Software Algnment for Trackng Detectors V. Blobel Insttut für Expermentalphysk, Unverstät Hamburg, Germany Abstract Trackng detectors n hgh energy physcs experments requre an accurate determnaton of a large number of algnment parameters n order to allow a precse reconstructon of tracks and vertces. In addton to the ntal optcal survey and correctons for electroncs and mechancal effects the use of tracks n a specal software algnment s essental. A number of dfferent methods s n use, rangng from smple resdual-based procedures to complex fttng systems wth many thousands of parameters. The methods are revewed wth respect to ther mathematcal bass and accuracy, and to aspects of the practcal realzaton. Key words: Track fttng, Track detector algnment, Global least-squares ft 1. Introducton Accurate algnment of trackng detectors s essental for mportant aspects of the physcs analyss. The large and accurate vertex detectors of present and future experments have a potental measurement precson of a few µm. The precson from mechancal mountng and e.g. Laser beam algnment s worse than the ntrnsc resoluton, and a hgh precson algnment usng tracks s requred. The general purpose of nstrument calbraton s explaned n the statement: Instrument calbraton s ntended to elmnate or reduce bas n an nstrument s readngs over a range for all contnuous values. For ths purpose, reference standards wth known values for selected ponts coverng the range of nterest are measured wth the nstrument n queston. Then a functonal relatonshp s establshed between the values of the standards and the correspondng measurements. [1] Software algnment/calbraton of HEP track detectors s based, after the use of survey data and correctons for electroncs and mechancal effects, manly on track resdual mnmzaton. Real reference standards wth known values do not exst and thus the algnment data may be ncomplete, wth several degrees of freedom undefned. Algnment/calbraton requres to understand the detector functonal relatonshp and to optmze thousands or tens of thousands of parameters. The goal s to reduce the χ 2 of the track fts, n order to mprove track and vertex recognton, and to ncrease the precson of reconstructed tracks and vertces, elmnatng or reducng bas n detector data. Emal address: volker.blobel@desy.de V. Blobel. URL: www.desy.de/ blobel V. Blobel. Preprnt submtted to Elsever Scence 4 February 2007
Dsplacement [cm] 0.2 0 Shfts from resduals -0.2 0 5 10 Index of detector plane Dsplacement [cm] 0.2 0 Shfts from resduals -0.2 0 5 10 Index of detector plane Fg. 1. Algnment of the planes of a toy detector. The true shfts are gven by the large crcles. Results of the hstogram algnment after 30 teratons are shown by ponts wth error bars, on the left after the frst attempt wth 10 free parameters, and on the rght wth two parameters fxed at zero. 2. Toy detector algnment A popular algnment method n HEP s based on resdual hstograms. Tracks are ftted usng a prelmnary algnment. Hstograms of ht resduals are generated and analysed, and the offsets observed n the hstograms are used to adjust the algnment. Ths method s tested below n a smple MC study. A toy track detector s assumed wth 10 drft chamber planes, 1 m hgh, 10 cm dstance, no magnetc feld, wth a nomnal accuracy of σ 200µm and an effcency ɛ = 90% plane 7 wth reduced accuracy and effcency. The chambers are dsplaced vertcally by a certan shft of 0.1cm. 10 000 tracks wth a total of 82 000 hts are generated and avalable for algnment. The frst algnment attempt s based on the dstrbuton of ht resduals: a straght lne s ftted to the track data, and the resduals = measured vertcal coordnate mnus ftted coordnate are hstogrammed, separately for each plane. The mean value of the resduals s taken as correcton to the vertcal plane poston, and the procedure s repeated teratvely. There are large changes n the frst teraton, small changes n the second teraton, and almost no change afterwards. The result after 30 teratons s show n Fgure 1 left. The reason for non-convergence to the true shfts s smple: two degrees of freedom are undefned, a smultaneous