P. Pegon & Ph. on & Ph. CaCapéran - CEA/CEAClub Cast3M, 21th of November 2008 An amazing...
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Club Cast3M, 21th of November 2008Club Cast3M, 21th of November 2008
An amazing optimisation problemAn amazing optimisation problemAn amazing optimisation problemAn amazing optimisation problem
P. Pegon & Ph. P. Pegon & Ph. CapéranCapérangg ppEuropean Laboratory for Structural AssessmentEuropean Laboratory for Structural Assessment
Joint Research CentreJoint Research CentreIspraIspra, Italy, Italy
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An amazing An amazing optimisationoptimisation problemproblem
OUTLINE:
• Background: some results from photogrammetry
Th (VERY) i lifi d ti i ti bl• The (VERY) simplified optimisation problem
• EXCE, steepest descent, conjugate gradient & BFGS
• The Gauss-Newton method (+ new operator GANE)The Gauss Newton method (+ new operator GANE)
• Conclusions, further works & bibliography
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An amazing An amazing optimisationoptimisation problemproblem
OUTLINE:
• Background: some results from photogrammetry
Th (VERY) i lifi d ti i ti bl• The (VERY) simplified optimisation problem
• EXCE, steepest descent, conjugate gradient & BFGS
• The Gauss-Newton method (+ new operator GANE)The Gauss Newton method (+ new operator GANE)
• Conclusions, further works & bibliography
![Page 4: P. Pegon & Ph. on & Ph. CaCapéran - CEA/CEAClub Cast3M, 21th of November 2008 An amazing optimisation problemAn amazing optimisation problem P. Pegon & Ph. on & Ph. CaCapéran European](https://reader031.fdocuments.in/reader031/viewer/2022041904/5e6231ff994574257f2cb7e6/html5/thumbnails/4.jpg)
Background: some results from Background: some results from photogrammetryphotogrammetry
The wall The cameraThe structureESECMASE
20cmm
4096 grey levels, S/N 64 dB, 1536x1024 pixels
Kodak CCD KAF 1602EKodak CCD KAF 1602E
+-1.3mm/pixel
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Current tiI iti l t
Background: some results from Background: some results from photogrammetryphotogrammetry
Current state
i
Set of M interpolation
Reference stateInitial t0
Set of M interpolation functions on M reference
sub-windowsSet of M current sub-windows
under tracking
Given one reference, C3 interpolation function,
( )0W X
Consider its corresponding current discrete function
( )iw x
Optimisation on the Quadratic Error:
( ) ( )( )( )2
∑C3 Analytic Template { } { }1.. ... ..i i
n nP P +
( )With respect to { }nP
( ) ( )( )( )2
01..
( ) ) ;k kk K
E w x W T x=
= −∑P P( )0 ( ; )kW T x P
Tracking MethodCoordinate Transformation
parametrised by jdijd( );T X P
{ }nP
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Background: some results from Background: some results from photogrammetryphotogrammetry
Comparison with Heidenhain measurement (K14 - 0.16g)
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Background: some results from Background: some results from photogrammetryphotogrammetry
Displacement on targets
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Background: some results from Background: some results from photogrammetryphotogrammetry
Displacement on texture
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Background: some results from Background: some results from photogrammetryphotogrammetry
Displacement fields (K12 - 0.12g)
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An amazing An amazing optimisationoptimisation problemproblem
OUTLINE:
• Background: some results from photogrammetry
Th (VERY) i lifi d ti i ti bl• The (VERY) simplified optimisation problem
• EXCE, steepest descent, conjugate gradient & BFGS
• The Gauss-Newton method (+ new operator GANE)The Gauss Newton method (+ new operator GANE)
• Conclusions, further works & bibliography
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The (VERY) simplified The (VERY) simplified optimisationoptimisation problemproblem
In the elastic case…
65 bricks , 1,...,65 0.3iE i ν= =
Joints 66 66
66 ,2 (1 )n s
E EE K K
e e ν→ = =
+
Identification of ??? E
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The (VERY) simplified The (VERY) simplified optimisationoptimisation problemproblem
d is imposed
d is measured
1+
=EThe reference solution is obtained with and( )+ +d E ( ) 5%σ + =E1EThe reference solution is obtained with and ( )d E ( ) 5%σ E
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The (VERY) simplified The (VERY) simplified optimisationoptimisation problemproblem
{ }* argmin ( )F= EE EFind
( ) ( ) ( )T( )F = =E f E f E f E( ) ( ) ( )( )( )
( ) ( )+ += −f E d E d EWhere
(first) problem !!! ( ) ( )λ=d E d E
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The (VERY) simplified The (VERY) simplified optimisationoptimisation problemproblem
Numerical derivation of the gradients
