SuperRes
The general idea behind the super resolution method that Alexey Lukin et al. explained in their paper is to treat upscaling as inverse downscaling. So the aim is to find a high resolution image which, after downscaling, is equal to the low resolution image.
The problem is that this is usually not well defined. For instance let's take the simplest possible example i.e. an image consisting of just 1 pixel, with value X. And say we want to find an image consisting of 2 pixels with values which we'll denote A and B. Now we need to decide on a downsampling algorithm from the image of 2 pixels to the image of 1 pixels. The obvious choice is to just average the two pixels, i.e. X = (A+B)/2. So if we had the values of A and B then we could find X but we're working backwards so we know the value of X and we want to find the values of A and B. However for any value of A there is a value of B such that (A+B)/2 = X, so there's no way to decide upon a solution. This can be solved by requiring the values of A and B to be close to each other, in which case the unique(!) solution is to make A and B equal X.
The SuperRes algorithm works similar to this except with more pixels, it requires the image to be "regular" (pixel close to each other should have "similar" values) and instead of requiring the result to be exactly equal to the original image after downscaling we simply require it to be "faithful". This method has an enormous amount of flexibility since we can choose which downscaling method to use, how to measure "regularity", and how to measure "faithfulness".
There are 3 parameters: Passes, Strength and Softness.
Strength and passes control how strong the effects are and how many times it will apply them, respectively. If you set the value for strength too high you risk 'overshooting' the optimal value, which will not only make the algorithm converge slower, but will also cause artefacts. Using more passes at a lower strength will give a nicer result but requires more power.