Transcript Numerical Optimization and applications (MA2600)

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
Part 1: three DFO problems
(i) 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
(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
(ii) Construction of an optical fiber with optimal properties
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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:
L = 0.5 mm
(reflectivity spectrum)
>
The index variation can be optimized in order to give the desired reflectivity
spectrum: inverse problem
Numerical Optimization and applications, ECP 2013
(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
(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
(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
(iv) Car shape optimization
Traînée aérodynamique
Résistance au roulement
Masse
at 20 km/h, oil consumption is due to :
Résistance au
roulement
26%
Traînée
aérodynamique
74%
among which, 65% to 70 %
depends on the exterior shape…
…among which 90
on the rear shape
% depends
• Goal: find the optimal rear shape of a car with respect to its drag coefficient
Numerical Optimization and applications, ECP 2013
(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
Part 2: two DFO algorithms
(i) Nelder Mead algorithm (1965)
(ii) Multi Direction Search method (1989)
Numerical Optimization and applications, ECP 2013