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Pedro Alonso et al.

Table 1: Execution time (sec.) with di erent number of cores and problems sizes (n).

For the solution of the two triangular linear systems conducing to the solution of (1) we also implemented a routine that works in parallel. We will denote as GSA to the whole algorithm that solves the linear system (1) by performing the Cholesky factorization of T .

3Results

The experimental results shown in this section were obtained on a board with two Intel R Xeon R E5420 processors with 4 cores and 6 Mb. of cache memory each, thus resulting in a total of 8 cores for the tests.

In general, “fast” algorithms lack of a large throughput per data. Furthermore, Schur– type algorithms are rather irregular, i.e., the concurrency drops in the last steps of the triangularization process due to the diminishing number of blocks that can be computed in parallel. Nevertheless, as Table 1 shows, the algorithm manages to reduce time when the number of cores increases. We used compilers in Intel R Composer XE 12.1 distribution and routine dgemm in MKL 10.3 for the matrix product. This matrix multiplication is threaded so it leverages the available number of cores. The reduction in time achieved with the algorithm depends on the block size ν. As long as the block size is large more e ciency is obtained with the matrix-matrix multiplication with the count of cores. This is what it can be seen if we compare, e.g. for the largest problem size, the time between using 4 or 8 cores.

Although the heaviest step of the algorithm is the triangularization, we have also obtained a certain level of parallelism in solving the two triangular systems.

Block–Toeplitz linear systems

4Conclusions

A parallel implementation of a “fast” algorithm has been presented in this paper. We rearranged operations so they could be cast in term of matrix-matrix multiplications. The more e cient the matrix product operation the faster our algorithm will be. The algorithm thus benefits not only of a level 3 operation but also of its multicore implementation.

Under certain conditions (problem size, block size, hardware, . . . ) our parallel implementation of the GSA well exploits a multicore by always producing the result with a given number of cores in less time than using a fewer number of cores.

References

[1]Pedro Alonso, Jos´e M. Bad´ıa, and Antonio M. Vidal. An e cient parallel algorithm to solve block–Toeplitz systems. The Journal of Supercomputing, 32:251–278, 2005.

[2]J. Chun, T. Kailath, and H. Lev-Ari. Fast parallel algorithms for QR and triangular factorization. SIAM Journal on Scientific and Statistical Computing, 8(6):899–913, November 1987.

[3]K. Gallivan, S. Thirumalai, and P. Van Dooren. On solving block toeplitz systems using a block schur algorithm. In Jagdish Chandra, editor, Proceedings of the 23rd International Conference on Parallel Processing. Volume 3: Algorithms and Applications, pages 274– 281, Boca Raton, FL, USA, August 1994. CRC Press.

Proceedings of the 12th International Conference on Computational and Mathematical Methods

in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

CMB Maps: a Bayesian technique

Pedro Alonso1, Francisco Arg¨ueso1, Raquel Cortina2, Jos´e Ranilla2 and

Antonio M. Vidal3

1 Department of Mathematics, Universidad de Oviedo, Spain

2 Department of Computer Science, Universidad de Oviedo, Spain

3 Department of Computer Systems and Computation, Universidad Polit´ecnica de Valencia, Spain

emails: palonso@uniovi.es, argueso@uniovi.es, raquel@uniovi.es, ranilla@uniovi.es, avidal@dsic.upv.es

Abstract

In this work we present a new e cient method to detect point sources in Cosmic Microwave Background maps and estimate their fluxes. The method uses previous information about the statistical properties of the sources, so that this knowledge can be incorporated in a Bayesian scheme. The experiments show that the method detects more sources than the results obtained using other strategies. Besides, the technique allows us to fix the number of detected sources in a non-arbitrary way.

Key words: E ciency, Cosmic Microwave Background, Bayesian

MSC 2000: 65F05, 65Y20, 68W10

1Introduction

The Cosmic Microwave Background (CMB) is a di use radiation which started to propagate freely in the early universe, about 400, 000 years after the Big Bang. The CMB is thus a fossil radiation which carries very important information about the fundamental properties of the universe and, in particular, about the chemistry of the early universe. Since its discovery in 1964 ([8]), the CMB has been detected and surveyed by instruments aboard balloons and satellites, such as the NASA satellites COBE in 1992 ([9]) and WMAP in 2003 ([10]).

CMB Maps: a Bayesian technique

The ESA satellite Planck was launched in 2009 and has been measuring the CMB fluctuations with unprecedented accuracy, resolution and frequency coverage. The first data obtained by Planck were published in 2011 ([1]). In order to extract all the relevant information from the CMB data, one must remove the contamination produced by extragalactic sources, e.g. galaxies that also emit microwave radiation which has a non-cosmological origin. Several techniques: matched filters, wavelets, etc. have been proposed in the literature (see [6] and [7] for detailed reviews) to deal with the detection problem.

The goal of this paper is to put forward a new e cient method to detect point sources in CMB maps and estimate their fluxes. The method uses previous information about the statistical properties of the sources, so that this knowledge can be incorporated in a Bayesian scheme ([3]). This means a new approach with respect to the tools considered for instance in ([2]), where such an information was not taken into account.

