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Gauge-invariant coarse-to-fine transformations for multigrid Monte Carlo algorithms in pure lattice gauge theories
Volume 210, number 1,2
PHYSICS LETTERS B
18 August 1988
GAUGE-INVARIANT COARSE-TO-FINE TRANSFORMATIONS FOR MULTIGRID MONTE CARLO ALGORITHMS IN PURE LATTICE GAUGE THEORIES Karsten D E C K E R
Institute for Computer Science and Applied Mathematics, University of Berne, CH-3012 Berne, Switzerland Received 27 April 1988
A class ofgauge-invariant coarse lattice to fine lattice transformations is constructed for pure lattice gauge theories. Coarse-tofine transformations will play an essential role in multigrid Monte Carlo updating algorithms which are believed to cope with the problem of critical slowing down, Efficiency estimates for multigrid Monte Carlo updating algorithms are given and preliminary results from the implementation of such an algorithm for four-dimensional U( 1) lattice gauge theory are reported.
It is well known that M o n t e Carlo ( M C ) simulations o f lattice gauge theories and statistical mechanical systems show critical slowing down when large lattices are simulated with p a r a m e t e r values which let the system develop collective b e h a v i o u r over distances of m a n y lattice spacings. The p h e n o m e n o n o f critical slowing down is clearly due to the locality property o f s t a n d a r d M C updating algorithms. Here, by local we mean interactions o f the scale o f one lattice spacing. While nonlocal generalizations o f stand a r d updating algorithms in principle should be able to avoid critical slowing down, they wil be difficult to design and efficient i m p l e m e n t a t i o n s on computers with vector and parallel architecture will be hard to achieve. Keeping in m i n d efficient i m p l e m e n t a t i o n s on m o d e r n computers, one conceivable a p p r o a c h to master critical slowing down is to carry through the updating process not only on the original lattice, but also on its block lattices. The essential idea is to use s t a n d a r d M C updating algorithms with simple local actions on all the block lattices to quickly establish the long-distance properties o f lattice field configurations, and to correct the short-distance properties on finer lattices with a p p r o p r i a t e l y chosen lattice spacings. In analogy to multigrid solvers for partial differential equations, where the error m o d e o f a cer~ Work supported in part by Schweizerischer Nationalfonds. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
tain wavelength is always eliminated on a grid with corresponding mesh-size, the method sketched above will be called M u l t i G r i d M o n t e Carlo ( M G M C ) updating. An i m p o r t a n t part o f any M G M C updating algor i t h m is the coarse-to-fine transformation, which is needed to m a p the long-distance properties o f lattice field configurations established on some coarse lattice to the next finer one. It is the aim o f this note to construct a class o f gauge-invariant coarse-to-fine transformations for pure lattice gauge theories. We consider a pair o f hypercubical lattices A B c (aBZ) a,
A c ( a ~ _ ) d,
(1)
which we call block lattice and (fine) lattice, respectively, a B a n d a are the corresponding lattice spacings which, for simplicity are assumed to fulfill a / a " = ½. d denotes the space-time dimension o f the lattices for which no specific value is assumed. (A ~, A) can be any pair o f direct successors o f lattices in the sequence o f lattices used in a M G M C updating algorithm. All quantities defined on A B will carry superscript 'B' to distinguish them from quantities defined on A which carry no superscript. Finally, our consideration will be valid for arbitrary gauge group G. We first discuss the geometrical relationship between block lattice and fine lattice. To each lattice site x B on A ~, there correspond 2 a sites y on the fine 207
V o l u m e 210, n u m b e r
1,2
PHYSICS
lattice which define the corners of a hypercube contained in A. Consequently, to each block lattice gauge field U,-,:, n in arbitrary but fixed direction #, there T correspond 2 J-1 pairs (U~.~, L,,+~,:,) of fine lattice gauge fields with y=
2x B ,
y=2xB+~a,
LETTERS
B
18 A u g u s t 1988
exp [Re ;/F ( V~ Uv+,~:, Uv:,) l
(5)
in the transformation. In addition we have to introduce fluctuations characteristic of the value of the coupling parameter fl at which the fine lattice gauge field configuration is to be equilibrated. This can be provided with a factor exp[S(fl) ] ,
21 = 1, ..., d,
21##,
where S(fl) is the action on the fine lattice. Altogether we propose to perform the coarse-to-fine transformation in a probabilistic way and to choose the fine lattice gauge fields Uv:,, Uv+~..... with probability
y=2xn(£, +£~)a, 2 j < 2 2 = 1 .... ,d, y=2x'+
2,¢#,
(£, +'~2 + / 3 ) a ,
2 1 < 2 2 < 2 3 - - - 1 ..... d,
2, va# .
