Gravitational lensing

We present a series of papers using analytic and numerical methods to study gravitational lenses, culminating with the development of LensMEM, an algorithm for inverting poorly resolved lens images using the maximum entropy method (MEM).;We begin with a study of quasar lensing statistics. We compute lensing cross sections for a range of lens parameters. The model predictions are in good agreement with the observations. We are able to constrain the size of lens core radii to ;We next consider lensing of cosmologically distant gamma-ray bursts. We find that for point-like lenses the quantity (1 + ;We then begin to develop LensMEM by producing a version of the algorithm in one dimension. This method uses an objective error measure and makes no assumptions about the source structure. We test the algorithm with simulations, and find that it accurately inverts ring-type images and is effective at solving for the lens parameters. When the source is larger than the lens critical radius. the reconstruction produces discontinuities at the boundaries separating different image multiplicities. We show that these errors are intrinsic to MEM when applied to the lensing problem and can be removed by filtering out the high frequency power.;We next develop a pixel-mapping scheme for inverting extended images without taking into account the finite resolution of the observations. We apply this method to the giant arc in Cl0024 + 1654 and produce very good inversions of the arc structure. We find that acceptable models require the two perturbing galaxies near the middle arc segment.;Finally, we combine this pixel-mapping technique with a two-dimensional expansion of the MEM equations to produce the complete LensMEM algorithm. We test the method with simulations and apply it to the radio ring MG1654 + 134. We obtain results in very good agreement with those obtained by Kochanek (1995) using a complementary inversion algorithm based on Clean.