Intelligent Water Drops with Perturbation Operators for Atomic Cluster Optimization R.M.T. Gamot, P.M. Rodger Centre for Scientific Computing, University of Warwick [email protected], [email protected] Overview The Intelligent Water Drops algorithm was modi- fied (MIWD) and adapted to allow it to determine the most stable configurations, for the first time, of Lennard-Jones (LJ), Binary LJ (BinLJ), Morse and Janus Clusters. The algorithm, referred as MIWD+PerturbOp, is an unbiased type of algorithm where no a priori cluster geometry information and construction were used during initialization. Cluster perturbation operators were applied to clusters gen- erated by MIWD to further generate lower energies. A limited-memory quasi-Newton algorithm, called L- BFGS, was utilized to further relax clusters to its nearby local minimum. Basic Properties of IWD a) A B i j i j b) A B i j i j c) i j i j A B m n m n Figure 1: A path measures quality of connectivity between particles. (a) An IWD gathers soil (brown ellipse) as it flows from particle i to particle j while path(i,j) loses an amount of soil; (b) Soil gathered increases with IWD velocity; (c) An IWD travelling on a path with lesser soil, path(m,n), will gather more soil and higher velocity. (d) The algorithm pro- gressively builds the cluster by choosing the connectivity with desirable measures. FlowChart Modifications to IWD 1. The probability of choosing a path depends on amount of soil and the potential energy. p IWD i,j = f (soil(i,j ))η (i,j ) ∑ kV IWD a f (soil(i,j ))η (i,j ) η (i, j )= 1 2+V type (r i,j ) V M = e a(1-r i,j ) (e a(1-r i,j ) - 2) V LJ (r i,j )=4ε i,j (( σ i,j r i,j ) 12 - ( σ i,j r i,j ) 6 ) V Janus (r i,j )= V LJ (r i,j )MV ang (Ω i , Ω j , ˆ r i,j ) MV ang (ˆ r i,j , Ω i , Ω j )= f (Ω i ) f (Ω j ) f(Ω i )= -exp θ 2 i,j 2σ 2 + exp (θ i,j -180) 2 2σ 2 2. An appropriate heuristic undesirability factor, HUD, is chosen to fit the LJ cluster optimization. HUD i,j =2+ V type (r i,j )+ μr i,j + β (max(0,r 2 i,j - D 2 )) 2 3. Worst iteration agent, TIW, affects the soil content as well. soil i,j = (1+ ρ)soil i,j + P i,j P i,j = ρ( soil IWD N -1 ) 4. L-BFGS was used as a relaxation algorithm for IWDs. On LJ Clusters Figure 2: Five independent LJ 98 test runs (color lines) (10,000 iterations/run) for Chen bounding volume showing decline in cluster energy. Figure 3: Cubic Bounding volume and Grow Etch pertur- bation operator combination shows energy decline as tested on LJ 38 . Runs of MIWD alone shows improvement as iterations progress (Fig. 2). Final runs for MIWD+GrowEtch, utilizing spherical bounding volume for scattering of initial sites (Fig. 3), agrees with high-accuracy to (Cambridge Cluster Database) CCD results of up to 104 atoms. Com- pactness measures (Fig. 4) of this study versus CCD results show high-accuracy. Rotation and translation reveal that chiral clusters were gener- ated (Fig. 5). MIWD+GrowEtch achieved rela- tively high-success rates for difficult clusters com- pared to Basin-Hopping with Occasional Jumping (BHOJ)(Table 1). N MIWD+ BHOJ Energy GrowEt 38 100% 96% -173.928426591 75 50% 5% -397.492330983 76 20% 10% -402.894866009 77 10% 5% -409.083517124 98 75% 10% -543.665360771 102 35% 16% -569.363652496 103 40% 13% -575.766130870 104 15% 12% -582.086642068 Table 1: Good success rates with all "difficult" LJ clus- ters. Figure 4: Compactness of clusters MIWD+GrowEtch versus CCD. Figure 5: Row 1 : Overlayed clusters showing unmatched positions. Row 2 : Rotated and translated clusters showing matching configurations. On Binary LJ and Morse BINARY LJ : Tested for up to 50 atoms on 6 instances of σ BB =1.05 - 1.30. MIWD+Knead rediscovered the global minima (GM) for most of the clusters except for N = 41,43, 45 -49 for σ BB = 1.05 and N = 47 for σ BB = 1.10. MIWD+CutSpliceVar rediscovered most of the GM except for N = 30-32 for σ BB = 1.30, N = 35 for σ BB = 1.05, 1.15, N = 36, 39-50 for σ BB = 1.05 and N = 47, 49-50 for σ BB = 1.10. Combination of perturbation operators (Com- biOp) in Phase 2 (CutSplice+Knead, Cut- Splice+H1L2, CutSplice+H2L1, Knead+H1L2 and Knead+H2L1) were further done. Combina- tions were able to arrive at the GM except for N = 45 for σ BB = 1.05 (Fig. 6). MORSE : Tested for up to 60 atoms on 2 values of