Analog computers perform a few, specific operations at full-speed that take quite a few components to do in digital with them only running at a specific, clock rate. Combining them with digital computers that feed them problems and measure the answers can be quite powerful. See especially the math-coprocessor at first link. The neural simulation at second is illustrative where one wafer (300+ chips) of analog NN's took 294,912 cores to simulate. That's full-custom, high-end digital stuff rather than the average stuff. Analog also gets used a lot in mixed-signal ASIC's where specific things are cheaper or lower power per unit in analog so they mix digital and analog.

https://www.schneier.com/blog/archives/2015/07/friday_squid_...

https://news.ycombinator.com/item?id=11731697

EDIT to add: The paper below is a short history of them that gives examples whose benefits are consistent with other stuff I linked to. Some really cool stuff there, too. :)

http://oro.open.ac.uk/5795/1/bletchley_paper.pdf

Yes, it's possible to do incredibly fast signal processing calculations using interfering laser beams.

http://www.nextbigfuture.com/2014/08/startup-with-optical-so...

In practice, it's not easy to compete with the economies of scale of conventional chips, though.

A modern GPU (10 TFlops) can simulate a million nodes at 1 MHz, far more than any analog computer. The article was written in 2005, when that might still have been true.

Sure - http://mythic-ai.com

Infinite amount.