[SciPy-dev] SciPy improvements

Ondrej Certik ondrej@certik...
Thu Apr 12 18:31:35 CDT 2007


thank to all of you for your thorough response.

1) The DE code from Robert:


looks incredibly simple, I'll check it on my problem, if it behaves
better than my ugly one. But that's the thing - I didn't find any
mention about it in the documentation, otherwise I would be using your
code, because it's simpler (=better). Also I didn't find it in google.

I'll check the simplex method from SciPy - when I was using it, I just
needed a simple script, not a whole dependence on SciPy and it was
very difficult to get just the simplex algorithm out of SciPy.

2) I am sending my Broyden update methods in the attachement together
with tests (you need py.test to run them). I am using a test from some
article, I think from Vanderbilt. However my code is giving a little
different results. I'll polish it and send it as a patch to SciPy in
the trac, as directed, so you can just look how I do it. But when you
do it, you will see there is really nothing to it - it's very trivial.

However, my code doesn't use linesearch. Also I am curious how the
BFGS method is going to work when I implement it - it's very similar
to Broyden, except the update of the Jacobian is a little different.

Could you Michael please also rewrite your code to Python?


It would be nice to have all of it in SciPy with the same interface.

BTW - why are you using matlab at all? To me, the python with numpy is
better than anything else I've programmed in, including matlab.

3) about the logistics transforamtion  - I was sceptical too, until I
tried that and it was converging faster by a factor of 7x on my
problem (chemev). So for me it's enough justification, but of course I
am not saying that it must converge faster for any problem.

However, I implemented it as a wrapper function above all the
unconstrained algorithms with the SciPy interface, so the user is not
forced to use it - he can just try it and see, as I did. I'll post it
to the cookbook.

4) About the petsc - I know it's another dependence. However, I
noticed you are using umfpack in SciPy. So why not petsc? I think it
contains much more (sometimes better) solvers (depends on the
problem). It's seems logical to me, to either use nothing, or the best
library available, which I believe is petsc.

5)documentation: the front page is quite fine, however the
documentation needs complete redesign in my opinion. First - I believe
the numpy should be separated from SciPy and have it's own page
(numpy.org), but if you think it belongs under the hood of scipy.org,
then ok.

So, I'll copy the page:


into some new one, and redesign it as I would like it to be, and then
you'll tell me what you think about it. The same with other pages if
I'll get a better idea about them. This way I shouldn't spoil anything
in case you wouldn't like it. Because I don't have just couple of
small fixes.


On 4/12/07, Michael McNeil Forbes <mforbes@physics.ubc.ca> wrote:
> On 11 Apr 2007, at 1:56 PM, Matthieu Brucher wrote:
> > 2) Nonlinear solvers
> >
> > I have written these nonlinear solvers for the problem R(x) = 0, where
> > x and R has a dimension "n":
> >
> >     broyden1 - Broyden's first method - is a quasi-Newton-Raphson method
> for
> >         updating an approximate Jacobian and then inverting it
> >     broyden2 - Broyden's second method - the same as broyden1, but updates
> the
> >         inverse Jacobian directly
> >     broyden3 - Broyden's second method - the same as broyden2, but instead
> of
> >         directly computing the inverse Jacobian, it remembers how to
> construct
> >         it using vectors, and when computing inv(J)*F, it uses those
> vectors to
> >         compute this product, thus avoding the expensive NxN matrix
> >         multiplication.
> >     broyden_generalized - Generalized Broyden's method, the same as
> broyden2,
> >         but instead of approximating the full NxN Jacobian, it construct
> it at
> >         every iteration in a way that avoids the NxN matrix
> multiplication.
> >         This is not as precise as broyden3.
> >     anderson - extended Anderson method, the same as the
> broyden_generalized,
> >         but added w_0^2*I to before taking inversion to improve the
> stability
> >     anderson2 - the Anderson method, the same as anderson, but formulated
> >         differently
> >     linear_mixing
> >     exciting_mixing
> >
> > I use them in the self-consistent cycle of the Density Functional
> > Theory (so I use a terminology of DFT literature in the names of the
> > methods).
> Could the part that computes the step be separated from the function iself
> and the optimizer ? I'm trying to "modularize" non linear solvers so as to
> select more efficiently what is needed - kind of optimizer, kind of step,
> kind of stopping criterion, ... -
> Matthieu
> It should be possible to modularize these with a step class that maintains
> state
> (the Jacobian, or its inverse etc.).
> (Where is the latest version of your optimization proposal?  I have not had
> a chance
> to look at it yet, but have been meaning to and would like to look at the
> latest version.
> Perhaps we should make a page on the Wiki to collect suggestions and code
> samples.)
> I have been meaning to get a good Broyden based algorithm coded for python
> for a
> while.  I have a MATLAB version of a globally convergent Broyden
> implementation
> using a linesearch as a base with a couple of unusual features that might be
> useful (not specific to Broyden based methods).  (The code is based on the
> presentation of the algorithm given in Numerical Recipies with some
> modifications
> suggested by Dennis and Schnabel's book and is partially documented using
> noweb.)
> http://www.phys.washington.edu/users/mforbes/projects/broyden/
> 1) Variable tolerances.  The idea is to quickly estimate the starting
> Jacobian
>    with low tolerance calculations and then improve the tolerances as the
> code
>    converges to the solution.  This is useful if the function R(x) is
> computed
>    with numerical integration or some similar technique that is quick for
> low
>    tolerances but expensive for high tolerance functions.  (I have rarely
> seen
>    this technique in generic optimization algorithms, but found it
> invaluable for
>    several problems.)
> 2) Real-time bailout.  This allows you to compute for a specified length of
> time
>    and then return the best solution found within that time frame.  Most
> algorithms
>    simply count function calls.
> 3) Selective refinement of the Jacobian as the iteration proceeds.  This
> amounts
>    to monitoring the condition number of the Jacobian and recomputing parts
> of it
>    selectively if it becomes ill-conditioned.  (For this I update the QR
>    factorization of J rather than maintaining inv(J)).
> These things all slow down the fundamental algorithm, but are very useful
> when the
> function R(x) is extremely expensive to evaluate.
> Michael.
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