This thesis by Dmitry M. Malioutov focuses on the application of sparse signal reconstruction techniques to passive source localization using sensor arrays. The core of the thesis is the formulation of the source localization problem as an inverse problem with sparsity-enforcing regularization. The approach involves reformulating the problem in an optimization framework using an overcomplete basis and applying sparsifying regularization to focus signal energy and achieve high resolution. The thesis develops numerical methods for enforcing sparsity using $\ell_1$ and $\ell_p$ regularization, with the second-order cone programming framework used for $\ell_1$ regularization and half-quadratic regularization for $\ell_p$. It also proposes methods for using multiple time samples of sensor outputs and automatically selecting the regularization parameter. Extensive numerical experiments demonstrate the advantages of the proposed approach, including super-resolution, robustness to noise and limited data, and robustness to correlated sources. The thesis further extends the approach to self-calibration of sensor position errors. Additionally, the thesis includes theoretical analysis of the noiseless signal representation problem using overcomplete bases, discussing the uniqueness of solutions to the $\ell_0$ problem and the equivalence of $\ell_0$, $\ell_1$, and $\ell_p$ problems under certain conditions.This thesis by Dmitry M. Malioutov focuses on the application of sparse signal reconstruction techniques to passive source localization using sensor arrays. The core of the thesis is the formulation of the source localization problem as an inverse problem with sparsity-enforcing regularization. The approach involves reformulating the problem in an optimization framework using an overcomplete basis and applying sparsifying regularization to focus signal energy and achieve high resolution. The thesis develops numerical methods for enforcing sparsity using $\ell_1$ and $\ell_p$ regularization, with the second-order cone programming framework used for $\ell_1$ regularization and half-quadratic regularization for $\ell_p$. It also proposes methods for using multiple time samples of sensor outputs and automatically selecting the regularization parameter. Extensive numerical experiments demonstrate the advantages of the proposed approach, including super-resolution, robustness to noise and limited data, and robustness to correlated sources. The thesis further extends the approach to self-calibration of sensor position errors. Additionally, the thesis includes theoretical analysis of the noiseless signal representation problem using overcomplete bases, discussing the uniqueness of solutions to the $\ell_0$ problem and the equivalence of $\ell_0$, $\ell_1$, and $\ell_p$ problems under certain conditions.