1. A method of using a preconditioned optimization strategy in generating a magnetic resonance (MR) image with an accelerated compressed sensing reconstruction, the method comprising: a) acquiring k-space MR data with a magnetic resonance imaging system; b) reconstructing an MR image from the acquired k-space, with an accelerated compressed sensing reconstruction process that utilizes a computer processor in communication with the magnetic resonance imaging system by: i) implementing the use of an alternative-direction-method-of-multiplier (ADMM) strategy in the reconstruction of the generated MR image that initially decomposes the optimization of the ADMM strategy into sub problems before reconstructing the MR image with the computer processor in communication with the magnetic resonance imaging system; ii) after completing step i), further decomposing at least one of the subproblems of step i) by the use of a Woodbury matrix identity having a diagonal preconditioner (PC) that is based on non-Toeplitz matrix models and iii) after completing step ii), solving each of the decomposed sub problems with the computer processor in communication with the magnetic resonance imaging system, whereby the accelerated compressed sensing reconstruction process of steps i), ii) and iii) are utilized in order to generate, provide on a display, and/or store in a memory, the magnetic resonance image resulting from performing this Woodbury preconditioned ADMM optimization strategy.