(2)
Here .~ denotes the unit vector in direction )~. We next discuss the general properties which any coarse-to-fine transformation has to fulfill. If we demand gauge-invariance on both the block lattice and the fine lattice, the coarse-to-fine transformation has to be gauge-invariant, too. Gauge-invariant quantities are easily constructed from combinations of block lattice and fine lattice gauge fields. In view of the use in the coarse-to-fine transformation, we suggest
;(~(v .B U,.+,~.:. U,,:.),
(3)
with W B :=
B* Bt BI U,<.+ 52 (U~..v • g.rB+~a;t~
+ ir IB
B
gxB+lia; u
B* ,~..._~ . . . .~JB+ . . . ~..~U,.~+(,., ~).:~)
(4)
and )ff is the group character in the fundamental representation. This choice introduces enough averaging over those block lattice gauge fields which are relevant for the determination of the value of some fine lattice gauge field and is still almost local in the sense defined above. This second property is important in order to obtain a sufficient degree of computational parallelism in the coarse-to-fine transformation algorithm. We now come to the explicit construction of the transformation itself. The primary objective of the transformation is to map the properties of the coarse lattice gauge field configuration to a gauge field configuration living on the next finer lattice. This can be done in a probabilistic way by inclusion of a factor 208
(6)
P,( U,,:.; U,,+,~.:.) : = e x p [ C R e Zf(
V.BU,,+r,.:. U~,:,~)] exp [S(fl)
], (7)
where we have introduced an additional tunable parameter C, which can be used to optimize the relative strength of the two competing factors in the transformation. Such a parameter seems to become important in any attempt to optimize the coarse-to-fine transformation. The properties of the coarse-to-fine transformation will substantially influence the overall efficiency of M G M C updating algorithms. We recall that the final aim is to generate uncorrelated gauge field configurations on a large lattice at almost critical parameters as quickly as possible. In a coarse-grained pseudocode representation, the structure of a full M G M C updating algorithm employing all the block lattices is presented in fig. 1. As can be easily concluded from the figure, the total work to be spent in a M G M C updating algorithm for the generation of a new uncorrelated gauge field configuration is composed of the decorrelation work on the coarsest lattice available, the equilibration work on the intermediate and finest lattices, and the coarse-to-fine transformation work. To estimate the costs for the decorrelation process, we first note that for blocking factor two, the number of operations needed to perform one full updating sweep on the next coarser lattice is always reduced by a factor 2 -J, if we make the reasonable assumption that we use the same updating algorithm on both lattices. In addition, the correlation length in lattice units
Volume 210, number 1,2
PHYSICSLETTERSB
generate next appropriate, statistically uncorrelated gauge field configuration on the coarsest lattice available perform coarse-to:fine transformation to the next finer block lattice perform updating ('equilibration') sweeps on this block lattice with appropriate action to establish the correct properties on the length scale given by the lattice constant of this block lattice perform coarse-to-fine transformation to the finest lattice perform updating ('equilibration') sweeps on the finest lattice to establish the correct properties on the length scale a Fig. 1. Coarse-grainedpseudocoderepresentation of a full MGMC updating algorithm. is reduced by a factor of two in the transition from any lattice to the next coarser one. Since the autocorrelation time rc typically behaves like rc ~ {2 where is the correlation length corresponding to the lightest state of the theory, this behaviour together with the first observation implies that in a MGMC updating algorithm the decorrelation process itself will be