Techniques to make use of large distributed computing facilities allow for denser parameter scans of the dynamic aperture, i.e., the domain in phase space where bounded single-particle motion prevails. Moreover, one can also increase the number of 'seeds' each of which represents a possible realization of multipolar components around the machine. In this paper the dependence of the dynamic aperture on the step size of the grid of initial conditions and on the number of seeds is studied. Estimates on the accuracy of the dynamic aperture are derived and the definition of an improved protocol for numerical simulations is presented

Giovannozzi, Massimo

McIntosh, Eric

CERN Document Server

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsPARAMETER SCANS AND ACCURACY ESTIMATES OF THEDYNAMICAL APERTURE OF THE CERN LHCTechniques to make use of large distributed computing facilities allow for denser parameter scans of thedynamic aperture, i.e., the domain in phase space where bounded single-particle motion prevails. Moreover,one can also increase the number of ’seeds’ each of which represents a possible realization of multipolarcomponents around the machine. In this paper the dependence of the dynamic aperture on the step size of thegrid of initial conditions and on the number of seeds is studied. Estimates on the accuracy of the dynamicaperture are derived and the definition of an improved protocol for numerical simulations is presented.Large Hadron Collider ProjectAbstract M. Giovannozzi, E. McIntoshLHC Project Report 925CERN,CH-1211 Geneva 23,SwitzerlandGeneva, June 2006CERN, Geneva, SwitzerlandPresented atEPAC'06, Edinburgh, UK, June 26-30, 2006PARAMETER SCANS AND ACCURACY ESTIMATES OF THE DYNAMICAPERTURE OF THE CERN LHCM. Giovannozzi, E. McIntosh, CERN, Geneva, SwitzerlandAbstractTechniques to make use of large distributed computingfacilities allow for denser parameter scans of the dynamicaperture, i.e., the domain in phase space where boundedsingle-particle motion prevails. Moreover, one can also in-crease the number of ’seeds’ each of which represents apossible realization of multipolar components around themachine. In this paper the dependence of the dynamic aper-ture on the step size of the grid of initial conditions and onthe number of seeds is studied. Estimates on the accuracyof the dynamic aperture are derived and the definition of animproved protocol for numerical simulations is presented.INTRODUCTIONThe dynamic aperture (DA), i.e. the amplitude of the re-gion in phase space where stable motion occurs, is a keyquantity in the evaluation of the performance of the futureLHC. Therefore, an accurate numerical estimate is manda-tory as well as a good knowledge of the error associatedwith the protocol used to compute the DA (see Ref. [1] fora detailed account on the subject). The computation of sucha quantity relies on numerical simulations, performed withthe MAD-X [2] and/or the Sixtrack [3] codes. For the caseof the LHC studies, the number of turns N is equal to 105.A polar grid is defined in the physical space (x, y), at a lo-cation where the optical parameters satisfy αx = αy = 0,where 5 angles, corresponding to different transverse emit-tances ratio y/x, are considered. Along each of these ra-dial directions a number of initial conditions are consid-ered, namely 30 initial conditions are uniformly distributedover an amplitude range of 2σ (each initial condition is infact split into two nearby conditions to allow chaos detec-tion by means of the computation of the maximal Lyapunovexponent [4, 5]). The momentum off-set of the initial con-ditions is set to 0.75 × 10−3, corresponding to 3/4 of thebucket half-height.The use of such an approach should guarantee an accu-racy in the computation of the DA of about 0.5σ [6]. It isworth mentioning that it is customary to express the DA inunits of transverse beam size, i.e. sigmas.Indeed, the need of taking into account the influence ofrandom magnetic errors, requires that the DA computationis repeated for a number of different sequences of gener-ated error so as to evaluate minimum, maximum, and aver-age values of the DA over the ensemble of realizations ofrandom errors. An analysis of the statistical error is carriedout in Ref. [7] showing that one needs an unbiased sampleof 60 realizations of the LHC with magnetic field errors tofind, with a 95 % confidence level, that only 