Noncritical multigrid Monte Carlo: O(3) non-linear σ model

110

Nuclear Physics B (Proc. SuppL) 2~ (t95~-i ~ ; ~ t i $

NONCRITICAL MULTIGRID MONTE CARLO : 0 ( 3 ) NON-LINEAR ~ MODEL Martin HASENBUSCH and Steffen MEYER Fachbereich Physik der Universit~;t Kais*~rslautern, Germany Gerhard MACK II. Institut fi]r Theoretische Physik der Universit~t Hamburg, Germany We describe a multigrid Monte Carlo technique for asymptot~aity rice lattice field theories like the O(n) non-linear cr model and the SU(N) x SU(N) chiral model in two d i ~ . We study this multigrld Monte Carlo algorithm with a smooth interpolat;on operator in the t w o - d i m l 0 ( 3 ) ~rmodel. On lattices up to 256 × 2~6 we achieve almost complete elimination o f critical slo~ng down. With piecewise-constant interpolation we fimJ a remaieing oitical slo~ing down w~h a dynamic critical exponent o f z ~. 1. 1.

INTRODUCTION

The phenomenon o f c~tical s l o ~ n g down is not restricted to Monte Carlo simulations. To speed up the convergence rate in the soiutt~0~ o f large systems o f linear equations by iterative methods the multigr;,d method was introduced and has become a widely used algorithm in many F,elds o f numerical analysis during the last decade- Multigrid algorithms overcome sio~ng down bY updating on various length scales. The basic idea o f the mu~grkJ approach when applied to study the continuum limit o f lattice field theory is to map a nearly critical system into a noncritical theory to overcome critical slowing down. This can be achieved by a deconplieg o f the various length scales. O~--'ecan expect that this crucially depends on the form o f the interpolation operator At [~zUiee'~9 we reported first results o f a multigrld Monte Carlo study with a smooth interpolation operator for the ~b4-theory in 4 dimensions 1 We present here a modified muitigrid Monte Carlo algorithm for asymptotically free non-linear models with O(N) or SU(N) × SU(N) symmetry and apply it to the 0 ( 3 ) non-linear ~ model. Our formulation is particular useful when one wants to study the role of different interpolation operators in the multigrid cycle.

2.

MULTIGRID M O N T E CARLO ALGORITHM Updates o . the rations levels o f a multigrid systern (:an be interpreted as non-local updates o f the fundamental fletd. For the sake o f siraplicity we will employ this po~ut o f view in the following. Let us first ~ c u ~ the solution o f the Laplace equation with a multigrid algo~thm. Tile solution o f the Laplace equation in two dimens~ns with a S-point crlseretisation i~_the minimum o f the energy functional 1

E

= ~

~ ( ÷ ~ - ÷j)2

(2.z)

with ~bi real. Starting ~rom some trial solution one performs cycles o f non-local updates ~b'; = 4~i + ~ - ~

(2.2)

with • a real number, such that - ~ = 0. The real kernel ~ is only nonzero within a LB × LB Mock. This allows for different choices of kernels A~. We have investigated the step function kernel and a smooth interpolation kernel. The latter has the property that Ai is zero on the boundary o f the block and grows continuously untill the centre o f the block is reached. The multigrid V-cycle starts with local updates and then performs a sequence o f updates with blocks of size 2 n. For the Laplace problem eq.(2.1) there exist proofs that slowing down is completely eliminated

0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-Holland)

M. Haser~b~cb er al. /.Sfo~criticM multigrid Mor~te C~rto

111

f ~ _~m~lLh kermds, where~s no s~:h prc~E is known f ~ step fuv~lio~ ke~rals. A ~cent n u r n e ~ | study d~ncms~ates ~ . b h~h accuracy that for the step func~c~ ~ the n u . ~ o f ~ o n s to reach a accura~ grows ~ the l~.a: s~ze L of the system 3 . Tb~s a~go,r~hm can eas~ be t r a ~ into a J,~poEs ~ for the G~u~sian mada which is ~ v e . by tt, e ~ k , . $:

A,, upaate ~

is l~,,m by

V ~ r e 1: Time hm~ry ~ ~ e s u s c q ~ t ~ X on a ~ b t t ~ e ~t 3' : 1 . 7 0 . 3gO0 ~ r a ~ n s a

~ is row a unKorm ~ e d random n m ~ bet from the ~eeva! (-~.~). T ~ ~ . c e is L ~ us creepier the change o f the eneq~ ~ for a m dlmensional sysl~m due to a non-kx:al

.pdate ~ ~,e

the give. ~ . ~ r , ~ .

÷ = ~ns~.

~the ground

For the s~ep function l~mel

b ~ h types of kemds. A c~ns~m aCCelmmCeraf~ requires a c~m~ant change o f the emerge. acc, r~tant a._~,.=-tan~rate requires• (x 1 / ' , ~ u for the s~p ~uncl~n kemd and e ~ndependmt o f L ~ forthesmoothla~ei. The ~ to non-rmear o m o d ~ m wh~h the state space at each ~ e is a compact l~e lFoup leads to update proposals

where A isa linearcombination of generators of the Lie group. 3.

MULTIGRID MONTE CARLO SIMULATION We did numerical studies o f the performance of the multigrid Monte Carlo algorithm with step function kernels 4and smooth kernels for the 0 ( 3 ) model defined by the action

s = - ~ ~ ~,. ~ (~)

(3.~)

wlvme ~ is a tl~ee cFmem~a~ ~ vector. W e c o ~ s k ~ oeay V-cycks. s~,ce t ~ ~ ~ ~ ~or a W - c ~ k ~ ik~ L d - L . w l ~ me cba,q~ d S ~ ~omp.t~ ~ ~ a.m~ d ~ ~damm~ taJ ~ L W e l ~ ~ ~ s m w ~ k ~ m d to be L a ~ a ~ W a ~ m on • L m ~ L ~ ~ ~ D;m~ let bounda~ ~ . - . . ~ m L A~ is mxmarmd s E b U I ~ ~

~ = l for ~ M o c k ~

Ls.

