 
Astron. Astrophys. 348, 3842 (1999)
1. Introduction
In the context of weak or statistical lensing, the problem of the
determination of the dimensionless mass density distribution
from a map of the reduced shear
has been considered in detail by
various authors, using either simulations 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 defined in terms of the
measured reduced shear (Kaiser
1995). In the ideal case where the measured shear
is just the true shear
, the vector field
can be shown to satisfy the
relation
where is the true dimensionless
mass map and . However, because of
statistical and measurement errors,
is not necessarily curlfree, and thus
can be determined only
approximately. In a separate paper (Lombardi & Bertin 1998b) we
have shown that
To these three formulations of the mass inversion problem there
correspond three practical methods to calculate
from a given set of data.

The first method considered is based on a direct calculation of the
kernel (Seitz & Schneider 1996).
Once this kernel has been calculated for a given field
, the mass inversion is
straightforward (note that the kernel
depends on the field of
observation). A problem with this method is that a calculation of
is expensive in terms of memory
requirements and computation time. In fact, in order to compute
on a square grid of
points,
must be calculated on a
multidimensional grid of points.
Moreover, multiplications are needed
to evaluate Eq. (2), and thus the method is of order
. Because of the large memory needed
to allocate , calculations can be
performed only with a limited value of N (typically
).

The introduction of a method that directly solves the Neumann
problem allows one to go beyond many of the limitations of the
method. Eqs. (3) and (4) can be
solved using an overrelaxation method (Seitz & Schneider
1998). In this case is calculated
directly, and thus we need to allocate only
real numbers. Moreover, the method
can be applied without difficulties to "strange" geometries
(while the previous method is
straightforward only when applied to rectangular or circular fields).
The overrelaxation method is quicker than the kernel method, being
approximately of order .

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 efficient from a computational
point of view, being of order .
Moreover, it is extremely easy to implement (two implementations for
rectangular fields written in
C and in IDL are freely available on request).
We should stress that, as proved in an earlier paper (Lombardi
& Bertin 1998b), the three formulations are mathematically
equivalent. Thus it would not be surprising to find that proper
numerical implementations perform, 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.
© European Southern Observatory (ESO) 1999
Online publication: July 16, 1999
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