,

1k

a bk


Arithmetico-Geometric mean: IV

We have a rapidly converging recursive function to compute sn(u;k) and K(k) for real u and 0<k<1

Values for arbitrary complex u and k can be computed from this using simple modular transformations

These are best done analytically and not numerically

With a little care about signs we can also compute cn(u;k) and

dn(u;k) which are then also needed

Testing for a=b in floating point arithmetic is sinful: it is better to continue until a ceases to decrease


Conclusions: II

One can compute matrix functions exactly using fairly low-order rational approximations

“Exactly” means to within a relative error of 1 u.l.p.Which is what “exactly” normally means for scalar computationsOne can also use an absolute error, but it makes no difference for the sign function!

For the case of the sign function we can evaluate the Золотарев coefficients needed to obtain this precision on the fly just to cover the spectrum of the matrix under consideration

There seem to be no numerical stability problems


Bibliography: I

Elliptic functionsНаум Ильич Ахиезер (Naum Il’ich Akhiezer), Elements of the Theory of Elliptic Functions, Translations of Mathematical Monographs, 79, AMS (1990). ISBN 0065-9282 QA343.A3813.


Bibliography: II

Approximation theoryНаум Ильич Ахиезер (Naum Il’ich Akhiezer), Theory of Approximation, Constable (1956). ISBN 0094542007.

Theodore J. Rivlin, An Introduction to the Approximation of Functions, Dover (1969). ISBN 0-486-64069-8.


Hasenbusch Acceleration

Hasenbusch introduced a simple method that significantly improves the performance of dynamical fermion computations. We shall consider a variant of it that accelerates fermionic molecular dynamics algorithms by introducing n pseudofermion fields coupled with the nth root of the fermionic kernel.


Non-linearity of CG solver

Suppose we want to solve A2x=b for Hermitian A by CG

It is better to solve Ax=y, Ay=b successivelyCondition number (A2) = (A)2

Cost is thus 2(A) < (A2) in general

Suppose we want to solve Ax=bWhy don’t we solve A1/2x=y, A1/2y=b successively?

The square root of A is uniquely defined if A>0This is the case for fermion kernels

All this generalises trivially to nth rootsNo tuning needed to split condition number evenly

How do we apply the square root of a matrix?


Rational matrix approximation

Functions on matricesDefined for a Hermitian matrix by diagonalisationH = UDU-1

f(H) = f(UDU-1) = U f(D) U-1

Rational functions do not require diagonalisation

Hm + Hn = U( Dm + Dn) U-1

H-1 = UD-1U-1

Rational functions have nice propertiesCheap (relatively) Accurate


No Free Lunch Theorem

We must apply the rational approximation with each CG iteration

M1/n r(M)The condition number for each term in the partial fraction expansion is approximately (M)So the cost of applying M1/n is proportional to (M)Even though the condition number (M1/n)=(M)1/n

And even though (r(M))=(M)1/n

So we don’t win this way…


Pseudofermions

We want to evaluate a functional integral including the fermionic determinant det M

1

det MM d d e

We write this as a bosonic functional integral over a pseudofermion field with kernel M -1


Multipseudofermions

We are introducing extra noise into the system by using a single pseudofermion field to sample this functional integral

This noise manifests itself as fluctuations in the force exerted by the pseudofermions on the gauge fieldsThis increases the maximum fermion forceThis triggers the integrator instabilityThis requires decreasing the integration step size

11

detnMnM d d e

A better estimate is det M = [det M1/n]n


Hasenbusch’s method

Clever idea due to HasenbuschStart with the Wilson fermion action M=1-HIntroduce the quantity M’=1-’HUse the identity M = M’(M’-1M)Write the fermion determinant as det M = det M’ det (M’-

1M)Introduce separate pseudofermions for each determinantAdjust ’ to minimise the cost

Easily generalisesMore than two pseudofermionsWilson-clover action


Violation of NFL Theorem

So let’s try using our nth root trick to implement multipseudofermions

Condition number (r(M))=(M)1/n

So maximum force is reduced by a factor of n(M)(1/n)-1

This is a good approximation if the condition number is dominated by a few isolated tiny eigenvaluesThis is so in the case of interest

Cost reduced by a factor of n(M)(1/n)-1

Optimal value nopt ln (M)

So optimal cost reduction is (e ln) /

This works!


RHMC technicalities

In order to keep the algorithm exact to machine precision (as a Markov process)

Use a good (machine precision) rational approximation for the pseudofermion heatbathUse a good rational approximation for the HMC acceptance test at the end of each trajectoryUse as poor a rational approximation as we can get away with to compute the MD force

saturday, 02 may 2015saturday, 02 may 2015saturday, 02 may 2015saturday, 02 may 2015 2005 taipei...

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