Strong gravitational lensing produces multiple distorted images of a background source when it is closely aligned with a mass distribution along the line of sight. The lensed images provide constraints on the parameters of a model of the lens, and the images themselves can be inverted providing a model of the source. Both of these aspects of lensing are extremely valuable, as lensing depends on the total matter distribution, both luminous and dark. Furthermore, lensed sources are commonly located at cosmological distances and are magnified by the lensing effect. This provides a chance to image sources that would be unobservable when viewed with conventional optics. The semilinear method expresses the source modeling step as a least-squares problem for a given set of lens model parameters. The blurring effect due to the point spread function of the instrument used to observe the lensed images is also taken into account. In general, regularization is needed to solve the source deconvolution problem. We use Krylov subspace methods to solve for the pixelated sources. These optimization techniques, such as the Conjugate Gradient method, provide natural regularizing effects from simple truncated iteration. Using these routines, we are able to avoid the explicit construction of the lens and blurring matrices and solve the least squares source optimization problem iteratively. We explore several regularization parameter selection methods commonly used in standard image deconvolution problems, which lead to previously derived expressions for the number of source degrees of freedom. The parameters that describe the lens density distribution are found by global optimization methods including genetic algorithms and particle swarm optimizers. In general, global optimizers are useful in non-linear optimization problems such as lens modeling due to their parameter space mapping capabilities. However, these optimization methods require many function evaluations and iterative approaches to the least squares problem are beneficial due to the speed advantage that they offer. We apply our modeling techniques to a subset of gravitational lens systems from the Sloan Lens ACS (SLACS) survey, and are able to reliably recover the parameters of the lens mass distribution with both analytical and regularized pixelated sources.

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