From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: How to compute gradient of functional ???
Date: 24 Sep 1999 13:37:02 GMT
Newsgroups: sci.math.num-analysis
Keywords: numerical computation of surface minimizing a functional [65J15]

In article <37EA6154.7859DC18@pi.tu-berlin.de>,
 Andreas Wick <wick@pi.tu-berlin.de> writes:
|> Hello,
|> 
|> I have to solve the following problem:
|> 
|> 
|> Given the functional
|> 
|> F( x , y ) = \int  f( dx/ds , dx/dt , dy/ds , dy/dt )  dsdt
|> 
|> f ... is a fouth order polynomal with only one extremum (minimum)
|> 
|> x(s,t), y(s,t) ... are single-valued mappings (s,t) -> (x,y) whose
|> jacobian is nowhere zero in A = \int dsdt
|> 
|> dx/ds ... partial derivative of x with respect to s
|> 
|> 
|> and some initial guess
|> 
|> x^0 (s,t) , y^0 (s,t)     in  A
|> 
|> with
|> 
|> x(s,t) = x^0 (s,t)    \
|>                                |    on the boundary of A
|> y(s,t) = y^0 (s,t)    /     i.e. the mapping is known only on the
|> boundary
|> 
|> 
|> Find the mapping
|> 
|> x(s,t) , y(s,t) that minimizes F
|> 
|> 
|> First of all: By means of variation calculus I can get the EULER
|> equations of this problem. But they
|> are too difficult to solve numerically. So I have to get the solution by
^^^^^^^^^^^^^ sounds not very promising
|> minimizing the functional with
|> a conjugate gradient type algorithm.
snip
there is a controversy whether first discretize (the integral,
say by finite elements) - then minimize  ( by an appropriate minimizer ,
in your case , sy a newton, quasi - newton, preconditioned
conjugate gradient method (in some R^N, N>>1) or 
first minimize (formally) then discretize. 
you obviously want to go the second path. look up papers
of Sachs and Kelley which describe how to do  that 
788.65067
Kelley, C.T.; Sachs, E.W.
Pointwise Broyden methods. (English)
[J] SIAM J. Optim. 3, No.2, 423-441 (1993).
Pointwise quasi-Newton methods update the coefficients of 
differential and integral operators in function spaces. 
This paper gives a general theory
of such methods and unifies it with the theory of Broyden's method 
in Hilbert space. In particular, a new superlinearly convergent method is
introduced for elliptic boundary value problems.
[ S.Zlobec (Montreal) ]
Schlüsselwörter: superlinear convergence; 
pointwise quasi-Newton methods; Broyden's method; Hilbert space

Kelley, C.T.; Sachs, E.W.; Watson, B.
Pointwise quasi-Newton method for unconstrained optimal control problems. II. 
(English)
[J] J. Optimization Theory Appl. 71, No.3, 535-547 (1991).
The necessary optimality conditions for an unconstrained optimal 
control problem are used to derive a quasi-Newton method, where the update
involves only second-order derivative terms. 
A pointwise update which was presented in part I of this paper 
[the first and second author, Numer.
Math. 55, No. 2, 159-176 (1989; Zbl. 661.65068)] 
is changed to allow for more general second-order sufficiency conditions 
in the control problem.
In particular, pointwise versions of the Broyden, PSB, and SR1 
update are considered. A convergence rate theorem is given for the Broyden and
PSB versions.

625.65104 
Kelley, C.T.; Sachs, E.W.
A quasi-Newton method for elliptic boundary value problems. (English)
[J] SIAM J. Numer. Anal. 24, 516-531 (1987).
The authors develop a quasi-Newton method based on the differential 
equation. As a consequence, sparsity properties of the discrete
approximation are preserved in the iterates. An additional 
advantage of the method is that because it is specifically designed 
for differential
equations, the updating algorithm incorporates information about 
the problems at hand and therefore can outperform more general quasi-Newton
methods. A proof of superlinear convergence is included, and 
several computational examples are given.
[ G.Hedstrom ]
Schlüsselwörter: quasi-Newton method; updating algorithm; 
superlinear convergence; computational examples

hope this helps
peter