shft and an overall shearng of the planes. In a second attempt, the dsplacement of two planes planes 3 and 9, assumed to be carefully algned externally, s fxed at zero. After large changes n the frst teraton there are smaller and smaller changes n the followng teratons. The result after 30 teratons n Fgure 1 rght shows that the method s convergng, but rather slow, because the determnaton of dsplacements s based on based fts. Can the bas of the track-ft results due to the ntally unalgned detector be avoded by a dfferent method? A correct approach would be the smultaneous ft of the global algnment parameters p global and all local track parameters q local, wth the model m = fq local, p global = q local 1 +q local 2 s + p global j for the measured value m, where p global j s the shft for plane j. Ths s a lnear least squares problem of 82 000 equatons measurements and 20 010 parameters, whch requres the soluton of a lnear equaton wth a 20010 20010 matrx. The specal structure of the matrx however allows the reducton to an 8 8 matrx for the nterestng plane-shft parameters, whch are easly determned wthout teratons, f planes 3 and 9 are fxed at dsplacement = 0. Ths matrx reducton s explaned n Chapter 4. The full covarance matrx s avalable after the ft and shows that the standard devatons of the shfts are around 3 µm. The method s easly extended to the determnaton of addtonal parameters e.g. the determnaton of correctons v drft /v drft of drft veloctes for each plane, usng the equatons m = q local 1 + q local 2 s + p global j + l drft, vdrft v drft wth a further reducton of the resduals. Ths mprovement would be rather dffcult to obtan wth a pure resdual-based method. 3. Specal algnment methods Hstogrammng. The basc dea of the hstogram method, already mentoned n the prevous secton, s to extract parameter correctons from the peak or mean or medan of resdual hstograms. The advantage of ths method s that almost no extra j 2
Fg. 3. Examples for resdual fts n the SLD vertex detector algnment, here as a functon of tan λ n a certan layer before the algnment [2]. Fg. 2. Plot of resduals as a functon of azmuthal angle before left and after rght algnment. a sngle step. Wth the algned geometry a one-ht resoluton of 4 µm n both the rz and rφ planes was found, close to the true ntrnsc CCD resoluton, and the desgn performance was acheved. The post-algnment RMS of the resdual dstrbutons was found to be around a factor of four mproved over the pre-algnment RMS values. Algnment usng a Kalman flter. A method for the estmaton of algnment parameters durng track reconstructon n parallel wth the track parameter estmaton usng the Kalman flter has been studed [3]. After each track ft, the current algnment parameters are updated usng an extenson of the standard Kalman flter, whch s a recursve least squares method. Equatons for updatng the algnment parameters p and ther covarance matrx V, as well as for the parameters of track k and ther covarance matrx V k are derved, whch decouple nto separate systems of equatons. The correcton to the algnment parameters s gven by code s necessary and the hstograms can be generated from n-tuples of the resduals. However, the resduals are taken from based fts and no precson algnment can be expected. In order to obtan convergence, many teratons would be necessary and therefore the method s extremely slow. The method s lmted to those parameters whch are drectly accessble from resduals hstograms. Furthermore t s not obvous how to fx undefned or badly defned degrees of freedom. Resduals hstograms can, however, be useful to detect large msalgnments see Fgure 2 of certan detectors, f also hts that are not assgned to the track are ncluded n the hstograms. The detecton of very large local msalgnments may be mpossble otherwse. Parametrzaton of resdual dependence. An nterestng method has been used for the nternal algnment of the SLD vertex detector[2]. Startng from tracks reconstructed n the central drft chamber, dfferent types of trackng constrants were classfed n the three planes of the vertex detector. For each type of resduals