Numerical Optimization and Applications - Lecture 2: Derivative Free Optimization (DFO)

1. Numerical Optimization and applications (MA2600) Lecture 2: Derivative Free Optimization (DFO) Laurent Dumas Laboratoire de Mathématiques de Versailles, Université de Versailles Saint Quentin en Yvelines http://dumas.perso.math.cnrs.fr/ecp2012.html Numerical Optimization and applications, ECP 2012

2. Part 1: three DFO problems (i) Construction of an optical fiber with optimal properties (ii) Debluring and denoising of a barcode image (iii) Car shape optimization Numerical Optimization and applications, ECP 2012

3. L = 0.5 mm (i) 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 2012

4. (i) 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 2012

5. (i) 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 2012

6. (ii) Debluring and denoising of a barcode image Code à 13 chiffres • Objectif: à partir d’une image floue et bruitée d’un code barre, être capable d’identifier ce code barre Numerical Optimization and applications, ECP 2012

7. Lignes de courant et tourbillons longitudinaux à l’arrière d’un véhicule expérimental type 206 (DRIA) …dont 65% à 70 % dépend de la forme extérieure… …dont 90 % de la forme arrière! (iii) Car shape optimization à 120 km/h, facteurs de la consommation d’un véhicule: • Objectif: obtenir la forme arrière optimale d’une automobile par simulation numérique. Numerical Optimization and applications, ECP 2012

8. (iii) 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 2012

9. Part 2: two DFO algorithms (i) Nelder Mead algorithm (1965) (ii) Multi Direction Search method (1989) Numerical Optimization and applications, ECP 2012