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**Numerical Optimization and applications (MA2600)**

Lecture 1: Derivative Free Optimization (DFO) Laurent Dumas & Zaid Dauhoo Laboratoire de Mathématiques de Versailles, Université de Versailles Saint Quentin en Yvelines Numerical Optimization and applications, ECP 2013

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**Part 1: three DFO problems**

Minimal molecular energy (ii) Construction of an optical fiber with optimal properties (iii) Debluring and denoising of a barcode image (iv) Car shape optimization Numerical Optimization and applications, ECP 2013

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**(i) Minimal molecular energy**

N=4 atoms N=7 atoms Goal: find the position of N atoms minimizing the Lennard Jones potential of the associated molecular: V( r )=1/r12 – 2/r 6 for 2 atoms at a distance r. Numerical Optimization and applications, ECP 2013

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**(ii) Construction of an optical fiber with optimal properties**

Such filters can be obtained by using an optical fiber called FBG (Fiber Bragg Grating) having a fast periodic modulation of its refractive index in the core: The index variation can be optimized in order to give the desired reflectivity spectrum: inverse problem L = 0.5 mm (reflectivity spectrum) Numerical Optimization and applications, ECP 2013

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**(ii) Construction of an optical fiber with optimal properties**

The refractive index of a FBG is expressed through a quasi-sinusoïdal function in the longitudinal direction z: n(z)=n0+dn(z) cos(2pz/L0) z [0, L] with the following notations: n0 : index refraction of the core L0: nominal period of the FBG dn(z): slowly varying amplitude (also called apodisation) The inverse-type optimization problem will consist in finding the ‘best’ apodisation function leading to the desired reflectivity spectrum. Numerical Optimization and applications, ECP 2013

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**(ii) Construction of an optical fiber with optimal properties**

The reflectivity spectrum is a function l R(l) =| r(l) |2 where r(l) = bB(0,l) / bF(0,l) In the above expression, the enveloppes of the forward and backward propagating waves are obtained by the resolution of the following system of coupled ODE’s: where , and Numerical Optimization and applications, ECP 2013

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**(iii) Debluring and denoising of a barcode image**

Code à 13 chiffres Goal: identify a barcode from a blurred barcode image Numerical Optimization and applications, ECP 2013

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**(iv) Car shape optimization**

at 20 km/h, oil consumption is due to : among which, 65% to 70 % depends on the exterior shape… …among which 90 % depends on the rear shape Goal: find the optimal rear shape of a car with respect to its drag coefficient Numerical Optimization and applications, ECP 2013

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**(iv) Car shape optimization**

Ford T: (1908) Hummer H2: (2003) Citroën SM: (1970) Peugeot 407: (2004) and… Tatra T77: (1935) Numerical Optimization and applications, ECP 2013

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**Part 2: two DFO algorithms**

(i) Nelder Mead algorithm (1965) (ii) Multi Direction Search method (1989) Numerical Optimization and applications, ECP 2013

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Principle of operation. Applied on typical single mode fibers for telecommunication applications Core diameter ~10μ and Δn/n <0.4% (where Δn = n2-n1)

Principle of operation. Applied on typical single mode fibers for telecommunication applications Core diameter ~10μ and Δn/n <0.4% (where Δn = n2-n1)

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