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# Nonlinear least square problem

 P: n/a 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. Apr 3 '08 #1 