Dear David,

Here is a reference realised by the researcher A. H. ISIK, I think that you can inspired from this work. I seal you in attached files in subject.

Best regard

thank you Mohamed,  I did notice the first paper a couple of years ago and in fact as I recall I contacted the author wondering if he was interested in my problem.  nothing came of it.  I have today tried to see how one might be able to use ANN in my above problem. 

as I understand from isik a certain amount of data must be taken presumably lens voltages vs spotsize in my situation and then ANN will find the minimum voltages giving the minimum spotsize, after training.  what training consists of I do not know.

my problem is locating the region of parameter space that might be hopeful in simplex leading to a minimum less than the previous mins. -- there are many!!

I must remark that I have no experience with ANN but I am not particularly hopeful in a multi parameter system that has many local minimums.

are you interested in this problem or the more general type that I suggested?

much thanks for your kind response,

david

hi Mohamed again.  I would like to come back to what I feel is an analogous problem, namely finding the tallest tree in a forest.  let us suppose we try to use ANN to help. so I must create a data set, and what would this data set be? the height of the tree and its location for all the trees in the forest?  but then I am done and the problem is solved for I have the tallest tree in the forest.  if it is in fact only the height on a grid, I might miss out on the tree that is the tallest.

although I admit I don't understand what isik is doing I do not see how that approach will help.

kind regards,

davide

Dear David,

I honestly think the problem not multi-parameter linear function optimization can be solved effectively using the simplex algorithm or at least not lead to satisfactory results. that is why the path traced by A. H. ISIK in my opinion encouraging.

I Conceil you one hand to boost demand for contact with ISIK can be through the Turkish Tubitek agency. On the other hand it will be necessary for you to swim in the ANN field and not the training part that discourage you because the literature in this area is in abundance. Good luck.

Best regards

hi Mohamed, the  problem is multi parameter and non linear, and I agree not solvable by simplex. I only use simplex to find a local minimum from the parameter set I am testing.

in the tree analogy we have to from being near the base sense whether the tree is tall.  in the case of trees you might suspect looking at the diameter might be a clue. in my lens design problem I am looking for clues. 

I enjoy swimming Mohamed but not when there does seem to be any reason to swim! from my comments above ANN does not look promising.

thank you for your regards,

I wish you well,

david

When at the base, there is no way to know if it is worthwhile to climb.  I suggest you try a so called surface-response methodology.  Moreover, i think your problem does not depend on the voltages but perhaps on their ratios.

Hope this helps. 

Maybe I am too old, or old-fashioned , but first of all, I would try to model the problem at hand  as a mathematical, nonlinear or linear optimization problem, and then I would like to  find an appropriate algorithm to solve the problem. In the discussion so far I do not see any attempt to come up with a mathematical model, that is meant to model the problem at hand! Nothing replaces the modeling phase, and without an appropriate amount of operations research training many are swimming in the dark!

I fully agree with you. However, my understanding of the problem is that the electrostatic potential in the multi-element is obtained by solving numerically an elliptic PDE and the trajectories are subsequently obtained by solving numerically a second-order ODE. Hence, finding the voltage setup for a given spotsize is an inverse problem that cannot be expressed in closed form. Hence, my suggestion to use a surface-response methodology as an attempt to build an approximate mathematical model.

let me try to comment. to h.e. my differential equation is laplaces equation in cylindrical coordinates which as you suggest is I believe an elliptic pde.  for the trajectories the space is 2d and the equations are:

(d^2 r(t))/(dt^2 )+beta*∂v(r(t),z(t))/∂r=0                     (4)

(d^2 z(t))/(dt^2 )+beta*∂v(r(t),z(t))/∂z=0

where v(r,z) is the potential at the point r,z.

the solution has lots of local min.

I have looked up surface-response methodology. from what I see must select a number of points (a selection of lens voltages) for each of these points evaluate the spot size.

after that i would be able to for each of these points us a  simplex algorithm using a small incremental change in each of the voltages.  this will give me a point close to the initial point and i will have two spot sizes.  it is a little unclear how i should  proceed using this information.

thank you both for your suggestions.

david

>>> I have looked up surface-response methodology. from what I see must select a number of points (a selection of lens voltages) for each of these points evaluate the spot size.

Yes, this is the initial step of SR.

Your objective function F (here, the spot size) depends on a set of parameters V (here, the voltages, more precisely their ratios). As  F is difficult to compute, the initial step of SR consists in using a set of values of F obtained for some selected values of V to build an approximate model M for the landscape of F in the parameter space. After that, this model will be used as a surrogate objective function M that will replace F to simplify and speedup the computations.

The next step consists in using your optimizer (here, the Nealder-Mead simplex method) to surf on the surface M to find the (local) optimum of M. In general, this optimum will not coincide with an optimum of F but will give you a worthwhile V which will allow you to get an additional value of F at this new point. Use this point to replace the worst one among the initial set. Update your model to get a new surrogate M. Find the new local optimum. etc.

Hope this helps

The last answer is a beauty in clarity!

thank you.  it is clearer than it has been.  I would only suggest that the spot sizes at the points themselves would be quite large that could be refined at the points themselves by finding the local min near the point.  one would then have the points for which a function could be fitted and proceed as you have suggested. 

again many thanks for your help.

david

thank you Isaac. unfortunately my lack of mathematical background will not enable me to go further.  but thank you for your suggestions.

kind regards,

david

Dear David,

If you do have a known set of inputs and desired outputs, you can train an Artificial Neural Network

do see my response to mohamid lafi.  I have let us say 6 inputs (the lens voltages) and one output (the spotsize)  I can get any number of points.  I do not see as I said previously how ann could help.

david

Sorry I missed out on your conversation thread with Mr Lafifi. From what I understand of the problem, you need to vary the input voltages, create a data set of 15 to 20 points, and then measure the output in terms of some discrete parameters to represent spot size, depending upon geometry of spot. This will train the ANN. Then vary the inputs over their complete range using a computer program and store the output predicted by ANN for each combination. From this stored data you can pick out which combination yielded maximum spot size. ANN are very efficient in predicting relationship of functions which are difficult to formulate analytically.

If you are interested in this problem I can supply you with 15 or 20 points (in the hyperspace) and then give you the spot sizes resulting from these points.

I could also supply you with any additional points you might wish.

it would be a collaborative effort, if you are interested. 

my present problem has either 6 or 10 independent lens voltages.

do let me know if you are interested.

you should also see my cma papers that I am attaching.

kind regards,

david