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arXiv:astro-ph/9906115v1 7 Jun 1999
A&A manuscript no.
(will be inserted by hand later)
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02(12.07.1; 03.13.1; 03.13.4)
ASTRONOMY
AND
ASTROPHYSICS
A fast direct method of mass reconstruction for gravitational
lenses
M. Lombardi1,2and G. Bertin2
1European Southern Observatory, Karl-Schwarzschild Straße 2, D 85748 Garching bei M¨ unchen, Germany
2Scuola Normale Superiore, Piazza dei Cavalieri 7, I 56126 Pisa, Italy
Received 25 March 1999; accepted 17 May 1999
Abstract. Statistical analyses of observed galaxy distor-
tions are often used to reconstruct the mass distribu-
tion of an intervening cluster responsible for gravitational
lensing. In current projects, distortions of thousands of
source galaxies have to be handled efficiently; much larger
data bases and more massive investigations are envisaged
for new major observational initiatives. In this article we
present an efficient mass reconstruction procedure, a di-
rect method that solves a variational principle noted in an
earlier paper, which, for rectangular fields, turns out to re-
duce the relevant execution time by a factor from 100 to
1000 with respect to the fastest methods currently used,
so that for grid numbers N = 400 the required CPU time
on a good workstation can be kept within the order of 1
second. The acquired speed also opens the way to some
long-term projects based on simulated observations (ad-
dressing statistical or cosmological questions) that would
be, at present, practically not viable for intrinsically slow
reconstruction methods.
Key words: cosmology: gravitational lensing – methods:
analytical – methods: numerical
1. Introduction
In the context of weak or statistical lensing, the problem of
the determination of the dimensionless mass density distri-
bution κ(θ) from a map of the reduced shear g(θ) has been
considered in detail by various authors, using either simu-
lations or analytical calculations (e.g., Bartelmann 1995,
Schneider 1995, Seitz & Schneider 1996, Squires & Kaiser
1996, Lombardi & Bertin 1998a, 1998b).
The mass inversion is usually performed starting from
the vector field ˜ u(θ) defined in terms of the measured re-
duced shear g(θ) (Kaiser 1995). In the ideal case where
the measured shear g(θ) is just the true shear g0(θ), the
vector field ˜ u0(θ) can be shown to satisfy the relation
˜ u0(θ) = ∇ln?1 − κ0(θ)?= ∇˜ κ0(θ) ,
Send offprint requests to: M. Lombardi
Correspondence to: lombardi@sns.it
(1)
where κ0is the true dimensionless mass map and ˜ κ0(θ) =
ln?1 − κ0(θ)?. However, because of statistical and mea-
surement errors, ˜ u is not necessarily curl-free, and thus
κ can be determined only approximately. In a separate
paper (Lombardi & Bertin 1998b) we have shown that
– The statistical errors on κ(θ) are minimized if this
function is calculated as
˜ κ(θ) = ¯ κ +
?
Ω
HSS(θ,θ′) · ˜ u(θ′)d2θ′, (2)
where ¯ κ is a constant (introduced to take into account
the sheet invariance), HSS(θ,θ′) is the noise-filtering
kernel (Seitz & Schneider 1996), and Ω is the field of
observation.
– The same mass map can be obtained by solving the
equations
∇2˜ κ = ∇ · ˜ u ,
∇˜ κ · n = ˜ u · n
where n is the unit vector perpendicular to the bound-
ary ∂Ω of the field of observation Ω. Hence, the kernel
HSS(θ,θ′) can be identified as the Green function of
this Neumann boundary problem.
– Equations (3) and (4) are precisely the Euler equations
associated with the functional
(3)
on ∂Ω ,(4)
S =1
2
?
Ω
??∇˜ κ(θ) − ˜ u(θ)??2d2θ .
In other words, the functional S is minimized when
˜ κ(θ) is calculated using Eq. (2) or, equivalently, by
solving Eqs. (3) and (4).
(5)
To these three formulations of the mass inversion problem
there correspond three practical methods to calculate κ(θ)
from a given set of data.
– The first method considered is based on a direct cal-
culation of the kernel HSS(Seitz & Schneider 1996).
