[Haskell-cafe] FRP, integration and differential equations.
Peter Verswyvelen
bugfact at gmail.com
Tue Apr 21 07:32:30 EDT 2009
Hey thanks for the Adam-Bashford tip, didn't know that one yet (although I
used similar techniques in the past, didn't know it had a name :-)
Well, solving the ODE is usually the task of a dedicated physics engine. But
IMHO with FRP we try to reuse small building blocks so we get very modular
systems; a big physics black box seems to be against this principle?
On Tue, Apr 21, 2009 at 1:24 PM, Paul L <ninegua at gmail.com> wrote:
> Adam-Bashford method can be easily implemented to replace Euler's. But
> to really get higher accuracy, one may need variable time steps and
> perhaps even back tracking, which is an interesting topic on its own.
> But my question is, is FRP really the right setting in which to
> explore a highly accurate ODE solver?
>
>
> On 4/21/09, Peter Verswyvelen <bugfact at gmail.com> wrote:
> > Well, the current FRP systems don't accurately solve this, since they
> just
> > use an Euler integrator, as do many games. As long as the time steps are
> > tiny enough this usually works good enough. But I wouldn't use these FRPs
> > to
> > guide an expensive robot or spaceship at high precision :-)
> >
> >
> > On Tue, Apr 21, 2009 at 11:48 AM, jean-christophe mincke <
> > jeanchristophe.mincke at gmail.com> wrote:
> >
> >> Paul,
> >>
> >> Thank you for your reply.
> >>
> >> Integration is a tool to solve a some ODEs but ot all of them. Suppose
> >> all
> >> we have is a paper and a pencil and we need to symbolically solve:
> >>
> >>
> >>
> >> /
> >> t
> >> de(t)/dt = f(t) -> the solution is given by e(t) = | f(t) dt +
> >> e(t0)
> >>
> /
> >> t0
> >>
> >> de(t)/dt = f(e(t), t) -> A simple integral cannot solve it, we need to
> >> use
> >> the dedicated technique appropriate to this type of ODE.
> >>
> >>
> >> Thus, if the intention of the expression
> >>
> >> e = integrate *something *
> >>
> >> is "I absolutely want to integrate *something* using some integration
> >> scheme", I am not convinced that this solution properly covers the
> second
> >> case above.
> >>
> >> However if its the meaning is "I want to solve the ODE : de(t)/dt =*
> >> something* " I would be pleased if the system should be clever enough to
> >> analyse the *something expression* and to apply or propose the most
> >> appropriate numerical method.
> >>
> >> Since the two kinds of ODEs require 2 specific methematical solutions, I
> >> do
> >> not find suprising that this fact is also reflected in a program.
> >>
> >> I have not the same experience as some poster/authors but I am curious
> >> about the way the current FRPs are able to accurately solve the most
> >> simple
> >> ODE:
> >>
> >> de(t)/dt = e
> >>
> >> All I have seen/read seems to use the Euler method. I am really
> >> interested
> >> in knowing whether anybody has implemented a higher order method?
> >>
> >> Regards
> >>
> >> J-C
> >>
> >>
> >> On Tue, Apr 21, 2009 at 5:03 AM, Paul L <ninegua at gmail.com> wrote:
> >>
> >>> Trying to give different semantics to the same declarative definition
> >>> based
> >>> on whether it's recursively defined or not seems rather hack-ish,
> >>> although
> >>> I can understand what you are coming from from an implementation angle.
> >>>
> >>> Mathematically an integral operator has only one semantics regardless
> >>> of what's put in front of it or inside. If our implementation can't
> >>> match
> >>> this
> >>> simplicity, then we got a problem!
> >>>
> >>> The arrow FRP gets rid of the leak problem and maintains a single
> >>> definition
> >>> of integral by using a restricted form of recursion - the loop
> operator.
> >>> If you'd rather prefer having signals as first class objects, similar
> >>> technique
> >>> existed in synchronous languages [1], i.e., by using a special rec
> >>> primitive.
> >>>
> >>> Disclaimer: I was the co-author of the leak paper [2].
> >>>
> >>> [1] A co-iterative characterization of synchronous stream functions, P
> >>> Caspi, M Pouzet.
> >>> [2] Plugging a space leak with an arrow, H. Liu, P. Hudak
> >>>
> >>> --
> >>> Regards,
> >>> Paul Liu
> >>>
> >>> Yale Haskell Group
> >>> http://www.haskell.org/yale
> >>>
> >>> On 4/20/09, jean-christophe mincke <jeanchristophe.mincke at gmail.com>
> >>> wrote:
> >>> > In a post in the *Elerea, another FRP library *thread*,* Peter
> >>> Verswyvelen
> >>> > wrote:
> >>> >
> >>> > *>I think it would be nice if we could make a "reactive benchmark" or
> >>> > something: some tiny examples that capture the essence of reactive
> >>> systems,
> >>> > and a way to compare each solution's >pros and cons.* *
> >>> > *
> >>> > *>For example the "plugging a space leak with an arrow" papers
> reduces
> >>> the
> >>> > recursive signal problem to
> >>> > *
> >>> > *
> >>> > *
> >>> > *>e = integral 1 e*
> >>> > *
> >>> > *
> >>> > *>Maybe the Nlift problem is a good example for dynamic collections,
> >>> but I
> >>> > guess we'll need more examples.*
> >>> > *
> >>> > *
> >>> > *>The reason why I'm talking about examples and not semantics is
> >>> > because
> >>> the
> >>> > latter seems to be pretty hard to get right for FRP?*
> >>> >
> >>> > I would like to come back to this exemple. I am trying to write a
> >>> > small
> >>> FRP
> >>> > in F# (which is a strict language, a clone of Ocaml) and I also came
> >>> across
> >>> > space and/or time leak. But maybe not for the same reasons...