a functonal form was derved and ftted to the measured resduals accumulated n n-tuples Fgure 3. In total 2108 coeffcents from 700 resdual fts were determned. Takng nto account n addton CCD shape correctons from the optcal survey data of the CCD surfaces and takng nto account the covarance matrces of the resdual fts, 866 algnment correctons were determned from 5026 coeffcents of the resdual fts, usng sngular value decomposton SVD technques for the least-squares ft mnmzaton n p = V D T W m f. The update requres the nverson of a matrx wth the dmenson of the measurement vector m as well as several matrx products: 1 W = V meas + HV k H T + DV D T, where V meas = covarance matrx of m, and D and H are the Jacobans of the track functon f w.r.t. the algnment and track parameters. The resultng algnment, consdered to be a sort of local algnment, gves the postons and orentatons of a set of detector elements wth respect to a fxed set of reference detectors. Convergence has 3
been studed n a MC smulaton wth sx detectors to be algned. Occasonal convergence to local mnma cannot be excluded; ths problem s solved by the ntroducton of annealng, gradually steppng up the weghts of the observatons n the course of the estmaton process. 4. Matrx methods for algnment There are large correlatons between the algnment parameters p, whch specfy the poston and orentaton of detector elements. The standard method for the determnaton of a large number of n correlated parameters s the soluton of a system of lnear equatons, derved from a least squares sum of resduals, whch has to be mnmzed. Correctons p are determned from the system of normal equatons C p = b wth a symmetrc n n matrx C and a n-vector b. The problem s characterzed not only by the large value of n, but also by the large amount of data from many events that have to be employed n the procedure n order to get accurate estmates for p. 4.1. Algnment parameters of a planar sensor Sx algnment parameters are requred for a complete algnment of a planar sensor lke a slcon pxel or strp detector. Coordnates and transformatons are as follows [4]. Local sensor coordnates q = u, v, w, defned w.r.t. a sensor and used for track reconstructon, and the coordnates r = x, y, z n the global detector system are related by the lnear transformaton q = R r r 0 wth a nomnal rotaton matrx R, and the nomnal poston vector r 0. The orgn of the u, v, w s at the center of the sensor; the u-axs along the precse coordnate and the v-axs along the coarse coordnate are n the sensor plane. After algnment the transformaton becomes q algned = R γ R β R α R r r 0 q wth the correcton vector q = u, v, w; R α, R β and R γ are rotaton matrces, defned by small angles of rotaton α, β and γ around the u-axs, the new v-axs and the new w-axs. In the small-angle approxmaton the correcton matrx for rotaton becomes 1 γ β R γ R β R α = γ 1 α. β α 1 Usually not all parameters u, v, w and α, β, γ are well-defned and t may be better to use a sub-set or selected lnear combnatons of the sx parameters. 4.2. Global degrees of freedom It s a trval fact, that e.g. a global translaton of the whole detector has no nfluence on the χ 2 of the track. Therefore, track resdual mnmzaton as the basc prncple n detector algnment s not suffcent to fx all global degrees of freedom. Undefned or weakly defned global degrees of freedom may ntroduce certan dstortons, whch do not affect track-ft χ 2 -values, but result n a bas of ftted track parameters. Ths has to be avoded. A general lnear transformaton wth a translaton vector and a 3 3 matrx R x d x x y = d y + R y, z z d z depends on 3 + 9 parameters; the matrx wth 9 parameters can be decomposed nto three rescalng factors f x, f y, f y, of coordnate axes, three rotatons D x, D y, D z and three shearngs T xz, T yz, T xy. At least some of these lnear-transformaton parameters have to be fxed n an algnment procedure. In addton, there are weakly defned nonlnear transformatons or deformatons. Examples are: so-called clockng,.e. a radus-dependent φ-shft, radal dstortons, telescope effect by a radus-dependent z-shft, and sagtta effects [5]. Some of the effects can be reduced by the smultaneous use of dfferent data sets n the algnment. The clockng effect wll be reduced by usng tracks wth a vertex constrant. Mass-constrants n twopartcle decays reduce sagtta effects. The tele- 4
