Laurent Dumas & Zaid Dauhoo Laboratoire de Mathématiques de Versailles,

1. 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 http://dumas.perso.math.cnrs.fr/ecp2013.html Numerical Optimization and applications, ECP 2013

2. 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

3. (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

4. L = 0.5 mm (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 (reflectivity spectrum) Numerical Optimization and applications, ECP 2013

5. (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

6. (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

7. (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

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

9. (iv) Car shape optimization Ford T: 0.8 (1908) Hummer H2: 0.57 (2003) Citroën SM: 0.33 (1970) Peugeot 407: 0.29 (2004) and… Tatra T77: 0.212 (1935) Numerical Optimization and applications, ECP 2013

10. Part 2: two DFO algorithms (i) Nelder Mead algorithm (1965) (ii) Multi Direction Search method (1989) Numerical Optimization and applications, ECP 2013