Tolerances for adaptive step size solvers

[MichaelClerx]

With ion channel fitting, we've been banging on about solver tolerances for a while:

See also https://mirams.wordpress.com/2018/10/17/ode-errors-and-optimisation/

And it occurred to me this is a general point for PINTS problems, so we should probably

1. check our toy models
2. Write some example

[MichaelClerx]

I'm trying to do a good drawing of this but failing :-D

Basically what happens is:

• you use parameters p, step size changes happen whenever derivatives get large, the first one happens at step 10, the second at step 20, the third at step 30, etc. until 100s of changes have been made
• you use parameters p + delta p; this makes the derivatives slightly steeper initially, causing the first step size change to happen at step 9. This has a knock-on effect, causing the second change to only be made at step 21, the third at step 28, etc.

The end result is that the steps you choose for integration vary sensitively with p, even if the simulated time series don't
So you end up with a simulation result that lies somewhere in the range allowed by the solver tolerance, but where it lies exactly can be very jittery, unless you set really tight tolerances

[MichaelClerx]

@DavidGavaghan suggests this could be a good student project / short paper