scope and sagtta effects are reduced or elmnated by the use of cosmcs wth magnetc feld large dstance to nteracton pont and wthout magnetc feld zero curvature. Cosmcs and beam halo muons mprove the algnment by ntroducng addtonal correlatons between the algnment parameters. Hgh-momentum tracks are preferred because of the larger predctve power compared to low-momentum tracks wth large multple scatterng. Algnment by tracks should be supplemented by the use of external nformaton e.g. from survey and Laser algnment. Fxng certan reference planes wll mprove the stablty of the algnment ft. Undefned degrees of freedom can be avoded by addng equalty constrant equatons, e.g. zero global dsplacement n the x drecton d x = x = 0 or zero rotaton of the whole detector. The standard technque s to ntroduce a Lagrange multpler λ after lnearzaton of the constrant gp + gt p = 0 wth vector g = gp/ p, extendng the matrx equaton to C global g T 0 g pglobal λ = bglobal gp. Another method s to modfy the soluton of the matrx equaton: sngular value decomposton or dagonalzaton methods allow to recognze weakly defned lnear combnatons and to remove ther effect see Secton 5. 4.3. Global mnmzaton Global algnment s based on the resduals obtaned from the measurement of a large number of tracks. The algnment parameters p are global parameters. Parameters q k for track k are local parameters. In the lnear approxmaton the measurement equaton for measurement m of the track k s wrtten as m = fq local, p global T + δ local q local k + d global T p global, 1 where fq local, p global s the track model predcton for m and d global and δ local are the dervatve vectors w.r.t. to the global and the local parameters. For a gven algnment, a sngle track ft for track k s performed by χ 2 - mnmzaton of the weghted sum of resduals r = m fq local, p global k usng the current algnment parameter values. The weght w s the nverse varance of the measurement m. Sngle track ft. The normal equatons of the least squares soluton for the track parameters are Γ k qk = βk 2 wth the symmetrc matrx Γ k and the vector β k, gven by the sums over all measurements of the track k: Γ k = w δ local δ local T β k = w r δ local. Track parameter correctons are obtaned teratvely by q k = Γ 1 k β k untl the vector β k becomes neglgble. The covarance matrx of the track parameters after convergence s V k = Γ 1 k. Smultaneous algnment and track ft. The socalled global χ 2 -functon s the least squares sum over a large number of tracks from dfferent data sets χ 2 = w r 2, data sets events tracks hts 3 whch s mnmzed ether wth respect to the algnment correctons p only soluton I, or wth respect to the correctons p and all track parameters q j soluton II. The normal equatons for the χ 2 -functon 3 of the algnment parameters and all track parameters are gven by C... G k... p b............ G T k 0 Γ k 0 q k = β k 4............ 5
where k s the track ndex. Submatrx C s a symmetrc n n matrx n = number of algnment parameters and there are K m addtonal rows and columns, f K tracks wth each m parameters contrbute. The complete matrx of Equaton 4 s huge, and cannot be stored n memory. In order to buld-up the matrx of Equaton 4 sngle tracks are ftted n a loop over all events from all data sets. The n n matrx C and the n-vector b of the global algnment parameters are formed, track by track, by a sum wth contrbutons from all measurements; the update formulae are, for each track, C := C + b := b + w d global d global T 5 w r d global. 