/ 99 10 50 2 v 1 2 8 O ct 1 99 9 Strong Lensing Reconstruction

We present a general linear algorithm for measuring the surface mass density 1 − κ from the observable reduced shear g = γ/(1 − κ) in the strong lensing regime. We show that in general, the observed polarization field can be decomposed into " electric " and " magnetic " components, which have independent and redundant solutions, but perfectly orthogonal noise properties. By combining these solutions, one can increase the signal-to-noise ratio by √ 2. The solutions allow dynamic optimization of signal and noise, both in real and Fourier space (using arbitrary smoothing windows). Boundary conditions have no effect on the reconstructions, apart from its effect on the signal-to-noise. Many existing reconstruction techniques are recovered as special cases of this framework. The magnetic solution has the added benefit of yielding the global and local parity of the reconstruction in a single step. Gravitational lensing is rapidly providing large, independent data sets which are direct measures of gravitational potentials. One would like to understand the properties of gravitational lensing reconstruction to optimize the solutions in a similar fashion as has been done for galaxy surveys (Vogeley and Szalay 1996). In the weak lensing regime, the reconstruction is linear, and optimization of signal-to-noise becomes a straightforward problem. When lensing becomes strong, the equations appear to become non-linear, and the solutions are more difficult to understand. Of particular interest is the understanding of error propagation, to quantify the confidence of the solution, and to optimize the signal-to-noise for a real, noisy data sets. The measurement of galaxy polarization, or reduced shear, yields two observable quantities, the ellipticity and orientation of galaxies at each pixel of the reconstruction. In the single source-lens plane lensing problem, the only unknown is the dimensionless surface mass density κ. One would thus expect, in general, two independent solutions to exist, each using only half the observable information. If the noise properties in each solution are understood, and orthogonal, the two solutions can be combined in an optimal fashion. It is the purpose of this paper to systematically construct such solutions. We also quantify the existing solution algorithms in the same framework, and discuss error properties.