interparticle force range (a =6, 14). MIWD+GrowEtch located the GM for most of the clusters except for N = 47, 55, 57, 58, 60 for a = 14 (Fig. 7). Figure 6: GM configurations generated from MIWD+CombiOp for selected Binary LJ Clusters. Figure 7: GM configurations from MIWD+GrowEtch for selected Morse Clusters. On Janus Clusters MIWD+CombiOP was applied on Janus clusters using the LJ potential as the patchy particles model but where anisotropic attraction and repul- sion is modulated by an orientational dependent term MV ang . Preliminary results were generated for cluster sizes N =3 - 30 (Fig. 8). MIWD with GrowEtch and Patch Orientation Mutation pro- duced the configurations with the lowest energies. Figure 8: Lowest Cluster Energies generated by MIWd+CombiOp for Janus clusters sizes N =3 - 30. Figure 9: Ob- served basic struc- tures in Janus Clus- ters. Figure 10: Janus cluster configurations with lowest energies. Remarks MIWD, together with a combination of pertur- bation operators, is a promising algorithm to find the lowest configurations of atomic clusters. Runs of the algorithm on known test systems such as LJ, Binary LJ and Morse clusters successfully re- discovered most of the putative global minima. Performance of the algorithm on small Janus clus- ters shows it is able to find relatively well struc- tured clusters. Acknowledgements Study is funded by Warwick Chancellor’s Scholarship (for- merly WPRS) and Centre for Scientific Computing. Com- puting facilities are provided by MidPlus Regional Centre of Excellence for Computational Science, Engineering and Mathematics under EPSRC grant EP/K000128/1. RMT Gamot is also supported by the University of the Philip- pines (UP) System under the UP Doctoral Studies Fund. References [1] Liu, D., Nocedal, J., Mathematical Programming B, 45, 503-528 (1989). [2] Locatelli, M., Schoen, F., Computational Opt and Applications, 21, 55-70 (2001). [3] Shah-Hosseini, H., Proc. Of IEEE Congress on Evolutionary Computation, 3226-3231 (2007). [4] Wales, D.J., Doye, J.P.K., Dullweber,A., Hodges, M., Naumkin, F.Y., Calvo, F., Hernandez-Rojas, J., Middleton, T.F., http://www-wales.ch.cam.ac.uk/CCD.html.
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IntelligentWaterDropswithPerturbationOperators forAtomicClusterOptimization
R.M.T. Gamot, P.M. RodgerCentre for Scientific Computing, University of [email protected], [email protected]
OverviewThe Intelligent Water Drops algorithm was modi-fied (MIWD) and adapted to allow it to determinethe most stable configurations, for the first time,of Lennard-Jones (LJ), Binary LJ (BinLJ), Morseand Janus Clusters. The algorithm, referred asMIWD+PerturbOp, is an unbiased type of algorithmwhere no a priori cluster geometry information andconstruction were used during initialization. Clusterperturbation operators were applied to clusters gen-erated by MIWD to further generate lower energies.A limited-memory quasi-Newton algorithm, called L-BFGS, was utilized to further relax clusters to itsnearby local minimum.
Basic Properties of IWDa)
A B
i j i j
b)
A B
i j i j
c)
i j i j
A B
m n m n
Figure 1: A path measures quality of connectivity betweenparticles. (a) An IWD gathers soil (brown ellipse) as it flowsfrom particle i to particle j while path(i,j) loses an amountof soil; (b) Soil gathered increases with IWD velocity; (c)An IWD travelling on a path with lesser soil, path(m,n), willgather more soil and higher velocity. (d) The algorithm pro-gressively builds the cluster by choosing the connectivity withdesirable measures.
FlowChart
Modifications to IWD1. The probability of choosing a path depends onamount of soil and the potential energy.pIWDi,j = f(soil(i,j))η(i,j)∑
kV IWDa
f(soil(i,j))η(i,j)
η(i, j) = 12+Vtype(ri,j)
VM = ea(1−ri,j)(ea(1−ri,j)−2)
VLJ(ri,j) = 4εi,j((σi,j
ri,j)12 − (
σi,j
ri,j)6)
VJanus(ri,j) = VLJ(ri,j)MVang(Ωi,Ωj , ri,j)MVang(ri,j ,Ωi,Ωj) = f (Ωi) f (Ωj)
f(Ωi) = −exp(θ2i,j2σ2
)+ exp
((θi,j−180)2
2σ2
)2. An appropriate heuristic undesirability factor,HUD, is chosen to fit the LJ cluster optimization.HUDi,j = 2 + Vtype(ri,j) + µri,j+
β(max(0, r2i,j −D2))2
3. Worst iteration agent, TIW, affects the soilcontent as well.soili,j = (1+ρ)soili,j+Pi,j Pi,j = ρ( soil
IWD
N−1 )4. L-BFGS was used as a relaxation algorithmfor IWDs.
