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ChrisRackauckas/DifferentialEquations.jl

Julia package for solving differential equations

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Christopher Rackauckas

@ChrisRackauckas

Well I think I am going to go ahead with wrapping only ODE.jl, and clean up my code to conform better with Julia standard (I realized that my camelCase isn't what's used here, so I'll fix that all up).

I will use that to set benchmarks, and when that's all together I'll contact you guys again.

Paweł Biernat

@pwl

cool, let's keep in touch!

Mauro

@mauro3

Tnx!

Christopher Rackauckas

@ChrisRackauckas

I am thinking that we shouldn't both need to write wrappers, and I would be willing to change my API in many ways to match what you guys think is reasonable.

Paweł Biernat

@pwl

that's great, we will let you know once we fix our docs, you could write some wrappers yourself if you are up for it:-)

Christopher Rackauckas

@ChrisRackauckas

So I'll get to there so you guys can take a look at it. If you think it's something that could be used for doing the rest of the wrapping (and we agree on whatever API changes), then we should merge into some org and get the rest of it together.

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

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