What's the alternative to Fast GI Approximation? Ticking this box seems to create a quite strong and brutally controllable fast approximation of GI. Fair and well labelled.
But how to do normal, fully calculated GI within Blender?
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As you can see in the manual Fast GI Approximation replaces indirect lighting with Ambient Occlusion. If it's off, normal rendering calculations are done. You don't need to do anything else than enable it if you want it or disable it if you don't get realistic enough results.
Rendering is done by tracing light paths from the camera to the light source with it interacting with the objects in your scene on the way. Every pixel gets many samples and the result is a value that is statistically likely for that pixel. This is where noise comes from if there aren't enough samples and that is what the samples settings are for. You also can control bounces of various types:
They define how many times light can interact with surfaces in your scene on that light path that is traced/calculated. If you wanted more accurate "Global illumination" you have to make sure that number of bounces is adequate. Keep in mind that it is really hard to see light that bounces off diffuse surfaces more than 5-6 times in reality and even in a room full of mirrors the view fades after a few hundreds of reflections, so you don't always need huge numbers of bounces to get realistic results. Light path going through transparent or transmissive surface also counts as "bounces", so if you want to see through many transparent surfaces, the number of those bounces needs to be at least equal to the number of surfaces light goes through.