( ) ( )( ) F E E E E F E E E EF δ δ+∂ Ε 1 1( ,..., ,..., ) ( ,..., ,..., )( )2
i n i n
i
F E E E E F E E E EFE E
δ δδ
+ − −∂ Ε ≈∂
And also (later)
1 1( ,..., ,..., ) ( ,..., ,..., )( ) i n i nE E E E E E E Eδ δ+ − −∂ Ε f ff 1 1( , , , , ) ( , , , , )( )2
i n i n
iE Eδ≈
∂
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An amazing An amazing optimisationoptimisation problemproblem
OUTLINE:
• Background: some results from photogrammetry
Th (VERY) i lifi d ti i ti bl• The (VERY) simplified optimisation problem
• EXCE, steepest descent, conjugate gradient & BFGS
• The Gauss-Newton method (+ new operator GANE)The Gauss Newton method (+ new operator GANE)
• Conclusions, further works & bibliography
![Page 16: P. Pegon & Ph. on & Ph. CaCapéran - CEA/CEAClub Cast3M, 21th of November 2008 An amazing optimisation problemAn amazing optimisation problem P. Pegon & Ph. on & Ph. CaCapéran European](https://reader031.fdocuments.in/reader031/viewer/2022041904/5e6231ff994574257f2cb7e6/html5/thumbnails/16.jpg)
EXCE, EXCE, steepeststeepest descentdescent, , conjugateconjugate gradient & BFGSgradient & BFGS
EXCE
The EXCELL operator computes the minimum of a function F(Xi),the usemethod is the well-known MMA (Method of Moving Asymptotes) proposed by K.Svanberg. The purpose is to find the minimum of a function F(Xi) with i=1,N given that :
- the relations Cj(Xi) < Cjmax (j>0 and j=1,M) must be satisfied
- there are relations on each unknown Ximin < Xi < Ximax
The functions F and Cj are defined by the function values and by their derivatives at the starting point X0.
( ) , ( )i j iF E C E ?????
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EXCE, EXCE, steepeststeepest descentdescent, , conjugateconjugate gradient & BFGSgradient & BFGS
( ) ( ) ( )F + += −E d E d E
EXCE
1st case 1
( ) ( ) ( )
( ) ( ) ( ) 0
( ) 0 ( ) 0
C
C C
σ σ +
+ +
= − <
= − < = − <
E E E
E E E E E E2 3( ) 0 , ( ) 0C C= < = <E E E E E E
( ) ( ) ( )F + += −E d E d E2nd case
21 2
( ) ( ) ( )
( ) ( ) ( ) 0 , ( ) ( ) ( /100) 0
F
C C orσ σ+ ++= − < = − <
E d E d E
E E E E E E E
3rd case … but no4
( ) ( )
( ) ( ) ( ) / ( ) 1 0 10
F
C
σ+ + + + −
=E E
E d E d E d E convincing results41( ) ( ) ( ) / ( ) 1.0 10C + + + + −= − < ×E d E d E d E
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EXCE, EXCE, steepeststeepest descentdescent, , conjugateconjugate gradient & BFGSgradient & BFGS
Descent method: general format
D t di ti ithiE 0i T F∇ <EDescent direction with
Line search Robust line search???
idE 0id F∇ <
EE
{ }0argmin ( )i i idFα α= +E ELine search Robust line search???
and loop on until convergence
{ }0argmin ( )dFαα α> +E E
i1i i idα+ = +E E E
Meaning of convergence???
Here we concentrate on !!!!!!!iEHere we concentrate on !!!!!!! dE
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EXCE, EXCE, steepeststeepest descentdescent, , conjugateconjugate gradient & BFGSgradient & BFGS
Steepest descent iist F= −∇
EE
( ) ( ) / ( )+ + + +−d E d E d E
i
The convergenceThe convergence can be very slow!s
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EXCE, EXCE, steepeststeepest descentdescent, , conjugateconjugate gradient & BFGSgradient & BFGS
Conjugate gradient : the descent directions are mutually orthogonal
(with precise line search)1
121
( )i
i i T ii i i i i igc gc
iFγ γ
−−
−
−= + = −∇ =E
g g gE g E ggg
( ) ( ) / ( )+ + + +−d E d E d E
i
Surprisingly slow convergences
i
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EXCE, EXCE, steepeststeepest descentdescent, , conjugateconjugate gradient & BFGSgradient & BFGS
BFGS: the Hessian matrix is iteratively constructed
1 ...ii i i ibfgs bfgs bfgs bfgsF −= − ∇ = +
EE H H H
( ) ( ) / ( )+ + + +−d E d E d E
i
BFGS=Broyden-Fletcher-Goldfarb-ShannoSurprisingly slow
convergences
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EXCE, EXCE, steepeststeepest descentdescent, , conjugateconjugate gradient & BFGSgradient & BFGS
Summary:
• Very slow convergence
• “High” level asymptotic convergence plateau
Conclusion: we need to do better!Conclusion: we need to do better!
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An amazing An amazing optimisationoptimisation problemproblem
OUTLINE:
• Background: some results from photogrammetry
Th (VERY) i lifi d ti i ti bl• The (VERY) simplified optimisation problem
• EXCE, steepest descent, conjugate gradient & BFGS
• The Gauss-Newton method (+ new operator GANE)The Gauss Newton method (+ new operator GANE)
• Conclusions, further works & bibliography
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The GaussThe Gauss--Newton method (+ new operator GANE)Newton method (+ new operator GANE)
Write the problem as a least square problem…
{ }* argmin ( )F= xE EFind
( )( ) ( ) ( ) ( )2 2 T1 1 1( )2 2 2
m
iF f= = =∑E E f E f E f E( )( ) ( ) ( ) ( )
( )12 2 2
( ) ( )i=
+ += −
∑f E d E d E
Where
… and use it!
