Analyse numerique et optimisation : Une introduction a la modelisation mathematique et a la simulation numerique French

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J

V

J

α

C

0 < µ < 2α/C u0 ∈ K

2

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u

%9,*:10.1',*C      5 "  0.     .  ;<       ,   v → v − µJ  (v)       0 < µ < 2α/C 2 A ,6?6        ∃ γ ∈]0, 1[ ,  v − µJ  (v) − w − µJ  (w)  ≤ γv − w .

*    I   PK  8 0  , ;  )0+'+) CC '  
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F1 , . . . , FM   ,
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 Fi (u) i∈I(u)   

  
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λ1 , . . . , λM
 
J  (u) + i=1 λi Fi (u) = 0( 
 λi = 0  i ∈ / I(u) " 
  
(   
i ∈ {1, . . . , M }

LE O ÉC

T Y POL &

lim

ε→0

' 2 max (Fi (uε ), 0) = λi . ε

)9.0;5) CC

/  . ?   6   
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@      )  V = RN  9            /C   F  8    C 2  RN  RN    u  F     F ,6?6   

N H EC

F (u) = 0  F  (u)   . 0

T Y POL

 8   5#  .     v   

  F (u) = F (v) + F  (v)(u − v) + O u − v2 ,

ÉC

,6?6 

E L O

u = v − (F  (v))

−1

  F (v) + O v − u2 .

    /C   ?    8O  .        6

   *        u0 ∈ RN A    un+1 = un − (F  (un ))

−1

F (un )   n ≥ 0.

; "
0

u

u

; &
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N H EC

E U IQ

   8   .     ,    :            #  Xad 

T Y POL


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x ∈ Xad        ;)0+'+)

E L O

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;&
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T Y POL

N H EC

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,  #       .  G   c˜kN     .A  8  8      ,   *        0A   

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x1 + 2x2

⎧ ⎨ −3x1 + 2x2 + x3 = 2 −x1 + 2x2 + x4 = 4 ⎩ x1 + x2 + x5 = 5

CC C 
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min

x1 ≥0, x2 ≥0

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H C E T

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 x1 + x2 ≤ 1
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min

x1 ≥0, x2 ≥0, x3 ≥0, x4 ≥0

3x3 − x4


$


      
  


 
   


N H EC

x1 − 3x3 + 3x4 = 6 x2 − 8x3 + 4x4 = 4

T Y POL

E U IQ



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LE O ÉC

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 i ∈ C ( 


 
 k
 

k 
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v  ≥ v  @     ;"<         ) v = f (v)A .  6 )   .   f : RN → RN  3   5  

66     v ? ,  ,        ) *    ?  .        )A     . ?        A ,#   .  8  X  ?     ,  @     ( RN  ,     A )   A      A , 6   f : x → f (x)    *   f (φ) ≤ φA         f r+1 (φ) ≤ f r (φ)A     r ≥ 0    {f r (φ)}r≥0  P .    .  .  v  ∈ RN  ;       ,     .  −∞    :   v  A      . < @  f (v  ) = v       f  1,  A  v      )     f A    . 6  v  ≤ φA     f   A v  = f r (v  ) ≤ f r (φ)A     r ≥ 0A    , )     r ≥ 0A v  ≤ v  A  v         )  f  *   ,  5  A  8  .          6)  f A  .        .

H C E T

Y  " L O $"$ -
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{f r (φ)}r≥0

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T" -



      
  



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CC >  
  
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C ⊂ ∂J(x) 2  

0 ∈ ∂J(x) '    
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ÉC

min

t

     y ∈ Rn , t ∈ R, Ji (y) ≤ t, ∀i ∈ I ,

 
 
0 ∈ C   0


 
*   
( C = ∂J(x)

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N

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E U           '   5  B   &  I Q "          % ("  

   ( *    N  '   4     %     C  H "  "  %   %E %
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    '  4 M   '   &   0 #/   '       (  %    0 #N      4 $    "    '  4   %'     "    '   <   K  "     &  "  = 4    '  % '       "        - ! 

   ? A O  . 1 7 (  J  IGI  "  - !
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T Y POL

N H EC

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         (   &  A$         (    "              '  "   ;    '        ' "            %  "   (       % &  
  * + K  (   ("       "      
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N

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N H EC




E U IQ

#442': : '2: 2:4#2: !2 ; ' 0

0 ≥ −2 x − xK , y − xK  + θz2 .