Once this kernel has been calculated for a given field
Ω, the mass inversion is straightforward (note that
the kernel HSSdepends on the field of observation).
A problem with this method is that a calculation of
Page 2
2 M. Lombardi & G. Bertin: A direct method of mass reconstruction
HSSis expensive in terms of memory requirements and
computation time. In fact, in order to compute κ on a
square grid of N × N points, HSSmust be calculated
on a multidimensional grid of N ×N ×N ×N points.
Moreover, 2N4multiplications are needed to evaluate
Eq. (2), and thus the method is of order O?N4?. Be-
cause of the large memory needed to allocate HSS, cal-
culations can be performed only with a limited value
of N (typically N ∼ 50).
– The introduction of a method that directly solves the
Neumann problem allows one to go beyond many of the
limitations of the HSSmethod. Equations (3) and (4)
can be solved using an over-relaxation method (Seitz
& Schneider 1998). In this case ˜ κ(θ) is calculated di-
rectly, and thus we need to allocate only N × N real
numbers. Moreover, the method can be applied with-
out difficulties to “strange” geometries Ω (while the
previous method is straightforward only when applied
to rectangular or circular fields). The over-relaxation
method is quicker than the kernel method, being ap-
proximately of order O?N3?.
– A direct method to minimize the functional (5) will be
presented in this paper. As we will see, this method
has several advantages and turns out to be very ef-
ficient from a computational point of view, being of
order O?N2logN?. Moreover, it is extremely easy to
implement (two implementations for rectangular fields
Ω written in C and in IDL are freely available on re-
quest).
We should stress that, as proved in an earlier paper
(Lombardi & Bertin 1998b), the three formulations are
mathematically equivalent. Thus it would not be surpris-
ing to find that proper numerical implementations per-
form, for large values of the grid number N, in a similar
manner as far as accuracy and reliability are concerned.
In practice, for finite values of N, the third method turns
out to be characterized by small errors, often smaller than
those associated with the other two procedures.
2. A direct method to solve the variational
principle
Direct methods in variational problems are well-known
especially in applied mathematics (see, e.g., Gelfand &
Fomin 1963). Suppose that one can find a complete set
of functions {fα} on the domain Ω (the full definition of
“complete” will be given below), so that any function on
Ω can be represented as a linear combination of the form
˜ κ(θ) =
∞
?
α=1
cαfα(θ) . (6)
More precisely, we assume that for any function ˜ κ(θ),
there is a choice for the coefficients {cα} such that
?
Ω
α=1
????∇˜ κ(θ) −
∞
?
cα∇fα(θ)
????
2
d2θ = 0 .(7)
Let us now introduce a sequence of trial mass maps
˜ κ[n](θ) =
n
?
α=1
c[n]
αfα(θ) . (8)
We further require that the function ˜ κ[n]minimizes the
functional S: in other words, c[n]
so that the functional S has minimum value. This obvi-
ously happens when
1,c[n]
2,...,c[n]
n are chosen
∂S
∂c[n]
α
= 0for α = 1,2,...,n . (9)
Solving this set of n equations, we obtain the n coeffi-
cients c[n]
operation for a sequence of values of n, we find a sequence
of functions ˜ κ[n]. These functions, under suitable assump-
tions (verified in our problem), have the following proper-
ties (see Gelfand & Fomin 1963 for a detailed discussion):
(i) Let us call S[n]the value of S when ˜ κ is replaced by
the function ˜ κ[n]. Then, obviously, the sequence S[n]is not
increasing. (ii) If the set {fα} is complete, then the func-
tions ˜ κ[n]converge to the solution ˜ κ of the problem. This
method thus provides a way to obtain the function ˜ κ(θ)
with desired accuracy.
The method described here can be easily applied to
our problem. In fact, by expanding ˜ κ[n](θ) as in Eq. (8),
we find
α , and thus the function ˜ κ[n]. By repeating this
∂S
∂c[n]
α
=
?
