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Magnetic resonance imaging (MRI) is a non-invasive medical imaging modality that uses magnetic fields. Accurate estimates of these fields are often used to improve the quality of MR imaging techniques. Regularized estimators for such fields are robust and can provide high quality estimates but often at a significant computational cost. In this work, we investigate several of these estimators with a focus on developing novel minimization methods that reduce their computation times. First, we explore regularized receive coil sensitivity estimation by demonstrating the improved performance of regularized methods over existing, heuristic approaches and by presenting several algorithms, based on augmented Lagrangian methods, that minimize the quadratic cost function in half the time required by a preconditioned conjugate gradient (CG) method. Second, we present a general cost function that combines the regularized estimation of the main magnetic field inhomogeneity for both multiple echo time field map estimation and chemical shift based water-fat imaging. We present two methods, both based on optimization transfer principles, that reduce the computation time of this estimator by a factor of 30 compared to the existing separable quadratic surrogates method. We also evaluate the effectiveness of edge preserving regularization for field inhomogeneity estimation near tissue interfaces. Third, we present a novel alternating minimization method that uses augmented Lagrangian methods to accelerate the computation of the compressed sensing based water-fat image reconstruction problem by at least ten times compared to the existing nonlinear CG method. The algorithms presented in this thesis may also be applicable to other MRI topics including B1+ estimation, T1 estimation from variable flip angles, and R2* corrected or parallel imaging extensions of compressed sensing based water-fat imaging.