Hallo, sorry for multiposting, but I am really looking

for some hint to solve my problem. And no, I don't

use Matlab, but maybe the matlab people have an idea

nevertheless.

I have to solve a nonlinear least square problem.

Let me tell you some background first. Imagine

you have a tool to process some work piece, say

polishing some piece of glas. The tool behaves

different on different locations of the piece,

and I can describe that behaviour. Now the tool

shall smooth the surface of the workpiece.

Next I have information about the piece before

handling it. What I have to find is optimal

time curve for the tool to obtain a perfectly

smooth surface.

How to formulate the problem?

Given a time vector (t_j) I have a function

g which calculates the remaining error (e_i)

(e_i) = g(t_j)

The rest error is given at, say, 100 points,

(t_j) is searched at 200 points.

My idea was to make the (t_j) a function of

some few parameters (t_j) = h(p_k), say 15

parameters. So the concatenated function

(e_i) = g(t_j) = g(h(p_k)) =: f(p_k) is to be minimized.

in the sense (e_i)-c -Min, where c is a constant,

the end level of the surface.

To solve this problem I use a "C" implementation

of the Levenberg-Marquardt algorithm as you can find

it in the LevMar Package (

www.ics.forth.gr/~lourakis/levmar/).

The function g contains the information about the

tool and about the initial surface. For the function

h I tried several approaches, making the time a

cubic spline of a selected times, or making it some

polynmial or...

Now what is my problem? With the above I do find

solutions, however a lot of solutions seem to

give very similar remaining errors. The only problem

is that the corresponding time vectors, which are

(t_j_optimal) = h(p_k_optimal) look very different

from optimal solution to optimal solution.

In particular the optimization algorithm often prefers

solutions where the time vector is heavily oscillating.

Now this is something I _must_ suppress, but I have no

idea how. The oscillation of the (t_j) depend of

the ansatz of h, of the number of parameters (p_k).

If f would be a linear function, then the matrix

representing it would be a band matrix with a lot

of diagonals nonzero. How many depends on the

ratio tool diameter to piece diameter.

Now what are my question: Is the problem properly

formulated? Can I expect to find non-oscillating

solutions? Is it normal that taking more parameters

(p_k) makes the thing worse? What else should I

consider? Is this more verbal description sufficient?

Thank you very much in advance.