From: Russell Martin
Subject: Re: deconvolution (Fredholm integral equation)
Date: Tue, 20 Apr 1999 00:07:18 GMT
Newsgroups: sci.math.num-analysis
Keywords: Numerical solution of Fredholm integral equations first kind
Louis M. Pecora wrote:
>
> In article <371B14E5.41A7@sci.muni.cz>, Antonin Brablec
> wrote:
>
> > Hi,
> > we try to solve the deconvolution problem, namely to solve
> > the Fredholm integral equation first kind
> >
> > h(x) = \int_{-\infty}^{\infty} f(x)g(x-t) dt (LaTEX notation).
> >
> > We have measured data x_i, h_i = h(x_i) which are weighted by noise,
> > and we would like to compute the function f(x), g is given apparatus
> > function. This is ill conditioned inversion problem. Could anybody
> > send us the pointers or references to programs for solution of this
> > problem ?
>
> An old one, but a good one is,
>
> V.F. Turchin, V.P. Kozlov, and M.S. Malkevich, "The Use of
> Mathematical-Statistics Methods in the Solution of Incorrectly Posed
> Problems," Soviet Physics USPEKHI 13 (6), 681 (1971).
>
> I'm sure a search on ill-conditioned problems would turn up lots of more
> recent books.
>
Slightly newer and perhaps more accessible:
_Introduction to the Mathematics of Inversion in Remote Sensing and
Indirect Measurements_ by S. Twomey (original copyright 1977 by Elsevier
Scientific Publishing Co., now available from Dover Publications for
about $10 U.S.). This kind of thing has been studied extensively
for the purposes of, among other things, deriving temperatures from
satellite data for use in weather forecasting models. Dr. Susskind
and his colleagues at NASA's Goddard Space Flight Center has done
a lot of work in this area, as have many others I'm not familiar with.
Good luck,
Russell Martin
--
Russell Martin R. L. Martin and Associates, Consultants
russell.martin@mail.wdn.com in Science and Technology
http://www.rmartin.com
->Insert pithy quotation here.<-
==============================================================================
From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: deconvolution (Fredholm integral equation)
Date: 20 Apr 1999 15:01:05 GMT
Newsgroups: sci.math.num-analysis
In article <371B14E5.41A7@sci.muni.cz>,
Antonin Brablec writes:
|> Hi,
|> we try to solve the deconvolution problem, namely to solve
|> the Fredholm integral equation first kind
|>
|> h(x) = \int_{-\infty}^{\infty} f(x)g(x-t) dt (LaTEX notation).
|>
|> We have measured data x_i, h_i = h(x_i) which are weighted by noise,
|> and we would like to compute the function f(x), g is given apparatus
|> function. This is ill conditioned inversion problem. Could anybody
|> send us the pointers or references to programs ...
snip
unfortunately, neither netlib nor HotGAMS contain any pointers to software
for fredholm equations of the first kind.
but some sources of text may help you:
Christophr T.H. Baker: the numerical treatment of integral equations,
Clarendon Press 1977
Johann Baumeister: stable solution of inverse problems.
Vieweg & Sons ISBN 3-528-08961-X.
see also papers in
W. Engl and C.W.Groetsch (eds): Inverse and illposed problems,
Acad Press 1987.
Maybe some of the specialists in the field, which of course have their software
running, is willing to help you.
E.g. Heinz Engl from Linz University (who also wrote recently a nice book
on the subject, but unfortunately in German) could help you:
His email is
engl@indmath.uni-linz.ac.at
hope this helps
peter
==============================================================================
From: borchers@nmt.edu (Brian Borchers)
Subject: Re: deconvolution (Fredholm integral equation)
Date: 20 Apr 1999 16:04:28 GMT
Newsgroups: sci.math.num-analysis
One Fortran package for Fredholm Integral Equations of the first kind is
Stephen Provencher's CONTIN (http://www.provencher.de/contin.html)
If you're using MATLAB, you should look at Per Hansen's regularization
toolbox for MATLAB (http://www.netlib.org/numeralgo/na4)
--
Brian Borchers borchers@nmt.edu
Department of Mathematics http://www.nmt.edu/~borchers/
New Mexico Tech Phone: 505-835-5813
Socorro, NM 87801 FAX: 505-835-5366