> >>> >
> >>> > Thinking about these problems and after some trials and errors, I
> came
> >>> to
> >>> > the following conclusions:
> >>> >
> >>> > I believe that writing the expression
> >>> >
> >>> > e = integral 1 *something*
> >>> >
> >>> > where e is a Behavior (thus depends on a continuous time).
> >>> >
> >>> > has really two different meanings.
> >>> >
> >>> > 1. if *something *is independent of e, what the above expression
> means
> >>> is
> >>> > the classical integration of a time dependent function between t0 and
> >>> t1.
> >>> > Several numerical methods are available to compute this integral and,
> >>> > as
> >>> far
> >>> > as I know, they need to compute *something *at t0, t1 and, possibly,
> >>> > at
> >>> > intermediate times. In this case, *something *can be a Behavior.
> >>> >
> >>> > 2. If *something *depends directly or indirectly of e then we are
> >>> > faced
> >>> with
> >>> > a first order differential equation of the form:
> >>> >
> >>> > de/dt = *something*(e,t)
> >>> >
> >>> > where de/dt is the time derivative of e and *something*(e,t)
> >>> indicates
> >>> > that *something* depends, without loss of generality, on both e and
> t.
> >>> >
> >>> > There exist specific methods to numerically solve differential
> >>> > equations
> >>> > between t0 and t1. Some of them only require the knowledge of e at t0
> >>> (the
> >>> > Euler method), some others needs to compute *something *from
> >>> intermediate
> >>> > times (in [t0, t1[ ) *and *estimates of e at those intermediary
> times.
> >>> >
> >>> > 3. *something *depends (only) on one or more events that, in turns,
> >>> > are
> >>> > computed from e. This case seems to be the same as the first one
> where
> >>> the
> >>> > integrand can be decomposed into a before-event integrand and an
> >>> after-event
> >>> > integrand (if any event has been triggered). Both integrands being
> >>> > independent from e. But I have not completely investigated this case
> >>> yet...
> >>> >
> >>> > Coming back to my FRP, which is based on residual behaviors, I use a
> >>> > specific solution for each case.
> >>> >
> >>> > Solution to case 1 causes no problem and is similar to what is done
> in
> >>> > classical FRP (Euler method, without recursively defined behaviors).
> >>> Once
> >>> > again as far as I know...
> >>> >
> >>> > The second case has two solutions:
> >>> > 1. the 'integrate' function is replaced by a function 'solve' which
> >>> > has
> >>> the
> >>> > following signature
> >>> >
> >>> > solve :: a -> (Behavior a -> Behavior a) -> Behavior a
> >>> >
> >>> > In fact, *something*(e,t) is represented by an integrand
> >>> > function
> >>> > from behavior to behavior, this function is called by the
> >>> > integration method. The integration method is then free to
> >>> pass
> >>> > estimates of e, as constant behaviors, to the integrand function.
> >>> >
> >>> > The drawbacks of this solution are:
> >>> > - To avoid space/time leaks, it cannot be done without side
> >>> effects
> >>> > (to be honest, I have not been able to find a solution without
> >>> > assignement). However these side effects are not visible from outside
> >>> > of
> >>> the
> >>> > solve function. ..
> >>> > - If behaviors are defined within the integrand function, they
> >>> > are
> >>> not
> >>> > accessible from outside of this integrand function.
> >>> >
> >>> > 2. Introduce constructions that looks like to signal functions.
> >>> >
> >>> > solve :: a -> SF a a -> Behavior a
> >>> >
> >>> > where a SF is able to react to events and may manage an internal
> >>> state.
> >>> > This solution solves the two above problems but make the FRP a bit
> >>> more
> >>> > complex.
> >>> >
> >>> >
> >>> > Today, I tend to prefer the first solution, but what is important, in
> >>> > my
> >>> > opinion, is to recognize the fact that
> >>> >
> >>> > e = integral 1 *something*
> >>> >
> >>> > really addresses two different problems (integration and solving of
> >>> > differential equations) and each problem should have their own
> >>> > solution.
> >>> >
> >>> > The consequences are :
> >>> >
> >>> > 1. There is no longer any need for my FRP to be able to define a
> >>> Behavior
> >>> > recursively. That is a good news for this is quite tricky in F#.
> >>> > Consequently, there is no need to introduce delays.
> >>> > 2. Higher order methods for solving of diff. equations can be used
> >>> (i.e.
> >>> > Runge-Kutta). That is also good news for this was one of my main
> >>> > goal
> >>> in
> >>> > doing the exercice of writing a FRP.
> >>> >
> >>> > Regards,
> >>> >
> >>> > J-C
> >>> >
> >>>
> >>
> >>
> >> _______________________________________________
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> >> Haskell-Cafe at haskell.org
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> >>
> >>
> >
>
>
> --
> Regards,
> Paul Liu
>
> Yale Haskell Group
> http://www.haskell.org/yale
>
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