2Description of the problem

In a region of the sky, we assume to have an unknown number of point-like sources, whose emission is mixed with that of the CMB, f (x, y). A model for the total emission as function of the position (x, y) is given by

α=1

where δ(x, y) is the 2D Dirac delta function, the pairs are the locations of the point sources in our region of the celestial sphere, and aα are their fluxes. We observe this radiation through an instrument, with beam pattern b(x, y), and a sensor that adds a random noise n(x, y) to the signal measured.

Therefore, the output of our instrument is:

n

where the point sources and the CMB have been convolved with the beam. In our application, we are interested in extracting the locations and the fluxes of the point sources and consider the rest of the signal as just a disturbance superimposed to the useful signal. If c(x, y) is the signal which does not come from the point sources, model (2) becomes

n

If our data set is a discrete map of N pixels, the above equation can easily be rewritten in vector form, by letting d be the lexicographically ordered version of the discrete map

P. Alonso, F. Argueso,¨ R. Cortina, J. Ranilla, A.M. Vidal

d(x, y), a be the n-vector containing the positive source intensities aα, c the lexicographically ordered version of the discrete map, c(x, y), and φ be an N × n matrix whose columns are the lexicographically ordered versions of n replicas of the map b(x, y), each shifted on one of the source locations. Equation (3) thus becomes

Looking at Eqs. (3) and (4), we see that our unknowns are the number n, the list of locations (xα, yα), with α = 1, . . . , n and the vector a. It is apparent that, once n and (xα, yα) are known, matrix φ is perfectly determined. Let us then denote the list of source locations by the n × 2 matrix R, containing all their coordinates.

If we want to adopt a Bayesian approach, we must write the posterior probability density of our unknowns. According to Bayes theorem, this posterior probability can be written as

where p(d|n, R, a) is the likelihood function, derived from our data model.

For the CMB plus noise we can assume that c is a Gaussian random field with zero

1)Both R and a depend on n through the number of their elements. On the other hand, fluxes and positions are, in principle, independent. Thus, we can write

2)A priori, it is reasonable to assume that all the possible combinations of locations occur with the same probability. Therefore

since N !/(n!(N −n)!) is the number of possible distinct lists of n locations in a discrete N -pixel map.

3)It has been checked (see [3]) that the flux distribution can be modeled by a Generalized Cauchy Distribution

p −γpa

CMB Maps: a Bayesian technique

with p a positive number. This distribution obviously assumes that the fluxes of the di erent sources are mutually independent. In order to work with non-dimensional magnitudes, we define xα = aα/a0; we also assume that we will detect point sources above a minimal flux ai, that leads to the following normalized distribution

where B is the incomplete beta function. The values of a0, p and γ, can be determined by fitting this formula to the point source distribution given by the De Zotti counts model (see [5]).

4)We assume that the number of sources in a given sky patch follows a Poisson distribution, with a known average number of sources λ

A detailed account of all these assumptions can be seen in [3].

Finally, by putting together all these prior distributions and the likelihood, we can write the negative log-posterior

with M = a20φtξ−1φ and e = a0φtξ−1d. We assume that we know a0, p, xi, γ and λ, so that the unknowns are: the normalized fluxes x, the number of point sources n and the positions of the point sources through the matrix φ(R).

3Description of the Algorithm

In [2] an algorithm with a high degree of parallelism that improves, from the computational point of view, the classical approaches for detecting point sources in CMB maps was presented.

In this work, our goal is to find the number of sources and their fluxes and positions by maximizing the posterior probability distribution, or equivalently and more simply, minimizing the negative log-posterior. In conclusion, we search the number of sources, their

P. Alonso, F. Argueso,¨ R. Cortina, J. Ranilla, A.M. Vidal

fluxes and positions, rendered more probable by the data, taking also into account the prior distributions.

Therefore, regarding the flux we minimize the negative log-posterior with respect to x, by taking the derivative and equating to zero, we obtain

In order to fix the positions, we assume that the point sources are in the local maxima of e (see [3]). To determine the number of sources, we sort these local peaks from top to bottom and solve (13) successively adding a new source. At the same time, we calculate (12) and select the number of sources which produces its minimum value.

For the sake of simplicity, we redefine φ ≡ a0 φ and consider p = 1, this last assumption is justified by the flux distribution. Besides, taking into account that the thresholding process has been described in [2], in the following we will describe the algorithm without explaining that process again.

The system of non-linear equations we want to solve is

with M = φtξ−1φ and e = φtξ−1d. Therefore, the non-linear system can be expressed in matrix form as

with v = (1/(1 + x1), 1/(1 + x2), . . . , 1/(1 + xn))t.

We consider that the matrices M , φ and ξ are of order N , while the vectors e and d have N rows. As ξ RN ×N is a symmetric positive definite matrix, Cholesky decomposition can be used to obtain a lower triangular matrix such that: ξ = LLt ([4]). Hence, vector e can

with Z = L−1φ.