performed at practically no cost. Along the same line of reasoning, the equilibration work on all the block lattices will also be negligible as compared to the equilibration on the largest lattice: if Wis the equilibration work needed on the largest lattice, the total equilibration work needed on all the lattices will be bounded by W / ( 1 - 2 - a ) . Here we have implicitly assumed that on all block lattices we can work with simple actions very similar to the one used on the largest lattice. We will comment on this assumption in the last paragraph of this paper. Altogether, we conclude that the efficiency of a MGMC updating algorithm depends on the number of final equilibration sweeps on the largest lattice and on the efficiency of the coarse-to-fine transformation. Since the number of equilibration sweeps on the largest lattice can be reduced, if the quality of the coarse-to-fine transformation is improved, the efficiency of a MGMC updating algorithm will almost
18 August 1988
exclusively be determined by the properties of the coarse-to-fine transformation. We now shortly report on our numerical studies. Based on the coarse-to-fine transformation constructed above, we have implemented a two-lattice MGMC updating algorithm for four-dimensional U( 1 ) lattice gauge theory. We used a 164 and an 84 lattice, respectively and applied a standard Metropolis [ 1 ] updating algorithm on both lattices. The longdistance effective action needed in the updating procedure on the 84 block lattice was determined previously in a numerical fashion [2] using the Monte Carlo renormalization group with Swendsen b = 2 blocking [3] and Wilson's two-lattice matching method. Using our two-lattice MGMC updating algorithm, so far we generated gauge field configurations for a single value of the coupling parameter, fl= 1.0080, close to the transition coupling determined by two different groups [4,5] to be flT=l.0106_+0.0018 and flT=l.0111_+0.0005, respectively (for this value of the coupling parameter, the autocorrelation time for a standard, one-lattice Metropolis algorithm was determined to be rc ~ 400 updating sweeps, corresponding to ~ 700 s CPU time on a Cray X-MP). The preliminary analysis of our data which we have obtained without any attempt to optimize the many free parameters available even in our two-lattice MGMC updating algorithm allows the conclusion that our two-lattice algorithm generates uncorrelated 164 gauge field configurations about four times.faster than a standard, one-lattice Metropolis algorithm. Details will be published elsewhere. To summarize, we have presented the construction of a class of gauge-invariant coarse-to-fine transformations for pure lattice gauge theories for arbitrary space-time dimension and gauge group. These transformations are one important building block for MGMC updating algorithms. The success of these algorithms will crucially depend on the ability to optimize the many tunable parameters and to provide the other basic building block, which is the calculation of the (approximate) long-distance effective action. We believe that for MGMC updating algorithms it is sufficient to determine these effective actions in some approximation which only provides the correct long-distance physics and that simple local actions with appropriately determined coupling parame209
Volume 210, number 1,2
PHYSICS LETTERS B
t e r ( s ) c a n b e used. E f f o r t s i n t h i s d i r e c t i o n will b e discussed in a following paper.
References [1 ] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 ( 1953 ) 1087.
210
18 August 1988
[2] K. Decker, A. Hasenfratz and P. Hasenfratz, Nucl. Phys. B 295 [FS21] (1988) 21. [3] R.H. Swendsen, Phys. Rev. Lett. 47 ( 1981 ) 1775. [4] T. Jersak, T. Neuhaus and P.M. Zerwas, Phys. Lett. B 133 (1983) 103. [ 5 ] R. Gupta, M.A. Novotny and R. Cordery, The nature of the phase transition in d = 4 U( 1 ) lattice gauge theory, Northeastern University preprint NUB#2654 ( 1985 ).