On LJ Clusters
Figure 2: Five independent LJ98 test runs (color lines)(10,000 iterations/run) for Chen bounding volume showingdecline in cluster energy.
Figure 3: Cubic Bounding volume and Grow Etch pertur-bation operator combination shows energy decline as testedon LJ38.Runs of MIWD alone shows improvement asiterations progress (Fig. 2). Final runs forMIWD+GrowEtch, utilizing spherical boundingvolume for scattering of initial sites (Fig. 3),agrees with high-accuracy to (Cambridge ClusterDatabase) CCD results of up to 104 atoms. Com-pactness measures (Fig. 4) of this study versusCCD results show high-accuracy. Rotation andtranslation reveal that chiral clusters were gener-ated (Fig. 5). MIWD+GrowEtch achieved rela-tively high-success rates for difficult clusters com-pared to Basin-Hopping with Occasional Jumping(BHOJ)(Table 1).
N MIWD+ BHOJ EnergyGrowEt
38 100% 96% -173.92842659175 50% 5% -397.49233098376 20% 10% -402.89486600977 10% 5% -409.08351712498 75% 10% -543.665360771102 35% 16% -569.363652496103 40% 13% -575.766130870104 15% 12% -582.086642068
Table 1: Good success rates with all "difficult" LJ clus-ters.
Figure 4: Compactness of clusters MIWD+GrowEtchversus CCD.
Figure 5: Row 1 : Overlayed clusters showing unmatchedpositions. Row 2 : Rotated and translated clusters showingmatching configurations.
On Binary LJ and MorseBINARY LJ : Tested for up to 50 atoms on 6instances of σBB = 1.05 − 1.30. MIWD+Kneadrediscovered the global minima (GM) for mostof the clusters except for N = 41,43, 45 -49for σBB = 1.05 and N = 47 for σBB = 1.10.MIWD+CutSpliceVar rediscovered most of theGM except for N = 30-32 for σBB = 1.30, N =35 for σBB = 1.05, 1.15, N = 36, 39-50 for σBB= 1.05 and N = 47, 49-50 for σBB = 1.10.
Combination of perturbation operators (Com-biOp) in Phase 2 (CutSplice+Knead, Cut-Splice+H1L2, CutSplice+H2L1, Knead+H1L2and Knead+H2L1) were further done. Combina-tions were able to arrive at the GM except for N= 45 for σBB = 1.05 (Fig. 6).MORSE : Tested for up to 60 atoms on 2values of interparticle force range (a = 6, 14).MIWD+GrowEtch located the GM for most ofthe clusters except for N = 47, 55, 57, 58, 60 fora = 14 (Fig. 7).
Figure 6: GM configurations generated fromMIWD+CombiOp for selected Binary LJ Clusters.
Figure 7: GM configurations from MIWD+GrowEtch forselected Morse Clusters.
On Janus ClustersMIWD+CombiOP was applied on Janus clustersusing the LJ potential as the patchy particlesmodel but where anisotropic attraction and repul-sion is modulated by an orientational dependentterm MVang. Preliminary results were generatedfor cluster sizes N = 3−30 (Fig. 8). MIWD withGrowEtch and Patch Orientation Mutation pro-duced the configurations with the lowest energies.
Figure 8: Lowest Cluster Energies generated byMIWd+CombiOp for Janus clusters sizes N = 3 − 30.
Figure 9: Ob-served basic struc-tures in Janus Clus-ters.
Figure 10:Janus clusterconfigurations withlowest energies.
RemarksMIWD, together with a combination of pertur-bation operators, is a promising algorithm to findthe lowest configurations of atomic clusters. Runsof the algorithm on known test systems such asLJ, Binary LJ and Morse clusters successfully re-discovered most of the putative global minima.Performance of the algorithm on small Janus clus-ters shows it is able to find relatively well struc-tured clusters.
AcknowledgementsStudy is funded by Warwick Chancellor’s Scholarship (for-merly WPRS) and Centre for Scientific Computing. Com-puting facilities are provided by MidPlus Regional Centreof Excellence for Computational Science, Engineering andMathematics under EPSRC grant EP/K000128/1. RMTGamot is also supported by the University of the Philip-pines (UP) System under the UP Doctoral Studies Fund.
References[1] Liu, D., Nocedal, J., Mathematical Programming B, 45, 503-528 (1989).[2] Locatelli, M., Schoen, F., Computational Opt and Applications, 21, 55-70 (2001).[3] Shah-Hosseini, H., Proc. Of IEEE Congress on Evolutionary Computation, 3226-3231 (2007).[4] Wales, D.J., Doye, J.P.K., Dullweber,A., Hodges, M., Naumkin, F.Y., Calvo, F., Hernandez-Rojas, J., Middleton, T.F.,http://www-wales.ch.cam.ac.uk/CCD.html.
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