A guaranteed convergence analysis for the projected fast iterative soft-thresholding algorithm in parallel MRI

•Analyze convergence criteria of the parallel imaging version pFISTA•Recommend step sizes to ensure algorithm convergence and to enable fast and promising reconstructions.•The recommended step sizes are close to the hand-tuned optimal step sizes.•pFISTA with the recommended step size outperforms other state-of-the-art algorithms in terms of fast speed and/or low reconstruction errors.