From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: interpolation & integration?
Date: 5 Oct 2000 10:15:45 GMT
Newsgroups: sci.math.num-analysis
Summary: [missing]

In article <qgog104e8h.fsf@umpa05.gwdg.de>,
 Joerg Behrens <jbehren@gwdg.de> writes:
|> Hello!
|> 
|> Given a smooth 3dim-function f(x,y,z) that is very expensive to
|> evaluate but has a cheap relation to its partial 1. and 2. derivatives,
|> i am interested in an effective method to calculate the integral over
|> a tetrahedron.
|> 
|> In the case that there is no effective use of derivatives in quadrature
|> schemes:
|> 
|> Is it reasonable to separate the problem into two independent parts:
|> 1. g = interpolation of f  &  2. integration of g?
|> 
|> This would make the integration part very fast and the whole problem lies
|> in the first step. Assuming that in this case the quadrature can be made
|> exact, the question is how well this scheme converges compared to direct
|> quadrature without derivatives.
|> 
most of the classical quadrature schemes are based on integrating an interpolating
polynomial exactly.there are of course interpolation schemes using partial 
derivatives. e.g. you could use interpolation by a quintic, using first and 
second order partial derivatives at the nodes and normal derivatives at the faces.
the integration error is the integral of the interpolation error , hence the total
error over all elements is of the same order as the interpolation error. 
a look at standard textbooks about multidimensional integration 
(krommer-ueberhuber; engels; stroud ) indeed does not reveal any formulas 
involving partial derivatives of the integrand.
there are some new papers: 

897.65012
Lyness, J.N.; Ruede, U.
Cubature of integrands containing derivatives. (English)
[J] Numer. Math. 78, No.3, 439-461 (1998). [ISSN 0029-599X] 

571.65016 
Wang, Chengshu
Calculation of numerical integration of multiple dimensions by dissection 
into simplices. (English)
[J] J. Comput. Math. 2, 239-246 (1984).
The author describes a time-honored method for integrating over an 
n-dimensional polytope by dissecting it into disjoint n-dimensional simplices 
and using
the same quadrature rule on each. The rule he uses is based on the 
Taylor expansion and involves only the first two or three partial 
derivatives at the
centroid of the simplex. The theory for both aspects 
(dissection and integration) is taken from standard sources in the 
western literature. The author's
contribution seems to be two numerical examples, over polytopes of 4 and 8 
dimensions respectively of the function $\vert x\vert\sp 2$, for which 
his rule is
exact. He notes how much more efficient it is to use a quadrature rule 
which should give the exact result, than to use Monte Carlo integration.
[ J.N.Lyness ]

hence your idea to use the cheap partial derivatives for forming a high order
interpolation process and then to integrate exactly might be a good one.

hope that helps
peter