5 % of thetotal number of all possible LHC realizations have a DAlower than the lowest one found by particle tracking. Asfar as the LHC model is concerned, the optics version V6.4is used for the discussion presented in the next sections,where the magnetic errors, both systematic and random,are assigned to the main dipoles, cold separation dipolesand main quadrupoles, all at injection energy. The errorvalues are collected in the error table number 0210. Eventhough this configuration is certainly not the most up-to-date, it represents a realistic test-bench for the study on DAfocusing on the accuracy of the DA estimate on the numberof space phase angles and on the number of realizations.The major computational tasks for 1000 angles, 6 ampli-tudes and 60 seeds, some 360,000 two hour jobs, were runon the CERN Modular Physics Screensaver (CPSS [8])system using WINDOWS desktop PCs (see also Beam–Beam simulations [9]).DA DEPENDENCE ON PHASE SPACEANGLEA special series of simulations was performed by in-creasing the number of phase space directions ϑ up to 1000.For each value of the total number of angles N , the actualvalues are given by:ϑN,i =90◦ iN + 11 ≤ i ≤ N. (1)   	 Figure 1: Minimum DA as a function of number of phasespace angles.The graph shows three different regimes:• Up to 10 Angles: Clearly the sampling of the anglesis not good enough and the DA decreases logarithmi-cally. As most of our tracking has been done with 5angles one has to reduce the stated DA by the ratio of10.5 to 11.1, i.e, roughly 5%.• 10 to 50 Angles: In this regime the long-term min-imum is reached but there is a very large variationof about 8% to larger DA values. Originally, it washoped that 30 angles might be sufficient. However,this analysis shows that regime 2 is not good enough.• More than 50 Angle: As of 50 angles very little vari-ation can be reported ∼ 1%. The minimum itself isnot falling any further. It can therefore be concludedthat a value between 50 and 100 angles is sufficient todetermine the minimum over angles. !" "!!"        ! "   Figure 2: Average (red markers) and minimum DA (greenmarkers) as a function of phase space angles.This graph 2 shows that the minimum of 60 seeds as a func-tion of phase space angle is following on average a straightline with high R2 value. Not surprising, there is quite somevariation and there are slight indications of the effect of res-onances where the DA changes up to 1σ. It is interestingto note that each individual seed does not show this nicelinear behavior: in Fig. 3 the result of two seeds are showntogether with the minimum over all seeds. The individualseeds depict very large variations with respect to the phasespace angle.!"        ! "  Figure 3: Minimum DA over sixty realizations (greenmarkers) compared to DA of two individual realizations(red and blue markers), all as a function of phase spaceangles.The average values of the DA (Fig. 2) has much less mi-croscopic variations but the average DA does not follow astraight line at all. It seems to us that this smooth behav-ior of the minimum DA as function of phase space angleis revealing an important underlying feature of the motion.Further studies would be required to fully understand if itcan be exploited for simplifying the tracking effort.DA DEPENDENCE ON THE NUMBER OFREALIZATIONSA special series of numerical simulations was performedfor a maximum number of 600 realization of the LHC lat-tice, while scanning the transverse phase space by meansof five angles, only. The resulting data were analyzed inview of determining the minimum number of realizationsrequired to derive a reliable estimate of the DA. In Fig. 4the DA distribution for two values of the phase space angleis shown. For each plot the distribution computed using the600 realization is plotted together with the one obtainedfor only 60 realizations, i.e. the nominal number of real-izations for the standard protocol. The actual shape of the05010015020010.010.611.211.812.413.013.614.214.815.416.016.617.217.818.419.019.6Dynamic aperture at 15 degrees (σ)Frequency600 realisations60 realisations05010015020010.010.611.211.812.413.013.614.214.815.416.016.617.217.818.419.019.6Dynamic aperture at 30 degrees (σ)Frequency600 realisations60 realisationsFigure 4: Distribution of the DA for two values of the phasespace angle ϑ.distribution strongly depends on the phase space angle and,of course, the tails