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A Fast Iterative Multi-Grid Map-Making Algorithm for CMB Experiments

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A Fast Iterative Multi-Grid Map-Making Algorithm for CMB Experiments A&A 374, 358–370 (2001) Astronomy DOI: 10.1051/0004-6361:20010692 & c ESO 2001 Astrophysics MAPCUMBA: A fast iterative multi-grid map-making algorithm for CMB experiments O. Dor´e1,R.Teyssier1,2,F.R.Bouchet1,D.Vibert1, and S. Prunet3 1 Institut d’Astrophysique de Paris, 98bis, Boulevard Arago, 75014 Paris, France 2 Service d’Astrophysique, DAPNIA, Centre d’Etudes´ de Saclay, 91191 Gif-sur-Yvette, France 3 CITA, McLennan Labs, 60 St George Street, Toronto, ON M5S 3H8, Canada Received 21 December 2000 / Accepted 2 May 2001 Abstract. The data analysis of current Cosmic Microwave Background (CMB) experiments like BOOMERanG or MAXIMA poses severe challenges which already stretch the limits of current (super-) computer capabilities, if brute force methods are used. In this paper we present a practical solution for the optimal map making problem which can be used directly for next generation CMB experiments like ARCHEOPS and TopHat, and can probably be extended relatively easily to the full PLANCK case. This solution is based on an iterative multi-grid Jacobi algorithm which is both fast and memory sparing. Indeed, if there are Ntod data points along the one dimensional timeline to analyse, the number of operations is of O(Ntod ln Ntod) and the memory requirement is O(Ntod). Timing and accuracy issues have been analysed on simulated ARCHEOPS and TopHat data, and we discuss as well the issue of the joint evaluation of the signal and noise statistical properties. Key words. methods: data analysis – cosmic microwave background 1. Introduction in this massive computing approach, i.e. the MADCAP package (Borrill 1999) has been applied to the recent As cosmology enters the era of “precision”, it enters si- BOOMERanG and MAXIMA experiments (de Bernardis multaneously the era of massive data sets. This has in et al. 2000; Hanany et al. 2000) some faster and less con- turn showed the need for new data processing algorithms. suming solutions based on iterative inversion algorithms Present and future CMB experiments in particular face have now been developed and applied too (Prunet et al. some new and interesting challenges (Bond et al. 1999). If 2000). This is in the same spirit as (Wright et al. 1996), we accept the now classical point of view of a four steps and we definitely follow this latter approach. data analysis pipeline: i) from time-ordered data (TOD) Design goals are to use memory sparingly by handling to maps of the sky at a given frequency, ii) from frequency only columns or rows instead of full matrixes and to in- maps to (among others) a temperature map, iii) from a crease speed by minimising the number of iterations re- temperature map to its power spectrum C , iv) from the ` quired to reach the sought convergence of the iterative power spectrum to cosmological parameters and charac- scheme. We fulfill these goals by an iterative multi-grid teristics of the primordial spectrum of fluctuation, the ul- Jacobi algorithm. As recalled below, an optimal method timate quantities to be measured in a given model. The involves using the noise power spectrum. We have thus work we are presenting focuses on the first of these issues, investigated the possibility of a joint noise (and signal) namely the map-making step. evaluation using this algorithm. Up to the most recent experiments, maps could be made by a brute force approach that aims to solve di- Section 2 presents in detail the method and its imple- rectly a large linear problem by direct matrix inversion. mentation, while Sect. 3 demonstrates its capabilities by Nevertheless the size of the problem, and the required using simulations of two on-going balloon CMB experi- 1 2 O 3 ments, ARCHEOPS (Benoit et al. 2000) and TopHat . computing power, grows as (Npix), and the limits of this kind of method have now been reached (Borrill 2000) (even The results are discussed in Sect. 4, together with the O 2 problem of the evaluation of the noise properties, as well if the (Npix) scaling of a preconditioned conjuguate gradient is actually the appropirate scaling for map- as possible extensions. making only). Whereas the most efficient development 1 http://www-crtbt.polycnrs-gre.fr/archeops/ Send offprint requests to:O.Dor´e, e-mail: [email protected] 2 http://topweb.gsfc.nasa.gov/ Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20010692 O. Dor´e et al.: MAPCUMBA: A fast iterative multi-grid map-making algorithm for CMB experiments 359 2. The method the least square solution which was used to analyse the COBE data (Jansen & Gulkis 1992), 2.1. Optimal map making −1 The relation between the sky map we seek and the ob- W = AT N −1A AT N −1 . (8) served data stream may be cast as a linear algebra system (Wright et al. 1996; Tegmark 1997; Bond et al. 1998). Let t In this paper we will actually deal only with this esti- and p indices denote quantities in the temporal and spatial mator. Nevertheless as a next iteration in the analysis domains, and group as a data vector, dt, and a noise vec- process, we could incorporate various theoretical priors tor the temporal stream of collected data and the detector by making P(x) more explicit. For example, it is often noise stream, both of dimension Ntod.Wethenhave assumed a Gaussian prior for the theory, i.e. P(x) ∝ −1 − T 0 0 h T i exp( xp Cpp0 xp /2) where Cpp = xpxp0 is the signal co- dt = Atpxp + nt, (1) variance matrix. In that case the particular solution turns out to be the Wiener filtering solution (Zaroubi et al. 1995; where A x is the signal vector given by the observation tp p Bouchet & Gispert 1996; Tegmark & Efstathiou 1996; of the unknown pixelised sky map, x , which has been p Bouchet & Gispert 1998): arranged as a vector of dimension Npix.TheNtod ×Npix “observation” matrix A therefore encompasses the scan- − −1 T −1 1 T −1 ning strategy and the beam pattern of the detector. W = C + A N A A N . (9) In the following, we restrict the discussion to the case when the beam pattern is symmetrical. We can therefore However this solution may always be obtained by a further (Wiener) filtering of the priorless maximum likelihood so- take xp to be a map of the sky which has already been convolved with the beam pattern, and A only describes lution, and we do not consider it further. how this sky is being scanned. For the total power mea- The prior-less solution demonstrates that as long as surement (i.e. non-differential measurement) we are inter- the (Gaussian) instrumental noise is not white, a simple ested in here, the observation matrix A then has a single averaging (co-addition) of all the data points correspond- non-zero element per row, which can be set to one if d ing to a given sky pixel is not optimal. If the noise exhibits and x are expressed in the same units. The model of the some temporal correlations, as induced for instance by a measurement is then quite transparent: each temporal da- low-frequency 1/f behavior of the noise spectrum which tum is the sum of the pixel value to which the detector is prevails in most CMB experiments, one has to take into pointing plus the detector noise. account the full time correlation structure of the noise. The map-making step then amounts to best solve for x Additionally, this expression demonstrates that even if given d (and some properties of the noise). We shall seek the noise has a simple time structure, the scanning strat- a linear estimator of xp, egy generically induces a non-trivial correlation matrix −1 AT N −1A of the noise map. xˆp = Wptdt. (2) Even if the problem is well posed formally, a quick look at the orders of magnitude shows that the ac- To motivate a particular choice of the Npix ×Ntod matrix W , a Bayesian approach is convenient. Indeed we are seek- tual finding of a solution is a non trivial task. Indeed, a brute force method aiming at inverting the full ma- ing the optimal solution to this inversion problem which − T −1 1 O N 3 maximises the probability of a deduced set of theory pa- trix A N A , an operation scaling as ( pix), is already hardly tractable for present long duration bal- rameters (here the map xp) given our data (dt)bymax- imising P(x|d). Bayes’ theorem simply states that loon flights as MAXIMA, BOOMERanG, ARCHEOPS or 7 5 TopHat where Ntod ∼ 10 and Npix ∼ 10 . It appears to- P | P P | (d x) (x)· tally impractical for PLANCK since for a single detector (x d)= (3) 9 7 P(d) (amid 10s) Ntod ∼ 10 and Npix ∼ 10 ! One possibility may be to take advantage of specific If we do not assume any theoretical prior,thenx follows scanning strategies, and actually solve the inverse of the a uniform distribution as well as d. Therefore, convolution problem as detailed in Wandelt & Gorski P(x|d) ∝P(d|x). (4) (2000). This amounts to deducing the map coefficients alm in the spherical harmonic basis through a rotation If we further assume that the noise is Gaussian,wecan of a Fourier decomposition of the observed data. The map write will then be a simple visualisation device, while the alm would be used directly for a component separation (as −1 P | ∝ − T 0 (x d) exp( nt Ntt0 nt /2) (5) in Bouchet & Gispert 1996; Tegmark & Efstathiou 1996; −1 ∝ − − T − 0 exp( (d Ax)t Ntt0 (d Ax)t /2) (6) Bouchet & Gispert 1998) and the CMB power spectrum ∝ exp(−χ2/2) (7) estimate.
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