Ω
∇fα(θ)·
? n
β=1
?
c[n]
β∇fβ(θ)− ˜ u(θ)
?
d2θ = 0 .(10)
The previous equation, for α = 1,2,...,n, represents a
linear system of n equations for the n variables?c[n]
solution is thus the set of coefficients to be used in Eq. (8).
However, we note that care must be taken in the choice
of a complete set of functions. Let us define, for the pur-
pose, the product ?v,w? between two generic vector fields
v(θ) and w(θ) as
α
?. Its
?v,w? =
?
Ω
v(θ) · w(θ)d2θ .(11)
As our problem involves ∇˜ κ, the completeness has to be
referred to the set of the gradients. In other words, the set
?fα
?∇fα,∇˜ κ? = 0
for every α implies ˜ κ(θ) = constant. It is easy to show
that this condition is equivalent to Eq. (7).
The direct method can be further simplified if a set of
functions {fα} can be taken to satisfy a suitable orthonor-
mality condition, so that the gradients of the functions
{fα} satisfy
?is complete if
(12)
?∇fα,∇fβ? = δαβ, (13)
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M. Lombardi & G. Bertin: A direct method of mass reconstruction3
where δαβ= 1 for α = β and 0 otherwise. Then Eq. (10)
can be rewritten simply as
c[n]
α= ?∇fα, ˜ u? .
Thus, with the use of an orthonormal set of functions
we have secured two important advantages: (i) The lin-
ear system (10) has been diagonalized, so that its solution
is trivial. (ii) The coefficients c[n]
that is, the coefficients of the exact solution are given by
cα= ?∇fα, ˜ u?.
Because of these advantages, whenever possible an or-
thonormal set of functions should be used. We note, how-
ever, that the orthonormality condition (13) depends on
the field of observation Ω. Even if the existence of an or-
thonormal set of functions is always guaranteed by the
spectral theory for the Laplace operator (see Brezis 1987),
for “strange” geometries, it may be non trivial to find a
complete orthonormal set of functions. In such cases, we
need to solve the linear system (10).
The direct method described above has several ad-
vantages with respect to the “kernel” method and to the
over-relaxation method: (i) The method is fast in the case
where an orthonormal set of functions can be found. In
fact, we need only to evaluate one integral for each coeffi-
cient cαthat we want to calculate. (ii) The method does
not require a large amount of memory: we need to retain
only the n values of the coefficients cα. (iii) The precision
of the inversion is driven in a natural way by the value of
n. Typically, the larger α is, the smaller the length scale
of fα(see below). (iv) In some cases, the decomposition
of the mass density ˜ κ(θ) in terms of the functions fαcan
be useful.
(14)
α no longer depend on n:
3. Rectangular fields
When the field Ω is rectangular, an orthonormal set of
functions can be written easily. Here we consider the spe-
cial case when Ω is a square of length π (in some suitable
units); any rectangular field can be handled in a simi-
lar manner. In the case considered, an orthonormal set of
functions is given by
fαβ(θ) = nαβcosαθ1cosβθ2, (15)
with (α,β) ∈ N2\(0,0). The normalization nαβis defined
as
√2
π
?
2
π
?
The function f00is not defined. Note that here we use two
indices for the set. Cosines must be used in order to have
a complete set (see Eqs. (7), (12), and Appendix A). Our
problem is solved in terms of the coefficients cαβ:
nαβ=
α2+ β2
for α = 0 or β = 0 ,
α2+ β2
otherwise .
(16)
cαβ= −nαβ
?
Ω
?α˜ u1(θ)sinαθ1cosβθ2+
β˜ u2(θ)cosαθ1sinβθ2
?d2θ , (17)
˜ κ(θ) =
?
αβ
cαβnαβcosαθ1cosβθ2. (18)
We now observe that the particular choice of the orthonor-
mal set {fαβ} allows us to use fast Fourier transform
(FFT) techniques to evaluate Eqs. (17) and (18). The use
of FFT makes the direct method very efficient: in particu-
lar the method becomes of order O?N2logN?. Moreover,
several optimized FFT libraries are available.
The optimal truncation for the series (18) is deter-
mined by the adopted grid numbers: for a grid of N × M
points, α should run from 0 to N − 1, and β from 0 to
M − 1 (this is standard practice for FFT libraries).
