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Astron. Astrophys. 348, 38-42 (1999)

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1. Introduction

In the context of weak or statistical lensing, the problem of the determination of the dimensionless mass density distribution [FORMULA] from a map of the reduced shear [FORMULA] 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 [FORMULA] defined in terms of the measured reduced shear [FORMULA] (Kaiser 1995). In the ideal case where the measured shear [FORMULA] is just the true shear [FORMULA], the vector field [FORMULA] can be shown to satisfy the relation


where [FORMULA] is the true dimensionless mass map and [FORMULA]. However, because of statistical and measurement errors, [FORMULA] is not necessarily curl-free, and thus [FORMULA] can be determined only approximately. In a separate paper (Lombardi & Bertin 1998b) we have shown that

  • The statistical errors on [FORMULA] are minimized if this function is calculated as


    where [FORMULA] is a constant (introduced to take into account the sheet invariance ), [FORMULA] is the noise-filtering kernel (Seitz & Schneider 1996), and [FORMULA] is the field of observation.
  • The same mass map can be obtained by solving the equations


    where [FORMULA] is the unit vector perpendicular to the boundary [FORMULA] of the field of observation [FORMULA]. Hence, the kernel [FORMULA] can be identified as the Green function of this Neumann boundary problem.
  • Eqs. (3) and (4) are precisely the Euler equations associated with the functional


    In other words, the functional S is minimized when [FORMULA] is calculated using Eq. (2) or, equivalently, by solving Eqs. (3) and (4).

To these three formulations of the mass inversion problem there correspond three practical methods to calculate [FORMULA] from a given set of data.

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

  • The introduction of a method that directly solves the Neumann problem allows one to go beyond many of the limitations of the [FORMULA] method. Eqs. (3) and (4) can be solved using an over-relaxation method (Seitz & Schneider 1998). In this case [FORMULA] is calculated directly, and thus we need to allocate only [FORMULA] real numbers. Moreover, the method can be applied without difficulties to "strange" geometries [FORMULA] (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 approximately of order [FORMULA].

  • 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 [FORMULA]. Moreover, it is extremely easy to implement (two implementations for rectangular fields [FORMULA] 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.

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999