6 In addton, there s for each track k a rectangular n m matrx G k, whch correlates the parameters of track k wth the algnment parameters. It s determned by a sum over all measurements of track k: w d global δ local T. G k = The matrx equaton 4 cannot be solved drectly because of ts sze. Two methods for the determnaton of the correctons p of the algnment parameters are n use, whch are descrbed n the followng. Soluton I. One possblty s to solve the huge system of lnear equatons 4 teratvely. Frst, all tracks are ftted usng the current values of algnment parameters to form the sums for C and b and, gnorng the matrces G k, the lnear system C p = b. 7 s solved for algnment correctons p. Ths method only takes nto account the drect correlaton between the dfferent algnment parameters, whereas the complete correlatons between the dfferent algnment parameters, medated by the tracks, are neglected. Therefore, the sngle-track fts and the soluton of Equaton 7 have to be repeated many tmes. Experence has shown that the teratve soluton s convergng, but many, perhaps several hundred, teratons are necessary to reach the fnal soluton. Applcatons. The method has been used succesfully n several experments, e.g. [6]. In the algnment of the upgraded vertex detector of Aleph at LEP2 [7] a global χ 2 nvolvng all 864 = 144 6 degrees of freedom s bult, usng sngle tracks and vertex constrants. Selected nformaton s used from the outer trackng. Precse faces measurements are used to reduce the degrees of freedom, whle allowng for parametrzed dstortons. The method s shown to provde accurate results, even wth a lmted number of events. 16 000 hadronc Z events are used. A sensor algnment wth repeated track fttng and resdual optmzaton has been developed for the CMS experment by [4]. As descrbed n Secton 4.1, sx parameters are requred for one planar sensor. Wthn soluton I, there are no correlatons between dfferent sensors, and therefore the soluton nvolves matrces whose dmenson s at most 6 6. The method was appled n a precson survey n a test beam setup wth several slcon strp detectors. The performance has also been tested n a smulated two-layer pxel detector, wth 144 sensors n layer 1 and 240 sensors n layer 2, each wth 6 parameters. Only a few teratons were necessary n the case where all sensors of the second layer were fxed. In the case of only a sngle fxed sensor and all remanng 383 sensors msalgned, the number of teratons needed to reach a reasonable precson for the 2298 ftted parameters vared between 20 and 100. Soluton II. A non-teratve determnaton of the algnment parameters s possble due to the specal structure of the huge matrx n Equaton 4. As far as the algnment s concerned, Equaton 4 can be reduced to an equaton for the algnment parameters p only, wth a modfed matrx C and vector b. Ths method has been developed for the Mllepede package [8] for algnment 1. As before, a loop over all tracks s performed wth 1 There are ndcatons that ths matrx-sze reducton n large least squares problems has been used n surveyng already n the 19th century. 6
sngle track fts, and the contrbuton from each track s added: C := C + T w d global d global Gk V k G T k 2000/11/28 17.38 b := b + 8 w r d global G k V k β k 9 usually the correctons to the vector b are zero, because the vectors β k for tracks are zero. No further nformaton from sngle tracks has to be stored. The matrx update ncludes the nformaton from the parameters of track k. In partcular, the complete correlatons between the dfferent algnment parameters are ncluded. Fnally, after accumulaton of all nformaton from all tracks the matrx equaton C p = b 10 has to be solved. Matrces C and vectors b from several data sets can be smply added to get a combned result. The method allows to ntroduce equalty constrants see Chapter 4.2. The algorthm to solve large least squares problems n a sngle step no teratons and takng nto account all correlatons s general and can be appled to other problems wth a large number of global parameters and a huge number of measurements wth local parameters. In the algnment, a few teratons may be necessary to check the code, because of lnear approxmatons made n the track model, or because of the lmted accuracy of the soluton of the large system of lnear equatons. Another reason for teratons may be the removal of outlers, applyng certan cuts whch are tghtened from teraton to teraton. Applcatons. The Mllepede package s used n several