4. Performance
Our method has been implemented in C and in IDL. The C
version uses the library FFTW (“Fastest Fourier Transform
in the West,” version 2.0.1) to perform discrete Fourier
transforms (DFT). This library, written by Matteo Frigo
and Steven G. Johnson, is considered the quickest DFT
library publicly available. The performance of our di-
rect method is compared with that of the over-relaxation
method, also implemented in C. The procedure used in the
tests is summarized in the following points:
1. A simple model for the dimensionless mass distribu-
tion has been chosen. Then the mass distribution ˜ κ0is
calculated on a grid of N × N points.
2. The associated field ˜ u0is calculated on the same grid
using a 3-point Lagrangian interpolation in order to
numerically evaluate the derivatives that are needed.
3. Noise is added to the vector field ˜ u0using an analytical
model for the noise derived earlier (Lombardi & Bertin
1998b). In practice, the various Fourier components
of the noise are added using a suitable model for the
power spectrum.
4. The resulting noisy ˜ u map is inverted using the over-
relaxation method and the present direct method. The
two dimensionless mass maps obtained are then com-
pared. Moreover, the inversion times are recorded.
The results obtained in the tests are the following:
– The two mass densities obtained are consistent with
each other.
– Because of the set of functions used, the errors pro-
duced by the direct method are larger on the bound-
ary of the field. For this reason, we suggest that a one
pixel strip around the field should be discarded. The
area discarded is very small.
– Some tests have been performed by providing ˜ u0to the
inversion procedures. This allows us to compare the
reconstructed mass density with the original map κ0.
From such tests we have noted that the discretization
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4 M. Lombardi & G. Bertin: A direct method of mass reconstruction
0.01
0.1
1
10
100
1000
10000
50100 200400
time (sec)
N
Fig.1. Execution time per call vs. grid number N. The
solid line refers to the direct method applied to “good
numbers” (values 2N−1 that can be factorized with small
primes), the long-dashed refers to the direct method ap-
plied to “bad numbers” (2N − 1 prime), and the short-
dashed line to the over-relaxation method.
errors of the direct method are slightly smaller than the
discretization errors of the over-relaxation method.
– The results of the two methods differ because of the
sheet invariance: in particular, the direct method al-
ways gives a “reduced” mass map ˜ κ with vanishing
total mass.
Regarding the second item, we note that the error is re-
lated to the finite sampling scale of the method; the er-
ror affects only the outermost pixel because of the proper
choice of the truncation (see comment at the end of
Sect. 3).
The measured execution times are plotted in Fig. 1 for
different values of N. These are the averaged CPU execu-
tion times for a single reconstruction on a SUN Ultra 1
workstation. From this figure it is clear that the direct
method is much faster than the over-relaxation method.
Here we should recall that, because of some character-
istics of the FFTW library, the execution time of the di-
rect method can change significantly even for neighbour-
ing values of N. In particular, the inversion is faster when
(2N −1) can be factorized with small prime numbers, and
is slower in other cases (see Fig. 1). For example, the ex-
ecution time (on a SUN Ultra 1) changes from 2.942 to
0.232 seconds when N changes from 121 to 122. Finally,
we observe that our implementation of the direct method
is not optimal: in fact, with a different (non-trivial) use
of FFT one might gain an additional factor of 4 on the
execution time.
Besides the appealing aspects of simplicity inherent to
the direct method described in this paper, we should note
that gaining three orders of magnitude in CPU time will
make it possible to undertake a few long-term projects
of simulated observations (in particular, with the goal of
a statistically sound investigation of the quality of mass
?
?
???
0.6
0.5
0.4
0.3
0.2
0.1
0
?
?
10’
8’
6’
4’
2’
0’
?
?
10’
8’
6’
4’
2’
0’
0.1
0.2
0.3
0.4
0.5
????
0.6
0.5
0.4
0.3
0.2
0.1
0
?
?
10’
8’
6’
4’
2’
0’
?
?
10’
8’
6’
4’
2’
0’
0
0.1
0.2
0.3
0.4
0.5
??
