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ChrisRackauckas/DifferentialEquations.jl

Julia package for solving differential equations

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Jun 24 22:14

ChrisRackauckas commented #878
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ChrisRackauckas on format2

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Christopher Rackauckas

@ChrisRackauckas

Just let me know if that "new" interface sol = ODE.solve(ode,stepper;opts...) ends up changing at all.

I'll be watching that dev branch.

Paweł Biernat

@pwl

@ChrisRackauckas you mentioned wave relaxation, do you have any good books/overview papers on that?

Christopher Rackauckas

@ChrisRackauckas

I know Burrage's book covers it.

I was talking with Kevin Burrage about it not too long ago. Let me find the reference.

I guess that tome is kind of expensive and out of print.

https://www.amazon.com/Sequential-Differential-Mathematics-Scientific-Computation/dp/0198534329/ref=sr_1_1?ie=UTF8&qid=1469460761&sr=8-1&keywords=parallel+methods+for+ordinary

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.

For example: http://www.sciencedirect.com/science/article/pii/089571779490099X

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.

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