PHYSICS LETTERS B
18 August 1988
GAUGE-INVARIANT COARSE-TO-FINE TRANSFORMATIONS FOR MULTIGRID MONTE CARLO ALGORITHMS IN PURE LATTICE GAUGE THEORIES Karsten D E C K E R
Institute for Computer Science and Applied Mathematics, University of Berne, CH-3012 Berne, Switzerland Received 27 April 1988
A class ofgauge-invariant coarse lattice to fine lattice transformations is constructed for pure lattice gauge theories. Coarse-tofine transformations will play an essential role in multigrid Monte Carlo updating algorithms which are believed to cope with the problem of critical slowing down, Efficiency estimates for multigrid Monte Carlo updating algorithms are given and preliminary results from the implementation of such an algorithm for four-dimensional U( 1) lattice gauge theory are reported.
It is well known that M o n t e Carlo ( M C ) simulations o f lattice gauge theories and statistical mechanical systems show critical slowing down when large lattices are simulated with p a r a m e t e r values which let the system develop collective b e h a v i o u r over distances of m a n y lattice spacings. The p h e n o m e n o n o f critical slowing down is clearly due to the locality property o f s t a n d a r d M C updating algorithms. Here, by local we mean interactions o f the scale o f one lattice spacing. While nonlocal generalizations o f stand a r d updating algorithms in principle should be able to avoid critical slowing down, they wil be difficult to design and efficient i m p l e m e n t a t i o n s on computers with vector and parallel architecture will be hard to achieve. Keeping in m i n d efficient i m p l e m e n t a t i o n s on m o d e r n computers, one conceivable a p p r o a c h to master critical slowing down is to carry through the updating process not only on the original lattice, but also on its block lattices. The essential idea is to use s t a n d a r d M C updating algorithms with simple local actions on all the block lattices to quickly establish the long-distance properties o f lattice field configurations, and to correct the short-distance properties on finer lattices with a p p r o p r i a t e l y chosen lattice spacings. In analogy to multigrid solvers for partial differential equations, where the error m o d e o f a cer~ Work supported in part by Schweizerischer Nationalfonds. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
tain wavelength is always eliminated on a grid with corresponding mesh-size, the method sketched above will be called M u l t i G r i d M o n t e Carlo ( M G M C ) updating. An i m p o r t a n t part o f any M G M C updating algor i t h m is the coarse-to-fine transformation, which is needed to m a p the long-distance properties o f lattice field configurations established on some coarse lattice to the next finer one. It is the aim o f this note to construct a class o f gauge-invariant coarse-to-fine transformations for pure lattice gauge theories. We consider a pair o f hypercubical lattices A B c (aBZ) a,
A c ( a ~ _ ) d,
(1)
which we call block lattice and (fine) lattice, respectively, a B a n d a are the corresponding lattice spacings which, for simplicity are assumed to fulfill a / a " = ½. d denotes the space-time dimension o f the lattices for which no specific value is assumed. (A ~, A) can be any pair o f direct successors o f lattices in the sequence o f lattices used in a M G M C updating algorithm. All quantities defined on A B will carry superscript 'B' to distinguish them from quantities defined on A which carry no superscript. Finally, our consideration will be valid for arbitrary gauge group G. We first discuss the geometrical relationship between block lattice and fine lattice. To each lattice site x B on A ~, there correspond 2 a sites y on the fine 207
V o l u m e 210, n u m b e r
1,2
PHYSICS
lattice which define the corners of a hypercube contained in A. Consequently, to each block lattice gauge field U,-,:, n in arbitrary but fixed direction #, there T correspond 2 J-1 pairs (U~.~, L,,+~,:,) of fine lattice gauge fields with y=
2x B ,
y=2xB+~a,
LETTERS
B
18 A u g u s t 1988
exp [Re ;/F ( V~ Uv+,~:, Uv:,) l
(5)
in the transformation. In addition we have to introduce fluctuations characteristic of the value of the coupling parameter fl at which the fine lattice gauge field configuration is to be equilibrated. This can be provided with a factor exp[S(fl) ] ,
21 = 1, ..., d,
21##,
where S(fl) is the action on the fine lattice. Altogether we propose to perform the coarse-to-fine transformation in a probabilistic way and to choose the fine lattice gauge fields Uv:,, Uv+~..... with probability
y=2xn(£, +£~)a, 2 j < 2 2 = 1 .... ,d, y=2x'+
2,¢#,
(£, +'~2 + / 3 ) a ,
2 1 < 2 2 < 2 3 - - - 1 ..... d,
2, va# .