experments H11997, [9], CDF2001 [10], Hera-B or s under test for others CMS, LHCb, PHENIX, ZEUS. Startng from a dfferent ansatz, essentally the same method to reduce the matrx sze was derved n the ATLAS algnment group convenor A. Hcheur; the conference report[5] descrbes tests n smulated experments and contans several formulae for vertex and mass constrants etc. A common algnment and detaled calbraton was performed n the H1 experment usng the RMSresduals vs drft Fg. 4. Resduals as a functon of the drft length n the H1 drft chamber. RMS values are reduced from 180 µm to 125 µm n the algnment wth Mllepede. Mllepede package for the slcon detector two planes and the drft chambers 56 planes, wth a total of 1 400 parameters usng 50 000 tracks [9]. For the two drft chambers, 14 global parameters representng an overall shft or tlt were ntroduced. Local varatons of the drft velocty v drft for cell halfs and layer halfs are observed and descrbed by 180 + 112 correctons, whch change wth the HV confguraton. For each group of 8 wres correctons to T 0 are ntroduced 330 correctons. The result s shown n Fgure 4, where the reducton of the mean track resduals from 180 µm to 125 µm s vsble. Because of CPU tme and memory constrants the present Mllepede package s lmted to algnment problems wth a number n of parameters up to fve or ten thousand. A new verson Mllepede II s beng developed applcable for n much larger than 10 000. In Mllepede II, the two tasks of the program are splt. Accumulaton of data s done by a small subprogram Mlle nsde the user program and the soluton s determned n a standalone program Pede. An effcent sparse-matrx storage scheme s dynamcally defned and dfferent soluton algorthms see Chapter 5 are avalable. The soluton s easly tested under dfferent condtons usng the accumulated data. Tests on 7
a standard PC wth 25 000 parameters from one mllon tracks took less than one hour. 5. Numercal lnear algebra In matrx methods the correctons p for algnment parameters are determned by the soluton of matrx equatons 7 and 10 wth a large n n matrx C. Double precson storage for matrx and vector s mportant already n the accumulaton phase. It s not a-pror clear whether a soluton wth acceptable accuracy can be obtaned. The accuracy depends on the algorthm and on the data: 1. Algorthm: Wth a stable algorthm, the computed soluton s the exact soluton of a nearby problem. Gaussan Elmnaton wth restrcted scan on the dagonal for the next pvot element s consdered to be a stable algorthm for postve defnte matrces. 2. Data: The system s called ll-condtoned, f small changes n the data can cause large changes n the soluton. Ths behavour s caused by varables whch are undefned or poorly defned or strongly correlated; all components of the soluton are affected n ths case. Ill-condtonng can be detected by small egenvalues of the matrx and by global correlaton coeffcents close to 1. Re-defnton of the algnment parameters may mprove the condton of the system. Matrx and vector should be approprately scaled wth consstent unts n data and varables, n order to reach smlar precson for all elements. A large fracton of the memory wll often be used for the matrx; specal storage technques for symmetrc matrces, band matrces and sparse matrces n general reduce the requred space. Most algorthms can work n-space,. e. no extra space s requred for the nverted or decomposed matrx. Four soluton methods are lsted below. Soluton wth matrx nverson. The standard method s the Gauss algorthm wth pvot selecton of the dagonal, wth a computng tme n 3. Problems wth n = several thousand are solved on a standard PC wthn a tme of the order of one hour. In practce, often at least a few parameters are badly defned dead or neffcent channels and standard matrx routnes wll fal. A smple method to avod such problems s to stop the nverson f no acceptable pvot s found,.e. the largest possble submatrx s nverted wth accurate results for the related parameters; the correctons for the remanng parameters are set to zero subroutne SPINV n Mllepede [8]. An advantage of the method s that all varances and