0 *  θ %   '    ! A "     xK "  %
        y ∈ K   x − y2 = x − xK 2 + xK − y2 + 2 x − xK , xK − y ≥ x − xK 2 ,

 "   % "  xK  '   ;  (   x   K 

C E T Y L PO

HN

E U IQ


" V $ )  @
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, 

V

(en )n≥1 (xn )n≥1 p

V

x2 = x, x =



V p x n=1 xn en xn = x, en 

x

; 

0   y  '  
  (en )n≥1   "  x − y <  OP  D(       
     Sp "  &     z ∈ V  *    Sp z = zW  : zW    ;  (       %  W     p   %  (en )1≤n≤p  0 %  ! (z − Sp z)  (  &      W       & Sp z $    "  z2 = z − Sp z2 + Sp z2 , #!  "   " 

E L O

n )n≥1

T Y POL

N H EC

E U IQ

Sp z ≤ z∀z ∈ V.

5

 Sp z      (en )1≤n≤p   "  (z − Sp z)  (  & (   (en )1≤n≤p   %
 *  " 

ÉC

Sp z =

p 

z, en en .

n=1

  p  

    Sp y = y  y    '  
  (en )n≥1    "  Sp x − x ≤ Sp (x − y) + y − x ≤ 2x − y ≤ 2.

$      %   Sp x % x .   %    "   #!   2 2 lim Sp x = x ,

p→+∞

"      "   *    

  C!   %

H C E T

N

E U IQ


,  ,  0  0  00 ,           = 0      .          :6  ,  ,  0  0  00

Y L O P

" V $ )  @
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 $//
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&" $ / #
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LE O ÉC


  

$

E U  *       I % 
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V

vn

[v1 , · · · , vn ] = [e1 , · · · , en ]

(en )n≥1

(en )n≥1

V

(vn )n≥1




(en )n≥1

%@*'1',* C C 

V

W

$ A  V $ W " " $"$  
&" $ $"$" C "

  AxW ≤ CxV

∀x ∈ V.

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 $ 
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AxW . x∈V,x=0 xV sup

N H EC

E U IQ

 .          . ,     ,      = 0 ;    ,        X   ,6         )6 " K $ ) " $-& $$ -  "   $ )  @
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&"

$ #!))
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&" $ 
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  L(x0 ) < α < L(x)

; "
0A R ∆k    6    ,  k    AA    uii      8 @   B = LD  C = D−1 U  . ) A = BC  ' A = A∗ A     C(B ∗ )−1 = B −1 (C ∗ ) 9 .  
&    C(B ∗ )−1         A     B −1 C ∗      8   9          1  A       B  C   HA      B −1 C ∗ ,       A ,6?6    C(B ∗ )−1 = B −1 C ∗ = IdA  C = B ∗  *   ,          '%#A     ,      8   A = B1 B1∗ = B2 B2∗ A ,R B2−1 B1 = B2∗ (B1∗ )−1  1  
&A       B2−1 B1 = D = diag(d1 , ..., dn )A     A = B2 B2∗ = B2 (DD∗ )B2∗  ' B2  . 0A  .  D2 = Id  di = ±1 @    :     ,      '%#    8  # 1 di = 1A      B1 = B2 


LE O ÉC

Y L O P

H C E T

N

E U IQ

    *  
 

N H EC




E U IQ

.-+5- 30.1';5) &) -. =.+1,0':.1',* &) 4,-):DBC 9   A     86

   '%# B   )   ,   A = BB ∗    A = (aij )1≤i,j≤n A B = (bij )1≤i,j≤n . bij = 0  i < j  *  1 ≤ i, j ≤ nA  . 

LE O ÉC

T Y POL aij =

n 



min(i,j)

bik bjk =

k=1

bik bjk .

k=1

9  )         A ;   A   .  

H    A  #  <        B  3  A  .     (j − 1)     B  8   (j − 1)     AA     j 6   A        1 2 j j−1 2   2 ajj = (bjk ) ⇒ bjj = 3ajj − (bjk )2 ai,j =

k=1 j 

k=1

bjk bi,k

⇒ bi,j =

ai,j −

j−1

k=1

k=1 bjk bi,k

bjj

E U IQ

  j + 1 ≤ i ≤ n.

N H EC

@     j 6   B  8    (j − 1)    3     A   G  A  A  #   )    .A              8 3  A  A ,  2  )    .A   .   ajj − j−1 (b ) ≤ 0       j A  k=1 jk  H    ,  

E L O

T Y POL

-(,0'149) *59%0';5)C ,    '%#   ,   8O  6

ÉC

        0      A       ?  8  B  +   ,   :  %     8    A    A  #  

( j = 1, n ( k = 1, j − 1

ajj = ajj − (bjk )2 +  
  k √ ajj = ajj ( i = j + 1, n ( k = 1, j − 1 aij = aij − bjk bik

+  
  k aij =

aij ajj

+  
  i +  
  j

Y L O P

H C E T

N

E U IQ

,931) &!,3%0.1',*:C *    ,:       '%# 

LE O ÉC

  0 ,  ;         <   ?  6



*   


 

