"Inverse problems have many applications. In this thesis, we focus on designing and implementing numerical methods for three inverse problems: gravity inversion, synthetic aperture radar, and travel-time tomography. We present extensive numerical examples to demonstrate that these algorithms are stable and efficient. In Chapter 2, low-rank approximation is incorporated into a local level-set method for gravity inversion. This change helps to reduce the computational time of the mismatch gravity force term on the boundary, and reduces the computational complexity from O(N3 ) to O(N2 ) in 2D and from O(N5 ) to O(N4 ) in 3D. Many numerical results show that the locations of unknown objects are accurately captured by this low-rank level-set method. In Chapter 3, both the wave equation and Radon transform are carried out as an approach to the synthetic aperture radar problem. The wave-equation-based method includes harmonic extension at terminal time, solving the wave equation backward using a perfectly matched layer, and Neumann iteration. These two methods provide comparable results and help to prove that a curved flight path is no better than a straight one. In Chapter 4, we implement the finite element method as a penalization-regularization-operator splitting method for travel-time tomography based on the eikonal equation. Both the travel time and slowness are recovered with this algorithm in both 2D and 3D. Finally, Chapter 5 contains our conclusions."--Page ii.
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