Next, we compute the QR decomposition of Z = QR, with Q RN ×N , orthogonal,

CMB Maps: a Bayesian technique

In order to solve (15) we will use the classical Newton-Raphson method and Armijo rule, obtaining a sequence

with v = (1/(1+x1), 1/(1+x2), . . . , 1/(1+xn))t and w = (1/(1+x1)2, 1/(1+x2)2, . . . ,1/(1+ xn)2)t.

We will use the solution of the linear system M x = e as our initial condition (x(0)). Thus, vector x(0) can be computed by solving the triangular linear systems Rty = φc2 and

Rx = y.

Step 5.1 Calculate x(0): Rt y = φ c2, R x = y

Step 5.2 Apply Newton-Raphson: x(k+1) = x(k) − DF (x(k))−1 F (x(k))

Step 6. Find the number of point sources that minimizes the negative log-posterior

Output x

Preliminary experiments show that the new method detects more sources than the results obtained in [2]. Furthermore, the technique allows us to fix the number of detected sources in a non-arbitrary way.

Acknowledgements

This work was financially supported by the Spanish Ministerio de Ciencia e Innovaci´on and by FEDER (Projects TIN2010-14971, TIN2008-06570-C04-02, TEC2009-13741 and CAPAP-H4 TIN2011-15734-E), Universitat Polit`ecnica de Val`encia through Programa de

P. Alonso, F. Argueso,¨ R. Cortina, J. Ranilla, A.M. Vidal

Apoyo a la Investigaci´on y Desarrollo (PAID-05-11) and Generalitat Valenciana through project PROMETEO/2009/013.

References

[1]Planck collaboration. P.A.R. Ade et al., Planck early results I. The Planck Mission, Astron. and Astrophys. 536 (2011) A1.

[2]P. Alonso, F. Argueso,¨ R. Cortina, J. Ranilla and A. M. Vidal, Detecting Point Sources in CMB Maps using an Eficient Parallel Algorithm, J. Math. Chem. 50

(2012) 410–420.

[3]F. Argueso,¨ E. Salerno, D. Herranz, J. L. Sanz, E. E. Kuruoglu and K. Kayabol, A Bayesian Technique for the Detection of Point Sources in CMB Maps, Mon. Not. Roy. Astron. Soc. 414 (2011) 410-417.

[4]G. H. Golub, C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 1996.

[5]G. De Zotti et al., Predictions for high-frequency radio surveys of extragalactic sources, Astron. and Astrophys. 431 (2005) 893-903.

[6]D. Herranz, P. Vielva, Cosmic Microwave Backgound Images, IEEE Signal Processing Magazine 27 (2010) 67-75.

[7]D. Herranz, F. Argueso¨ and P. Carvalho, Compact Source Detection in Multichannel Microwave Surveys: from SZ Clusters to Polarized Sources, Advances in Astronomy, 2012, in press.

[8]A.A. Penzias, R.W. Wilson, A Measurement of Excess Antenna Temperature at 4,080 Mc/s, Astrophys. J. 142 (1965) 419-421.

[9]G. Smoot et al., Structure in the COBE Di erential Microwave Radiometer First-Year Maps, Astrophys. J. 396 (1992) L1-L5.

[10]D.N. Spergel et al., First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters, Astrophys. J. Suppl. 148

(2003) 175-194.

Proceedings of the 12th International Conference on Computational and Mathematical Methods

in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

Least squares problem and QR decomposition of Vandermonde matrices

P. Alonso1, R. Cortina2, F.J. Mart´ınez-Zald´ıvar3 and Antonio M. Vidal4

1 Departamento de Matem´aticas, Universidad de Oviedo

2 Departamento de Inform´atica, Universidad de Oviedo

3 Departamento de Comunicaciones, Universitat Polit`ecnica de Val`encia

4 Departamento de Sistemas Inform´aticos y Computaci´on, Universitat Polit`ecnica de Val`encia

emails: palonso@uniovi.es, raquel@uniovi.es, fjmartin@dcom.upv.es, avidal@dsic.upv.es

Abstract

Vandermonde matrices appear in many fields of Science and Technology. This paper is motivated by the need to solve least squares problems with complex Vandermonde matrices, using the QR decomposition in Signal Processing applications such as beamforming problems and DOA (Direction of Arrival) analysis, to be executed in devices with low computation power as mobile devices. We present an analysis of existing algorithms that solve the linear least squares problem with this kind of matrices. Some of these algorithms have been extended to explicitly compute the QR decomposition. New algorithms have been developed starting from the existing ones, obtaining also an algorithm for updating of the QR decomposition when a new column is added to the Vandermonde matrix, and an incremental method for computing the QR decomposition starting from the updating algorithm.

Key words: Vandermonde, QR, DOA, Beamforming, Szeg¨o polynomials

1Introduction

Vandermonde matrices appear in many fields of Science and Technology. As an example, this kind of structured matrices can represent the steering matrix of signal sources in an