(2)
Here .~ denotes the unit vector in direction )~. We next discuss the general properties which any coarse-to-fine transformation has to fulfill. If we demand gauge-invariance on both the block lattice and the fine lattice, the coarse-to-fine transformation has to be gauge-invariant, too. Gauge-invariant quantities are easily constructed from combinations of block lattice and fine lattice gauge fields. In view of the use in the coarse-to-fine transformation, we suggest
;(~(v .B U,.+,~.:. U,,:.),
(3)
with W B :=
B* Bt BI U,<.+ 52 (U~..v • g.rB+~a;t~
+ ir IB
B
gxB+lia; u
B* ,~..._~ . . . .~JB+ . . . ~..~U,.~+(,., ~).:~)
(4)
and )ff is the group character in the fundamental representation. This choice introduces enough averaging over those block lattice gauge fields which are relevant for the determination of the value of some fine lattice gauge field and is still almost local in the sense defined above. This second property is important in order to obtain a sufficient degree of computational parallelism in the coarse-to-fine transformation algorithm. We now come to the explicit construction of the transformation itself. The primary objective of the transformation is to map the properties of the coarse lattice gauge field configuration to a gauge field configuration living on the next finer lattice. This can be done in a probabilistic way by inclusion of a factor 208
(6)
P,( U,,:.; U,,+,~.:.) : = e x p [ C R e Zf(
V.BU,,+r,.:. U~,:,~)] exp [S(fl)
], (7)
where we have introduced an additional tunable parameter C, which can be used to optimize the relative strength of the two competing factors in the transformation. Such a parameter seems to become important in any attempt to optimize the coarse-to-fine transformation. The properties of the coarse-to-fine transformation will substantially influence the overall efficiency of M G M C updating algorithms. We recall that the final aim is to generate uncorrelated gauge field configurations on a large lattice at almost critical parameters as quickly as possible. In a coarse-grained pseudocode representation, the structure of a full M G M C updating algorithm employing all the block lattices is presented in fig. 1. As can be easily concluded from the figure, the total work to be spent in a M G M C updating algorithm for the generation of a new uncorrelated gauge field configuration is composed of the decorrelation work on the coarsest lattice available, the equilibration work on the intermediate and finest lattices, and the coarse-to-fine transformation work. To estimate the costs for the decorrelation process, we first note that for blocking factor two, the number of operations needed to perform one full updating sweep on the next coarser lattice is always reduced by a factor 2 -J, if we make the reasonable assumption that we use the same updating algorithm on both lattices. In addition, the correlation length in lattice units
Volume 210, number 1,2
PHYSICSLETTERSB
generate next appropriate, statistically uncorrelated gauge field configuration on the coarsest lattice available perform coarse-to:fine transformation to the next finer block lattice perform updating ('equilibration') sweeps on this block lattice with appropriate action to establish the correct properties on the length scale given by the lattice constant of this block lattice perform coarse-to-fine transformation to the finest lattice perform updating ('equilibration') sweeps on the finest lattice to establish the correct properties on the length scale a Fig. 1. Coarse-grainedpseudocoderepresentation of a full MGMC updating algorithm. is reduced by a factor of two in the transition from any lattice to the next coarser one. Since the autocorrelation time rc typically behaves like rc ~ {2 where is the correlation length corresponding to the lightest state of the theory, this behaviour together with the first observation implies that in a MGMC updating algorithm the decorrelation process itself will be