covarances are avalable wth the nverse matrx, whch s the covarance matrx for parameters: V = C 1. The global correlaton coeffcent ρ j 1 ρ j = 1 V jj C jj can be calculated and gves a measure of the total amount of correlaton between the j-th parameter and all other varables. It s the largest correlaton between the j-th parameter and every possble lnear combnaton of all the other varables and has a range from 0 to 1. Values of the global correlaton coeffcent close to 1 mean a large correlaton and may ndcate that too many partally redundant parameters were ntroduced. The accuracy of matrx nverson s reduced n ths case. Sngular value decomposton and Dagonalzaton. These algorthms allow to recognze sngularty or near-sngularty of the matrx by the determnaton of sngular values or egenvalues, and ths allows to gnore the correspondng lnear combnatons of parameters. Dagonalzaton s the decomposton C = UDU T wth D dagonal dagonal elements are the egenvectors λ j, and matrx U square and orthogonal wth U U T = U T U = 1. The nverse s C 1 = UD 1 U T. The decomposton algorthms are teratve, wth a computng tme 10 tmes larger than for nverson, and addtonal space s needed for the n n matrx U. The soluton of C p = b can be wrtten n the form [ ] 1 p = U dag U b T. Insgnfcant lnear combnatons, whch could produce dstortons of the algnment, ndcated by small egenvalues, are suppressed by settng 1/λ = 0 n the above formula for egenvalues λ = 0 or small. Ths method s tested n the paper [5]; several tny values appear n the spectrum λ 8
of the egenvalues, and those sngular modes are suppressed to avod potental dstortons of the detector. Another soluton s to defne a vector q by q = [ dag 1 λ ] U T b and to compute the soluton p by [ ] 1 p = U dag q. λ By constructon, the vector q has a covarance matrx V [q] = 1, and ths allows to recognze and suppress nsgnfcant contrbutons, recognzed by a small value of q 1. Generalzed mnmal resdual method GM- RES. The fracton of non-zero off-dagonal elements n the large matrx of a typcal algnment problem s often rather small, of the order of a few percent. The approxmate soluton of a very large system of lnear equatons wth a sparse matrx can be obtaned by a certan algorthm for a quadratc mnmzaton problem, n analogy to the method of conjugate gradents. One example s MINRES [11], desgned to solve C p = b or mn C p b 2, where C s a symmetrc matrx of logcal sze n n, whch may be ndefnte and/or sngular, very large and sparse. The matrx s accessed only by means of a subroutne call whch must return the product y = Cx for any gven vector x. Ths teratve soluton s faster by several orders of magntudes compared to matrx nverson. An example of a compact storage, optmzed for the above product, s the row-ndex sparse storage [12] wth two arrays of n + qnn 1/2 + 2 words q = fracton of non-zero off-dagonal elements, one for real numbers and one for ntegers. 512 Mbytes of memory are suffcent for a matrx wth n = 100 000 for a value of q close to 1%. Cholesky decomposton. The Cholesky decomposton C = LDL T of the symmetrc matrx C s numercally extremely stable, and can be made n-space; matrx L s a left unt trangular matrx dagonal elements =1 and D s a dagonal matrx. The soluton of C p = L DL T p = b s obtaned by forward and backward substtuton. Wth a clever orderng of parameters the matrx C of algnment problems can be approxmated by a band matrx. An mportant property of the Cholesky decomposton s the fact that for band matrces wth band-wdth m the band structure s kept n ths decomposton and the computng tme s only m 2 n. The subset of elements of the nverse matrx correspondng to the band of the orgnal matrx C can be calculated quckly. Fast methods exst also for varable-bandwdth matrces sky-lne matrx, and for bordered band matrces arrow matrx, where the border addtonal full rows and columns can for example be due to Lagrange multpler constrants. Several matrx algebra lbrares exst, e.g. GNU Scentfc Lbrary GSL, Numercal Algorthms Group NAG, BLAS Basc Lnear Algebra Subprograms, and LAPACK Lnear Algebra PACKage. 