Magnetic resonance imaging (MRI) is one of the most powerful imaging techniques for examining the human body since it allows early and accurate diagnosis of diseases. Although high magnetic field systems (≥ 3 Tesla) enable increased spatial resolution, long scan times (i.e. 15 min for high-resolution (HR) imaging around 500 um isotropic) and motion sensitivity continue to impede the exploitation of HR-MRI. To circumvent that problem, Compressed Sensing (CS) was introduced, among other techniques, to reduce the acquisition time, taking advantage of the structure of MR images. However, the time gained on acquisition has been lost on reconstruction as sparse recovery amounts to iteratively solving a linear inverse problem, The motivation to reduce the time required for image reconstruction is the following: the MRI physician needs to analyze the reconstructed image to know if there has been some movement causing some motion artifacts, and therefore take the decision to rerun the exam. Thus, decreasing reconstruction time would mean reducing the overall duration of the MRI exam. The goal of this work was to focus on the different techniques and benchmark their speed and memory load against the vanilla FISTA, the Condat-Vu algorithm and the POGM' algorithm (i.e. POGM combined with adaptive restart) to solve the reconstruction problem. The most promising algorithm, greedy FISTA, applies the following modifications to the vanilla FISTA. These algorithms are all implemented in the Python library modopt for the sake of reproducibility. The criterion used to assert convergence is the objective function. Our analysis was performed on the 2D MR phantom image, which was non-uniformly Fourier transformed to pick up the k-space samples using the random variable density (VD) sampling scheme. We also tried the same analysis with the SPARKLING sampling scheme, which is a physically feasible accelerated sampling scheme. The orthogonal Daubechies 4 was chosen for the wavelet basis. This made the proximal of the regularisation term computable in closed form. The results show that the greedy FISTA compares to POGM' in terms of time. However, it theoretically uses twice less memory.