performed at practically no cost. Along the same line of reasoning, the equilibration work on all the block lattices will also be negligible as compared to the equilibration on the largest lattice: if Wis the equilibration work needed on the largest lattice, the total equilibration work needed on all the lattices will be bounded by W / ( 1 - 2 - a ) . Here we have implicitly assumed that on all block lattices we can work with simple actions very similar to the one used on the largest lattice. We will comment on this assumption in the last paragraph of this paper. Altogether, we conclude that the efficiency of a MGMC updating algorithm depends on the number of final equilibration sweeps on the largest lattice and on the efficiency of the coarse-to-fine transformation. Since the number of equilibration sweeps on the largest lattice can be reduced, if the quality of the coarse-to-fine transformation is improved, the efficiency of a MGMC updating algorithm will almost
18 August 1988
exclusively be determined by the properties of the coarse-to-fine transformation. We now shortly report on our numerical studies. Based on the coarse-to-fine transformation constructed above, we have implemented a two-lattice MGMC updating algorithm for four-dimensional U( 1 ) lattice gauge theory. We used a 164 and an 84 lattice, respectively and applied a standard Metropolis [ 1 ] updating algorithm on both lattices. The longdistance effective action needed in the updating procedure on the 84 block lattice was determined previously in a numerical fashion [2] using the Monte Carlo renormalization group with Swendsen b = 2 blocking [3] and Wilson's two-lattice matching method. Using our two-lattice MGMC updating algorithm, so far we generated gauge field configurations for a single value of the coupling parameter, fl= 1.0080, close to the transition coupling determined by two different groups [4,5] to be flT=l.0106_+0.0018 and flT=l.0111_+0.0005, respectively (for this value of the coupling parameter, the autocorrelation time for a standard, one-lattice Metropolis algorithm was determined to be rc ~ 400 updating sweeps, corresponding to ~ 700 s CPU time on a Cray X-MP). The preliminary analysis of our data which we have obtained without any attempt to optimize the many free parameters available even in our two-lattice MGMC updating algorithm allows the conclusion that our two-lattice algorithm generates uncorrelated 164 gauge field configurations about four times.faster than a standard, one-lattice Metropolis algorithm. Details will be published elsewhere. To summarize, we have presented the construction of a class of gauge-invariant coarse-to-fine transformations for pure lattice gauge theories for arbitrary space-time dimension and gauge group. These transformations are one important building block for MGMC updating algorithms. The success of these algorithms will crucially depend on the ability to optimize the many tunable parameters and to provide the other basic building block, which is the calculation of the (approximate) long-distance effective action. We believe that for MGMC updating algorithms it is sufficient to determine these effective actions in some approximation which only provides the correct long-distance physics and that simple local actions with appropriately determined coupling parame209
Volume 210, number 1,2
PHYSICS LETTERS B
t e r ( s ) c a n b e used. E f f o r t s i n t h i s d i r e c t i o n will b e discussed in a following paper.
References [1 ] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 ( 1953 ) 1087.
210
18 August 1988
[2] K. Decker, A. Hasenfratz and P. Hasenfratz, Nucl. Phys. B 295 [FS21] (1988) 21. [3] R.H. Swendsen, Phys. Rev. Lett. 47 ( 1981 ) 1775. [4] T. Jersak, T. Neuhaus and P.M. Zerwas, Phys. Lett. B 133 (1983) 103. [ 5 ] R. Gupta, M.A. Novotny and R. Cordery, The nature of the phase transition in d = 4 U( 1 ) lattice gauge theory, Northeastern University preprint NUB#2654 ( 1985 ).