N H EC

E U IQ

   0      n     0    ,  W 2    '%# J  0 ,  Nop 

ÉC

OLE

T Y POL Nop =

n 

⎛

⎝(j − 1) +

j=1

n 

⎞

j⎠ ,

i=j+1

 A     A  Nop ≈ n3 /6

0    J  8  (          #       ? B  B ∗   0 ,        Nop ≈ n2      '%#     . &)5> =,': 3-5: 0.3'&)     4       #   )    . W

.10'+): /.*&): )1 9.10'+): +0)5:):

N H EC

E U IQ

 ,     0    :   A    ,  +0)5:)         ?       A             /.*&) *     #    ;           )            &5 2)+1)50 '*'1'.- x0 ∈ Rn '
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$-0 -
 
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H C E T

N

E U IQ

@            :  .  ,     . ? ,  #      ,   ;.   1)   
"       #  0   (M ∗ + N )  
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N  = 1 − (M + N )w, w < 1,

N

E U IQ

 "       
 "   (  %             & 

                 (         %
 "                   %   (      *  %  %  *      J  (         

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LE O ÉC

Y L O P

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E L O

>)0+'+) C C ) A 
 
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 " 
 
  ω ∈ ]0, 2[(   %


   
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 0 < ω < 2

E U IQ

" $ )+" 
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 "#  0 $"
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N=

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Y L O P

H C E T

N

     0       .      6 A                      8  f (x) = 12 Ax · x − b · x          0  

LE O ÉC



*   





)99) C C ... ≤ λn ,

E U )) " $ " 0$
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 "#  H 0 $" )    $ ' 
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LE O ÉC

λ1 ≤ λ2 ≤

A

 λ1 ≤ 0 ≤ λn -
   α,  0 < λ1 ≤ 
$"  0 < α < 2/λn αopt =

... ≤ λn

ρ(M −1 N )

α

2 λ1 + λn

min ρ(M −1 N ) = α

λn − λ1 . λn + λ1

)9.0;5) C C λ1 ≤ ... ≤ λn < 0A       #  

 )   8    α  −α *      αopt A  #      ,     8     λn /λ1  A    I A    ,         cond2 (A)     A *  A      A  0    A      .        •

E U IQ

%9,*:10.1',*C 1,   
 $A            .       ρ(M −1 N ) < 1 @ M −1 N = ( Id − αA)A 

N H EC

ρ(M −1 N ) < 1 ⇔ |1 − αλi | < 1 ⇔ −1 < 1 − αλi < 1 , ∀i.

T Y POL

'      αλi > 0     1 ≤ i ≤ n *      .    A  . H     H     α       .     λ1 ≤ 0 ≤ λn A        α A   A   0 < λ1 ≤ ... ≤ λn A       ,  8  0 < α < 2/λn  *        αopt A       8  λ → |1 − αλ|      ] − ∞, 1/α]       [1/α, +∞[A 

ÉC

E L O

ρ(M −1 N ) = max{|1 − αλ1 |, |1 − αλn |}.

*  A        8  α → ρ(M −1 N )       , 6 2   αopt = λ1 +λ      
n



 5      I      I

      .         #         #    )    .  ,  ,              ;     0  .    A .   1)   

0  " *$' )  E C ' ) T "Y L O P $ $  -"  $$ ) 3,, 8 )
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)99) C C

A x0 ∈ Rn r0 = b − Ax0

(Kk )k≥0

xk ∈ [x0 + Kk−1 ]

n r0

k≥1

;

)0+'+) C C ) A 
 
: 
 
 
 ) (xk )0≤k≤n 

T Y POL



  %
 

     %
 * 
 5*   
rk = b − Axk
dk = xk+1 − xk  " 
 
 $
 

N: Kk
  *  +

ÉC

E L O

Kk = [r0 , ..., rk ] = [d0 , ..., dk ],

  
(rk )0≤k≤n−1
  %* 
rk · rl = 0   0 ≤ l < k ≤ n − 1,

  
(dk )0≤k≤n−1
 5*
    + A Adk · dl = 0   0 ≤ l < k ≤ n − 1.

 )     , .          I

       9 (A  ,     ,       rk   ? Kk−1 A       xk         .   8                . 

HN

E U IQ

" A $ " !"  *$ )"-' " x0 " (xk , rk , pk ) "  " *$ )
 
"$   $

0,3,:'1',* C C

C E T Y L " )  O P E OL

p0 = r0 = b − Ax0 ,

ÉC

⎧ ⎨ xk+1 = xk + αk pk 0≤k rk+1 = rk − αk Apk ⎩ pk+1 = rk+1 + βk pk

∈ Rn ,

;

 C ! .     %

E L O

T Y POL

min |λi − λ| ≤ ˆ vk+1 

1≤i≤m

|ek · y| , y

 "       

ÉC

LE O ÉC

Y L O P




H C E T

N

E U IQ

T Y POL

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E U IQ

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TU 3=N3 +BA 734/3/5 5A @+/ NEA "NJ    + .A /C N# ;
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