This dissertation is composed of two parts. In the first part techniques of band exclusion(BE) and local optimization(LO) are proposed to solve linear continuum inverse problems independently of the grid spacing. The second part is devoted to the Fourier phase retrieval problem. Many situations in optics, medical imaging and signal processing call for solutions of linear continuum inverse problems. Spectral estimation is an example among those. A commonly used method is to seek a discrete, approximate solution for the continuum problem by discretizing the problem on a finite grid, but meanwhile, a gridding error, roughly proportional to the grid spacing, arises in the discretization process. When the grid spacing is above the Rayleigh length, the gridding error can be as large as the data themselves, creating an unfavorable signal to noise ratio. To reduce the gridding error, one has to refine the grid. However, when the grid spacing is reduced below the Rayleigh length, sensing matrices become underdetermined and highly coherent, resulting in the failure of many existing compressive sensing(CS) algorithms. In order to fill the gap, we propose the techniques of BE and LO to deal with coherent sensing matrices on a fine grid. These techniques are embedded in the existing CS algorithms, such as Orthogonal Matching Pursuit(OMP) and Basis Pursuit(BP), and give rise to the modified algorithms, such as BLO-based OMP (also called BLOOMP) and BLO-based BP (also called BP-BLOT) respectively. We have proved that, under certain conditions, BLO-based OMP is capable of reconstructing sparse, widely separated objects within one Rayleigh length in bottleneck distance independent of the grid spacing. Detailed numerical comparisons with other algorithms designed for the same purpose, such as the Spectral Iterative Hard Thresholding (SIHT) and the analysis-based BP, demonstrate the superiority of BLO-based algorithms. The second part of this dissertation is mainly concerned with the Fourier phase retrieval problem: reconstructing an unknown image from its Fourier magnitude measurements. This problem arises frequently in a number of different imaging modalities including X-ray crystallography, coherent light microscopy, astronomy, etc. It is well known that traditional phasing methods have stagnation problems of outputting an image that is not fully reconstructed, due to non-convexity as well as non-absolute-uniqueness, with absolute uniqueness being referred to as uniqueness up to a constant global phase. In the phasing literature a lot of emphasis has been put on the algorithm designs or the utilization of a priori information in order to avoid stagnation. Instead, we attack the Fourier phase retrieval problem using well-designed illuminations. Specifically we explore a phasing method based on a random phase mask(RPM) that randomly modifies the phases of the original image. We demonstrate that the use of RPM in Fourier phasing not only results in an absolute uniqueness but also leads to superior numerical performances, including rapid convergence, much reduced data and noise stability. More importantly, Fourier phasing with RPM does not rely on accurate information of the phase mask. We show that nearly perfect recovery can be achieved in the case of phase-uncertain mask where one's estimates on the mask phases differ from the true mask phases within certain level. Absolute uniqueness results are generalized to the case of phase-uncertain mask, stating that under certain conditions both the image and the mask within the image support are uniquely determined up to a constant global phase with high probability. A numerical scheme alternating between the image update and the mask update is proposed to recover the image and the mask simultaneously. Our numerical simulations demonstrate that nearly perfect recovery can be achieved by RPM with high uncertainty in mask phases. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]