~ ~ ~ e n e q ~ p e r ~ e E ~ t h e

tha~ 6 ~or a~ c ~ ~ t he ~ e s ~ e e ~ e ~ d s c ~ d n ~ af~e~ the dysamk~ o ~ c a l ~ d the system_ ~

bt~e

~e

L = 2~

and a ~ , , ~

a reasonable ammmt o~ ~ tkne t o del~rmine the autoconda6on time f~r local upda6ng. But looking at the time history d the ~ , , , ~ . Jb i ~ X . Fq~. 1 . isi~stmcbve_ $ ~ f r o m a thernlaliz~d a : m f i ~ from a multis~d run th= system evolves for 3000 ;terat~ns using Metropolis updating on the fine lattice, fo.ovRd W 1000 multigrid updates. The values of X are strongly

112

M. Hasenbus~,h et al /Noncritical multigrid Mon~e Carlo

.5 &

T, I 1

~

J

IlI,~l

A

A

I

z~

I

I

1

~

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Figure 2: Acceptance rates CO) and stepsizes (E~) for step function kernels versus blocksize Z,a as explained in the text. Also shown are the acceptance rates ( A ) for smooth kernels for a fixed stepsize.The solid lines are explained in the text. correlated over thousands e f iterations when using local updating, while the values o f X generated by the mu]tlgrid updates resemble white noise around a mean value < X > : t262 5.6 Next we study the acceptance rotes for the Metropolis updating with step function kernels and smooth kernels. We adjust the stepe~e ~ t o ~ an acceptance rate around 500~Dfor the local update_ In Fig. 2 results are shown for a 642 lattice a t / ~ : 1.50. Using equal stepsize • for all length scale~ the acceptance rate for the smooth kernels (symbol A ) is independent o f the b]ocksize LB, while tile acceptance rate for the step function kernel (symbol O) isdropping from the initial value o f 50% for the local update to 8% f o r / - e = 32, indicating that collective moves of this size are rare. But the stepsize • can be adjusted by hand t o get an acceptance around 50% for all block sizes LB. The resulting stepsizes are also shown in Fig.2 (symbol [3). The stepsize • o f the local update reduces to c/7 for L e = 32. The drop o f acceptance rate and step size for piecewise constant interpolation shows the expected behaviour ~ z - ~ven • by the solid lines in Fig. 2 - as discussed above. The efficiency of a stochastic algorithm can be characterized by the autocorrelation time as deter-

01 0

200

400

6DO

Figure 3: Autocorrelat~on functions for suscel~bility X with step function kernels ( O ) and smooth kernels(E3) at ~ = 1.60. mined from the normalized a ~ a t i o n p(t)

( ~(A~) ~÷') - (A)~ - (n) ~

=

functiam

(3-2)

o f the observaMes A in L~e usual way. We calculated integrated autocorrelation times =

if

-

e(t)

(3.3)

a self-consistent truncation window o f width 4 r for the energy E and the suScel~bW~y X with the two different types o f kemeis~ In Fig.3 we give the autocorrelation functions for the susceptibility X with step function kernels and smooth keenels a t / 3 = 1.6. Both autocorrelatlon functions d o show a d e a r e x ~ - ~ m t i a l decay. Our final results concerning the performance o f the multigrid Monte Carlo updating with different types o f kernels are summarized in Fig.4. Assuming a relation ~- ec ~ we can fit the data in Fig.4 with a dynamical critical exponent z --- 1.2(3) for step function kernels and z = 0.2(1) for smooth kernels. For the selected couplings and lattice s~zes our data are gathered in the table. All our results do agree quite well with the data provided by 5,6 except that our value for the energy at L = 2,56 and J~ = 1.7 is in agreement with 6 but not with 5. We used a total amount o f 15 hours CPU time on a Cray.

M. lta~enbusch e~ aL / Noncdtical m~d~i~tid _~,fonteCairo

113

ACK NOWLEDGEM ENTS We thank the Deutsche Forschung~emeinschaft for support through grants Ma 493/8-3 and Me 567/3-3. One of us (S.M.) w~uld ~iketo thank the organizers Of Lwttl'cz"90 for their hospP~aiityand the stimulating a t ~ e at the conference. The comput/ni~ was done at HLRZ J~lich. REFERENCES I- G~ Mack arKi S. Meyer, Plud. Phy~ B (Proc. Suppl.) 17 ( 1990 ) 293 2_ G. Mack, m: Noe~m~d~t~ee ~ma~mm fiadd 16 2~ Fgure 4: Integrated a u t o ~ , ~ o n times for step ~.~o. ~ (o) and ~ k~ (~)

eemb~t~ cravat d o u g down for the 0(3) no~ I~maur o modal. Tile all~uJ;ihm ~s easy to [ m l ~ merit on a w ~ - ~ . We ~ exp¢~

int~-e~J~ resultsfee the ~ mettmds for o models ~

S/7(/~)

x

sU(t~) where ~

fn~w~ea ~ g

i~ tim sjtmme~ I F o ~ a~o~hms lead to

mnde~.

Table: Data from muitigrid Monte Carlo rum smooth kemds : lattice si~e L, coupllng/~, ~ i ~ tics star, energy E , ~___~cepti~ "bi[dy X and autocorrelation times ~r~ of X-

Pres~.Y~g88 3. J.Lmn. Diploma ~ . la~tem, 1991.

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