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The GaussThe Gauss--Newton method (+ new operator GANE)Newton method (+ new operator GANE)
The Gauss-Newton MethodSame cost as
computing s/ jF E∂ ∂
Using with( ) ( ) ( )δ δ+ +f E E f E J E E ( ) ( )i
ijj
fE
∂⎡ ⎤ =⎣ ⎦ ∂J E E
is given by
j
{ }argmin ( )gn Fδ δ= +EE E E ( )T Tgn F= − = −∇J J E J f
Use as a descent directiongnE
Warning: no indetermination if the columns of are linearly independent
Jindependent
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The GaussThe Gauss--Newton method (+ new operator GANE)Newton method (+ new operator GANE)
Operator GA(us-)NE(wtonTAB2=GANE TAB1;
CHPO1 RIGI1=GANE TAB1 'MATR';;
Description:____________
The function to minimize is 2F(X)=f(X).f(X), and J(X)=df/dX
The descent direction is H solution of:transpose(J)J H = -transpose(J)f
Contents:_________
TAB1 : Table of type 'VECTEUR' containingTAB1 . 0 = f(X)TAB1 . 1 = df/dX1
.TAB1 . n = df/dXn/
CHPO1 : -transpose(J)fRIGI1 : transpose(J)J
TAB2 : TABLE of type 'VECTEUR' containingTAB2 1 H1TAB2 . 1 = H1
. TAB2 . n = Hn
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The GaussThe Gauss--Newton method (+ new operator GANE)Newton method (+ new operator GANE)
( ) ( )λ=d E d EThere is a second problem!!! (first problem )( ) ( )λ=d E d EThere is a second problem!!! (first problem )
• is almost always rank 1 deficient (use of RESO ‘ENSE’)( )TJ J
{ }65∂ ∂∑f f
• For instance: ( for )
BUT
1ia =− { }1=E66166
( )ii i
aE E=
∂ ∂=∂ ∂∑f fE
66 65( )E g= E
• GANE can be used to find out the ( )
• Assuming with with
ia 65 65 6566
( )T
E∂=
∂fJ J a J
/ i ig E a∂ ∂ = 65J ia∂ ∂ ∂= +f f f66 65( )gAssuming with with
• Use GANE again for finding and complete with 65
661
gn i gnii
E aE=
=∑i ig 65
66i
i iE E E∂ ∂ ∂
65gnE
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The GaussThe Gauss--Newton method (+ new operator GANE)Newton method (+ new operator GANE)
And finally!
( ) ( ) / ( )+ + + +−d E d E d E
i
… with
i
7max( ) 10i iE E+ −− ≈( )i i
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The GaussThe Gauss--Newton method (+ new operator GANE)Newton method (+ new operator GANE)
66E+Substituting with (and still starting from ) 66 , 1,2, 4,8nE n+ = { }1=E
1n 1n =2n =
3n =4n =
WARNING
• Sometimes RESO takes care itself of the indetermination…7E E+ +
• with 7max( ) 10i iE E −+− ≈
E E≠E E
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An amazing An amazing optimisationoptimisation problemproblem
OUTLINE:
• Background: some results from photogrammetry
Th (VERY) i lifi d ti i ti bl• The (VERY) simplified optimisation problem
• EXCE, steepest descent, conjugate gradient & BFGS
• The Gauss-Newton method (+ new operator GANE)The Gauss Newton method (+ new operator GANE)
• Conclusions, further works & bibliography
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Conclusions, further works & bibliographyConclusions, further works & bibliography
Conclusions & further works
• Nothing is given for free!
• Cast3M demonstrates again its ability to help to understand (and solve) problems!(and solve) problems!
• Gauss-Newton is the root for other classical methods
• Damped Gauss-Newton = Levenberg-Marquardt methodT T (already in GANE)
• Powell’s Dog Leg Method
• Use of noisy experimental data
( )T Tlm Fμ+ = − = −∇J J I E J f
• Use of noisy experimental data
• Identification of non-linear models with more refined meshes
• Line search, convergence tests…, g
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Conclusions, further works & bibliographyConclusions, further works & bibliography
Short bibliography
• W.H. Press, S.A. Teukolsky, W.T. Vetterling & B.P. Flannery,Numerical Recipes : The Art of Scientific Computing, 1986u e ca ec pes e t o Sc e t c Co put g, 986
• H. Matthis & G. Strang,The Solution of Non-Linear Finite Element EquationsIJNME, 14, 1613-1626, 1979IJNME, 14, 1613 1626, 1979
• K. Madsen, H.B. Nielsen & O. TingleffMethods for Non-Linear Least Squares ProblemsTechnical University of Denmark 2004Technical University of Denmark, 2004
• T. Charras & J. KicheninL’optimisation dans CAST3M