?
??????
?
???
?0.0010
?0.0005
0
?0.0005
?0.0010
?
?
10’
8’
6’
4’
2’
0’
?
?
10’
8’
6’
4’
2’
0’
?0.0005
0
?0.0005
Fig.2. A typical result of mass reconstruction; at the
adopted distance for the lensing cluster, the side of the
square field, 10 arcminutes, corresponds to approximately
2.88 Mpc. From top to bottom, true dimensionless mass
distribution, reconstructed distribution (from the direct
method), and difference between maps of the variable ˜ κ
derived from direct and over-relaxationmethods. The very
small residuals show that the two methods are practically
equivalent in terms of accuracy.
reconstruction; but other objectives might be formulated,
e.g. in the cosmological context) that would remain prac-
tically out of reach for other intrinsically slow reconstruc-
tion methods.
5. Examples of simulated reconstructions
In addition to the reconstructions from “synthetic” data
as described in the previous Section, we have performed
several additional tests in order to demonstrate the relia-
bility of our method. The tests, designed with the aim
to reproduce the main features of a “real” reconstruc-
tion, have been made following a straightforward proce-
dure (see, e.g., Lombardi & Bertin 1999 for similar simu-
lations).
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M. Lombardi & G. Bertin: A direct method of mass reconstruction5
First we have generated a population of source galaxies
using a pseudo-random number generator. Here a source
galaxy is represented by its position and by its ellipticity
(see Eq. (11) of Lombardi & Bertin 1999, following Seitz &
Schneider 1997). Positions are drawn from a homogeneous
distribution (with a density of 70 galaxies per square ar-
cmin), while ellipticities are drawn from a truncated Gaus-
sian distribution with variance σ2= (0.3)2. Sources are
assumed to have all the same redshift zs= 1.5. Source el-
lipticities are then transformed into observed ellipticities.
For simplicity, the observed galaxy positions are assumed
to be equal to the source positions: in other words, no
depletion effects are included in the simulations.
Then the calculation of the observed ellipticities has
been done by referring to a cluster of galaxies placed
at zd = 0.3 with total mass inside the 10’ × 10’ field
1.5 × 1015M⊙. For the purpose of introducing the lens-
ing effects, we only need to specify the dimensionless pro-
jected mass map κ(θ). For simplicity, we have used a den-
sity distribution made of three symmetrical components;
each component is described by the analytical model out-
lined by Schneider et al. (1992, p. 244), which, at large
radii, is approximately isothermal.
By averaging the observed ellipticities, we have then
obtained a map of the reduced shear g(θ) and, from that
map, the vector field ˜ u(θ). The mass inversion has been
performed using the direct method and the over-relaxation
method. The two mass distributions have been then com-
pared.
An example of typical results is shown in Fig. 2. Here,
from top to bottom, we display the original cluster mass
distribution, the reconstruction obtained using the direct
method, and the residuals, i.e. the difference between the
reconstructed maps from the direct method and from the
over-relaxation method. As the figure clearly shows, dif-
ferences are mainly confined to the boundary of the field
where they are found to be of the order of 0.0002, well
below the statistical errors of the reconstruction. In the
inner field the differences are about one order of magni-
tude smaller. Note also that the wavy overall appearance
of the reconstructed map is normal for weak lensing re-
constructions, resulting from the relatively low number of
source galaxies involved (see Lombardi & Bertin 1998b for
a discussion of the statistical aspects of the problem).
Acknowledgements. We thank Luigi Ambrosio and Peter
Schneider for interesting discussions and suggestions. The
reconstruction code uses FFTW 2.0.1 by Matteo Frigo and
Steven G. Johnson. This work has been partially supported
by MURST and by ASI of Italy.
Appendix A: Completeness of {fαβ}
In this appendix we will verify explicitly that the set of
functions defined in Eq. (15) is complete, in the sense that
Eqs. (3) and (4) can be recovered. For the purpose, we
will apply the Fourier theorem (Brezis 1987). In the fol-
lowing, we will assume that ˜ u(θ) is a smooth vector field
(we stress that this condition is needed only for the proof
provided below; the method remains applicable to more
general cases).
