Numerical issues with direct `Fun` usage

[ChrisRackauckas]

Direct usage of Funs in OrdinaryDiffEq.jl works... but kind of.

After a few steps, the size of the Fun's coefficient array grows really large and numerical errors occur. On the same problem, the fixed size Fun does okay, so I think it must be due to how the array just keeps growing.

[ChrisRackauckas]

@dlfivefifty

[dlfivefifty]

Adaptive time-stepping is a delicate issue. I had an honours student look at this, and we found it can only be done reliably with L-stable schemes. This may be avoidable with smarter Plateaux detection.

It's also important that the solutions coefficients are chopped to avoid the length getting out of control.

[dlfivefifty]

Note that L-stable schemes add a nit of numerical diffusion, which may not be desirable.

[ChrisRackauckas]

This doesn't do adaptive timestepping. This is fixed timesteps, adaptive coefficient length.

It's also important that the solutions coefficients are chopped to avoid the length getting out of control.

This is the issue.

[ChrisRackauckas]

Note that L-stable schemes add a nit of numerical diffusion, which may not be desirable.

Sundials CVODE_BDF seems to do fine. You likely only need L-stability if you're near the stability limit.

[dlfivefifty]

I meant adaptive in space

[ChrisRackauckas]

I see. Gets confusing with the two meanings.

Yeah, I don't think it's the spatial adaptivity as much as it is the numerical issue with huge numbers of coefficients. It works fine for quite a few steps, but reliably fails at step ~12 no matter what, and the coefficient array is large with tons of <1e-16 coefficients. Is there a way to zero the coefficients when they are sufficiently small?

[dlfivefifty]

I think you always need L-stability, as in infinite-dimensions you want the operators to be bounded. That is, you want to invert more differential orders than you apply. (This is more than a numerical issue.)

[dlfivefifty]

Yes, chop(f,atol) and chop!(f,atol)

[dlfivefifty]

chop should probably be redesigned to mimi isapprox and allow both relative and absolute tolerances

[ChrisRackauckas]

Is there a way to set the Fun to "auto-chop" all coefficients below a certain tolerance?

[dlfivefifty]

This really should happen in . The only reason it doesn't is that I'm not sure about the implications.

One of the big missing features in ApproxFun that Chebfun has is plateau detection and chopping, to determine which coefficients are noise. I've been meaning to copy their (very heuristic) chopping algorithm.

[dlfivefifty]

The more the constructor is used the more important this is (which is the case with broadcasting)

[dlfivefifty]

Sorry, this really should happen in backslash

[ChrisRackauckas]

Backslash isn't used here. This is just testing a Runge-Kutta method. Fixed coefficient length works (with adaptivity in time). Same timesteps doesn't work with adaptivity in space, and the number of coefficients seems to explode. If there's a way that a+b for Funs applies some tolerance on the coefficients of the result and correctly chops, this would work without intervention.