forwarded message from Dherminder Kainth

forwarded message from Dherminder Kainth

[Brian Gough]

To : bjg at network-theory dot co dot uk
Subject : Re: quasi-random numbers in high dimensions
From : "Dherminder Kainth" <dherminder_kainth at hotmail dot com>
Date : Thu, 11 Oct 2001 13:11:13 +0000


Dear Brian,

It is relatively straightforward to extend the Sobol generator (for example) 
to higher dimensions by altering the dimension etc to beyond 40 and putting 
in more irreducible polynomials (see Numerical Recipes for further 
discussion). The polynomials can - if you are keen - be computed using eg 
Mathematica; alternatively go to the NR web page and look for contributed 
code - in particular a file by Peter Jaeckel. Here there is a list of all 
the polynomials - one simply has to transcribe them to the code.

Alternatively there is the SPRNG web side and a number of fortran 
implementations (use f2c to get a quick and dirty c translation).

A point of warning - quasi randoms are NOT good for integrating to dimension 
400 (basically to do with the Kloskwa Hlawka) bound.
Instead you must

a. Hope the "effective dimension" of the problem is < 400 ~20 is good (!). 
By that I mean most of the weight of the integral is in 20 generalised 
dimensions.

b. Then use Brownian bridging/ pseudospectral methods. This will generally 
work for most problems. For a further discussion of these points see R E 
Caflisch (on the web).



>From: Brian Gough <bjg@network-theory.co.uk>
>To: <vcrespi@ists.dartmouth.edu>
>CC: gsl-discuss@sources.redhat.com
>Subject: Re: quasi-random numbers in high dimensions
>Date: Thu, 11 Oct 2001 12:28:07 +0100 (BST)
>
>crespi@ists.dartmouth.edu writes:
>  > Dear Sir,
>  >
>  > 	I am a user of the gsl library and in particular I'm trying to
>  > use the quasi-random numbers generators. Unfortunately the highest
>  > dimension is 40 whereas I would need dimensions of about 400.
>  >
>  > Please, could you tell me whether you know of some specific free C code
>  > that I can use?
>  >
>  > Thank you in advance for your suggestions.
>  >
>  > Best Regards,
>  > Valentino Crespi
>
>I'm not familiar with that subject myself so I'm forwarding your
>message to the gsl-discuss list to see if the original author or
>anyone else has any suggestions.
>
>Brian Gough


_________________________________________________________________
Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp

Re: forwarded message from Dherminder Kainth

[Gerard Jungman]

Brian Gough wrote:
> 
> It is relatively straightforward to extend the Sobol generator (for example)
> to higher dimensions by altering the dimension etc to beyond 40 and putting
> in more irreducible polynomials

Hi. Yes, I just didn't have the patience to deal with it.
The way I implemented it was trivial, with the polynomials
just sitting in a static table. With that approach there
will always be somebody who wanted N+1.

The right thing to do is to create a dynamic facility,
so the user can generate tables for dimensions as high
as desired. I don't have the time to do that, and it
probably won't get done. If somebody wanted to contribute
something, that would be fine. But however it is done, it
should not be allowed to affect the performance of the
low dimension cases.

> A point of warning - quasi randoms are NOT good for integrating to dimension
> 400 (basically to do with the Kloskwa Hlawka) bound.
> Instead you must

Yes, exactly. I didn't see any point in higher dimension tables
for that reason, but I understand your point about the possibility
that the effective dimension is much lower. I didn't think about
that at the time.


Thanks.

--
G. Jungman