Let us consider a solution of the form (6). Then,
because of the orthonormality condition (13), we have
?∇fα,∇˜ κ? = cα= ?∇fα, ˜ u?, so that
?
Ω
?∇fα,∇˜ κ − ˜ u? =∇fα· (∇˜ κ − ˜ u)d2θ = 0 .(A.1)
Now we observe that the previous equation holds for
any α: then it holds also for any linear combination
f =?
0 =
Ω
?
∂Ω
αdαfαof {fα}. Thus
?
∇f · (∇˜ κ − ˜ u)d2θ = −
?
Ω
f∇ · (∇˜ κ − ˜ u)d2θ
+
f(∇˜ κ − ˜ u) · ndθ .(A.2)
In the last step we have integrated by parts (n is the unit
vector orthogonal to the boundary ∂Ω of Ω).
We now use this equation to show that the chosen set
of functions, described by Eq. (15), is complete, while, e.g.,
a similar set made of sine functions would not be complete.
By the nature of the chosen set of functions we already
know that we can properly represent any smooth function
f. Using this property, we want to show that the two terms
∇ · (∇˜ κ − ˜ u) and (∇˜ κ − ˜ u) · n entering in the r.h.s. of
Eq. (A.2) vanish on Ω and on ∂Ω respectively.
For the purpose, we observe that if cosines are used
as set of functions, we can “build” any function f pro-
vided that the function has periodic derivatives on the
boundary. In particular, if A ⊂ Ω is an arbitrary open
subset of Ω, there is a function f that is positive on A and
vanishes on Ω \ A. Now suppose per absurdum that the
solution obtained from the direct method does not satisfy
Eq. (3), so that, e.g., ∇·(∇˜ κ− ˜ u) > 0 on a point θ∗∈ Ω.
Then, for the sign persistence theorem, this quantity must
be strictly positive in a neighborhood A of θ∗. However,
if we take a function f which is positive on A and van-
ishes elsewhere, the rhs of Eq. (A.2) will be positive, while
the lhs vanishes, which is contradictory. This proves that
Eq. (3) is verified by cosines.
In a similar manner, now that we have “disposed of”
the first term, we observe that using cosines we can build
a function f that vanishes everywhere on the boundary
of ∂Ω except for a neighborhood. In other words, given
an open subset B ⊂ ∂Ω of the boundary ∂Ω, there is a
function f that is positive on B and vanishes on ∂Ω \ B.
Note that this property would not be satisfied if the set of
functions {fα} were based on sines. Using a proof similar
to the one given above, we obtain that (∇˜ κ − ˜ u) ·n must
vanish, thus leading to Eq. (4).
One might worry that, on the boundary, the chosen
set of functions of Eq. (15) has zero derivative in the di-
rection of n, i.e. ∇fαβ· n = 0. This might suggest that
Page 6
6 M. Lombardi & G. Bertin: A direct method of mass reconstruction
the boundary condition (4) cannot be reproduced. In re-
ality, although this is true pointwise, this does not affect
the convergence in L2that is relevant for our problem (see
Eqs. (7) and (12)). This important point is best illustrated
by the following example.
Suppose that we measure a constant field ˜ u(θ) = (1,0)
on a square field Ω of side π. The obvious solution for ˜ κ
in this case is ˜ κ(θ) = θ1. Suppose now that we try to
use a set of functions made of sines. Then the correspond-
ing coefficients cαβwould be proportional to the integrals
α?π
cient cαβ vanishes. This proves that a set based on sines
is not complete (condition (12) is not satisfied or, equiva-
lently, a curl-free vector field ˜ u leads to a vanishing mass
map). On the other hand, if we use the set (15), the co-
efficients cαβ do not vanish and the corresponding mass
distribution is given by
0θ1cosαθ1dθ1
?π
0sinβθ2dθ2= 0. Hence, every coeffi-
˜ κ(θ) = −
?
α odd
4
πα2cosαθ1.(A.3)
This function can be shown to reduce to ˜ κ(θ) = θ1−π/2.
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