for the two cases, namely 600 or 60 re-alizations, differ considerably.To determine how many realizations are required, it ispossible to compute the key descriptors of the distribution,i.e. minimum, average and maximum value, as a functionof the number of realizations (see Fig. 5). For the case ofϑ = 15◦ a strong dependence on the number of realizationsis found, and, the minimum DA settles down after 300 re-alizations. On the other hand for ϑ = 30◦ the dependenceis less pronounced. From these considerations it emergesthat the dynamics at 15◦ gives rise to a richer phenomenol-ogy, hence, in the following, the analysis will be limitedto this case. The next step consist in deriving the Cumu-lative Distribution Function (CDF) for the DA distribution.Given an ordered set of N values of the DA resulting fromthe numerical computation on N realizations of the LHC,the CDF can be computed asCDF (DAi) =iN, i < N. (2)10.010.511.011.512.012.513.013.514.00 100 200 300 400 500 600Number of realisationsDinamic aperture at 15 degrees (σ)Minimum Average MaximumStandard LHC simulations10.010.511.011.512.012.513.013.514.014.515.015.516.016.517.00 100 200 300 400 500 600Number of realisationsDinamic aperture at 30 degrees (σ)Minimum Average MaximumStandard LHC simulationsFigure 5: Distribution of the DA for two values of the phasespace angle ϑ.An example is shown in Fig. 6, where the CDF is shownas computed from a set of 60 or 600 realizations It is0.00.20.40.60.81.010 10.5 11 11.5 12 12.5 13 13.5 14Dinamic aperture at 15 degrees (σ)CDF60 realisations 600 realisationsFigure 6: CDF as a function of the DA for ϑ = 15◦ com-puted using 60 or 600 realizations of the LHC.clearly seen that the curve obtained using 600 realizationsis smoother and, hence, more accurate. From the knowl-edge of the CDF it is possible to extract the value of theprobability to obtain a DA smaller than, e.g., 12σ at 15◦.In particular, to assess the minimum number of realiza-tions required, it is possible to determine such a probabilityas a function of the number of realizations used. Such aquantity is plotted in Fig. 7. The probability is a decreas-ing function of the number of realizations. Furthermore,this quantity seems to indicate that the asymptotic value isreached for about 300 realizations, thus confirming the ob-servations made on other quantities.It is worth mentioning that Extreme Value Theory (EVT)could also be applied to study the statistical behaviour ofthe DA, similarly to what is presented in Ref. [10].0.000.050.100.150.200.250.300.350 100 200 300 400 500 600Number of realisationProb. (dynamic aperture < 12 σ)Figure 7: Probability for the DA at 15◦ to be smaller than12σ as a function of the number of realizations used tocomputed the CDF.CONCLUSIONSIn this paper a detailed analysis of the accuracy of thenumerical computation of the DA for the LHC was carriedout. As far as the dependence of the DA on the number ofphase space angles is concerned, the optimal value lies be-tween 50 and 100. As far as the dependence on the numberof realizations are concerned, the optimal value is around300. These figures have to be compared with the ones usedfor the standard protocol, namely 5 angles and 60 realiza-tions. Even with the increased computing power presentlyavailable, the optimal values derived in this paper seem tobe out of reach. However, it is clear that at least a moderateincrease to about 20 angles and 100 realizations seems fea-sible in terms of CPU-time and necessary to improve thereliability of the DA estimate.ACKNOWLEDGEMENTSThe contribution of F. Schmidt is warmly acknowl-edged.REFERENCES[1] J.-P. Koutchouk et al., in proceedings of the PAC99Conference, ed. by A. Luccio, W. MacKay, IEEE, Pis-cataway, NY, p. 372, 1999.[2] F. Schmidt, http://frs.home.cern.ch/frs/Xdoc/uguide.html.[3] F. Schmidt, http://frs.home.cern.ch/frs.[4] G. Benettin, et al., Meccanica 15, (1980) 21.[5] F. Schmidt, F. Willeke, F. Zimmermann, Part. Accel.35, p. 249 (1991).[6] M. Hayes, E. McIntosh, F. Schmidt, LHC-PROJECT-NOTE- 309 (2003).[7] H. Grote, Beam Physics Note 34 (1999).[8] E. McIntosh, A. Wagner, CHEP 2004, Oct. 2004, In-terlaken.[9] W. Herr, et al., this conference, MOPLS001.[10] R. Duperrier and D. Uriot, Phys. Rev. ST Accel.Beams, 044202 (2006).

Parameter scans and accuracy estimates of the dynamical aperture of the CERN LHC

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