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Paweł Biernat
@pwl
that is some obscure source
Christopher Rackauckas
@ChrisRackauckas
I'll probably pick up a copy myself, but I'll send Kevin an email and see if I can get a PDF, or what the "latest" reference is (I know this is from early 2000)
(Late 1990s)
Paweł Biernat
@pwl
cool, let me know if you get it
Christopher Rackauckas
@ChrisRackauckas
Well it's not obscure, you'd find it if you were reading literature and looked at citations.
But it's just out of print.
Paweł Biernat
@pwl
what I mean by it is that this doesn't seem like a popular/well known method if the only book about it is out of print
Christopher Rackauckas
@ChrisRackauckas
None of them really are.
I mean, you can find a bunch of articles on it.
Paweł Biernat
@pwl
have you tested it?
Christopher Rackauckas
@ChrisRackauckas
There are a bunch of articles which I think tend to show that Waveform Relaxation "kind of" came out on top of the other parallel methods (parallel extrapolation, special Runge-Kutta methods)
But it's supposedly a little difficult to program.
That's why I want to try it out.
I know Hairer has parallel extrapolation in ODEX
so there's no reason to code that
Paweł Biernat
@pwl
thanks!
Christopher Rackauckas
@ChrisRackauckas
The special Runge-Kutta methods you just need to use threading. They just turn it into solving Ax=b each step in some simplified way.
Christopher Rackauckas
@ChrisRackauckas
@pwl This is a good SE post for explaining the difference between the major groups., and here is another reference for the differences . We might think about writing a wrapper to PFASST. Actually I think it we wrap enough of them or put enough of these algorithms under the same roof we might be able to get a review article out of it.
In the end I am looking to try a few of them since I am interesting in which ones may be useful for stochastic differential equations.
But I don't really think that any program puts all of these together already, so this is something to be done.
While I'm at it, the other ODE methods I am looking into implementing are super high order RKs.
Let me know if you plan on implementing any of these so we don't double up!
Paweł Biernat
@pwl
have you looked at high order Taylor methods?
Christopher Rackauckas
@ChrisRackauckas
I mean, I know how they're derived.
Paweł Biernat
@pwl
There was some package on github (now removed), that implemented arbitrary order Taylor methods, which I hooked up to ODE to test the backend API and the method, they worked!
Christopher Rackauckas
@ChrisRackauckas
Did you have to provide the derivatives?
Paweł Biernat
@pwl
ForwardDiff does it for you
Christopher Rackauckas
@ChrisRackauckas
But for really high derivatives, that doesn't give more overhead?
Paweł Biernat
@pwl
it does, but on the other hand you can make longer stepsizes
Christopher Rackauckas
@ChrisRackauckas
Depending on the function being "nice"
Paweł Biernat
@pwl
so it sort of balances out
Christopher Rackauckas
@ChrisRackauckas
With automatic differentiation... that's interesting. I'd like to test that out vs other methods.
Paweł Biernat
@pwl
I tested it with a hyperbolic PDE that I work on and it worked as quick as RK
Christopher Rackauckas
@ChrisRackauckas
I thought of using symbolic differentiation before with Taylor methods.
Paweł Biernat
@pwl
at least the same order of magnitude
the fun thing with this method is that you can prescribe an error much lower then the machine epsilon
like reltol 1e-30 with Float64
Christopher Rackauckas
@ChrisRackauckas
Probably scales better than extrapolation.
Paweł Biernat
@pwl
but this is a niche application
Christopher Rackauckas
@ChrisRackauckas
Well you'd need to go to other numbers to avoid weird truncation error problems at that point.
But a battle between high-order extrapolation, RK16, and 16th order Taylor series via automatic differentiation.
I'd want to try that out just for fun.
People in astrodynamics may find that useful.
Mauro
@mauro3
@PerezHz looks into taylor methods in his PhD with Benet & Sanders: https://github.com/JuliaLang/ODE.jl/issues/91#issue-139761284
Christopher Rackauckas
@ChrisRackauckas
But @pwl said the package is no longer available?
Paweł Biernat
@pwl
unfortunately the package disappeared
I still have a hard copy on my hd
Christopher Rackauckas
@ChrisRackauckas
Please share :smile:
It will have to be updated for sure.