6. Algnment strateges and summary Algnment problems wth n = several thousand parameters have been successfully solved, ether usng rather specalzed methods, by algorthms requrng a large number of teratons or by global χ 2 -mnmzaton n a sngle step. The experence has shown that the ntegraton of algnment nto the reconstructon code and the use of fast algnment algorthms s of advantage; t allows a routne check of the tme stablty of algnment. The smultaneous use of several or all avalable types of events, physcs and background events, and of sngle tracks and tracks wth vertex and nvarantmass constrants can reduce or avod potental dstortons. The next generaton of experments, e.g. the AT- LAS and the CMS experment at the LHC, have a huge number of ndependent sensors wth an excellent spatal resoluton from about 10 µm to about 50 µm. An algnment precson for all sensors, wth up to n = 10 5 parameters CMS experment at LHC, below the ntrnsc resoluton s requred to get the necessary measurement accuracy for the physcs program at the LHC. 9
The best and most sutable strategy for ths large number of parameters s unknown at present. Both experments have formed algnment workng groups, where the mpact of ms-algnment s studed [13] and dfferent methods of algnment are developed n parallel and compared. The algorthms under study are rather smlar n the two experments. Both, smple straghtforward and perhaps robust methods and advanced methods are studed n the ATLAS Algnment group convenor A. Hcheur: Local χ 2 algnment: modules algned on an ndvdual bass wth 6 6-matrces, teratvely usng the algnment and refttng tracks, Global χ 2 algnment: smultaneous algnment and track fts wth the full Pxel + SCT Barrel and Endcaps [5], Algnment wth overlaps: relatve module to module msalgnment determned from overlap resduals, as well as n the CMS Algnment group coordnator O. Buchmüller Sensor algnment by tracks: teratve procedure that consders ndvdual measurement devces wth 6 6-matrces not takng nto account correlatons between measurement devces [4] [14], Mllepede II: upgraded verson of Mllepede; Kalman flter algnment [3]. Some theoretcal progress n algnment would be welcome to fnd a general strategy to suppress unwanted dstortons, whch can be used even for a very large value of n. However, there s confdence that good algnment strateges wll be avalable at the tme when they wll be needed. References Nuclear Instr. Methods A 510, 233 247 2003 [3] R. Frühwrth, T. Todorov and M. Wnkler, Estmaton of detector algnment parameters usng the Kalman Flter wth annealng, J.Phys. G: Nucl. Part. Phys. 29, 561 574 2003 [4] V. Karmäk, A. Hekknen, T. Lampén and T. Lndén, Sensor algnment by tracks, CMS Conference Report CHEP03, La Jolla, Calforna, March 24-28, 2003 [5] P. Brückman de Renstrom, S. Haywood: Least squares approach to the algnment of the generc trackng system, Phystat2005, Oxford. [6] A. Sopczak, Algnment of the D0 Vertex Detector, these proceedngs. [7] A. Bonssent et al., Algnment of the upgraded VDET at LEP2, ALEPH 97-116, 1997 [8] V. Blobel, Lnear Least Squares Fts wth a Large Number of Parameters, 2000, http://www.desy.de/~blobel ncludng Fortran code. [9] V. Blobel and C. Klenwort: A New Method for the Hgh-Precson Algnment of Track Detectors, Procedngs Phystat2002, Durham, arxv-hep-ex/0208021 [10] R. McNulty et al., A Procedure for the Software Algnment of the CDF Slcon System, CDF/DOC/TRACKING/GROUP/5700 2001 [11] C. C. Page and M. A. Saunders, Soluton of sparse ndefnte systems of lnear equatons, SIAM J. Numer. Anal. 124, 617 629 1975 [12] W. H. Press, S. A. Teukolsky, W. T. Vetterlng, B.P. Flannery, Numercal Recpes The Art of Scentfc Computng, Cambrdge Unv. Press, 1999 [13] N. De Flpps, Impact of CMS Tracker Msalgnment on Track and Vertex Reconstructon, these proceedngs. [14] T. Lampén, General Algnment Concept of CMS, these proceedngs. [1] Natonal Insttute of Standards and Technology, NIST/SEMATECH e-handbook of Statstcal Methods 2005 www.tl.nst.gov/dv898/handbook/ [2] D. J. Jackson, D. Su and F. J. Wckens, Internal algnment of the SLD vertex detector usng a matrx sngular value decomposton technque. 10