Re: [Haskell-cafe] FRP, integration and differential equations.

23 Apr 2009


      Just my two cents. The open source project Maxima is a very successful
math engine dedicated to solving ODE PDE and integration among many
other things. It is implemented in LISP.

Steve


On 4/21/09, jean-christophe mincke  wrote:
...
Peter, Paul,
...
But my question is, is FRP really the right setting in which to
explore a highly accurate ODE solver?
...
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?
I believe it is a good question. My answer is why not. In fact when I
discovered FRP I liked the way systems could be constructed from building
blocks as Peter says. I have a previous experience in designing simulators
for complex  dynamic systems and I often had to find solutions to build
simulators from a smal set of building blocks and still keep an adequate
accuracy.
Doing so often leads to ODEs of the form:
de/dt = f (de/dt, e, t)
It is often possible to solve such ODE providing that we make some
reasonable physical assumptions on the de/dt term appearing in f(). This
comes with an acceptable decrease in accuracy. But modelling physical
systems is always a matter of trade-off...
Thus, a big physic black box containing large monolithic set of equations is
not always needed.
Regards
J-C
On Tue, Apr 21, 2009 at 1:32 PM, Peter Verswyvelen wrote:
...
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  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  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@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  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 
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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--
Regards,
Paul Liu
Yale Haskell Group
http://www.haskell.org/yale
-- 
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Re: [Haskell-cafe] FRP, integration and differential equations.

Steve Lihn