Random Illumination Microscopy (RIM) is a recent super-resolved fluorescence imaging technique in which the sample is recovered iteratively by matching the empirical variance of low-resolution images obtained under random speckled illuminations with the expected variance model. RIM was shown theoretically to achieve a twofold resolution gain and its performances have proven very robust to deteriorated imaging conditions. However, the reconstruction algorithm suffers from a slow convergence which can prevent the method from being used to its full potential. Here, we show that a simple, noniterative, linear deconvolution of the empirical standard-deviation image using an appropriate kernel can be sufficient to obtain a satisfactory super-resolved reconstruction of the sample. This first estimate can be further improved with a new accelerated iterative strategy which convergence speed is about two orders of magnitude better than that of variance matching.