Estimation of the mehanial p rop erties of a solid elasti - PowerPoint PPT Presentation

Estimation of the me hani al p rop erties of a solid elasti medium in onta t with a �uid medium �doua rd Canot Jo elyne Erhel Nabil Nassif Samih Zein SA GE team INRIA/IRISA � (Rennes, F ran e) 1

Plan F o rw a rd and inverse p roblems Ba y esian inferen e Choi e of the estimato rs • The MCMC metho d • The SPSA metho d • Sensitivit y analysis • • • 2

The fo rw a rd p roblem ٠f Fluid à ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� Solid ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ٠������������������� ������������������� s ������������������� ������������������� 3 ������������������� ������������������� ������������������� �������������������

The fo rw a rd p roblem The ontinuous fo rmulation: ∂p  ∂t + c 2 f ρ f div v f = 0 (Ω f ) (1 . 1)  (1)       ∂v f    ρ f ∂t + ∇ p = 0 (Ω f ) (1 . 2)        A∂σ    ∂t − ǫ ( v s ) = 0 (Ω s ) (1 . 3)   ∂v s    ρ s ∂t − div σ = 0 (Ω s ) (1 . 4)         v s .n = v f .n (Γ) (1 . 5)        σ.n = − p.n (Γ) (1 . 6)   4

The fo rw a rd p roblem The dis rete fo rmulation: P n + 1 2 − P n − 1  2  + D f V n   M p f = 0 (2 . 1)  (2)   ∆ t    V n +1  − V n 2 + B Σ n +1 + Σ n   f P n + 1  f f  − D t  M f = 0 (2 . 2)    ∆ t 2     − B t V n +1 + V n Σ n +1 − Σ n  n + 1 f f + D t 2 M Σ s V = 0 (2 . 3) s    ∆ t 2    n + 1 n − 1    2 2 V − V   s s (J. Diaz and P . Joly + D s Σ n = 0 2005)   M s (2 . 4)    ∆ t        5

The fo rw a rd p roblem Simulations results: −6 x 10 10 1 layer solid medium 2 layers solid medium 3 layers solid medium 5 0 −5 6 0 50 100 150 200 250 300 350 400

The inverse p roblem : the o e� ients to re over ( λ, µ, ρ ) • θ Pressure measures • where u := { u ij = u ( x i , t j ) } , ǫ := { ǫ ij } ∼ N (0 , s 2 ) . y = u + ǫ 7

Estimato rs of θ 1. the exp e tation with resp e t to the p osterio r p robabilit y: � 2. the maximum E ( θ | y ) = a p osterio θ p ( θ | y ) dθ ri: θ ∗ = arg max p ( θ | y ) , θ ∈ D θ 8

The Ba y esian mo del and the inverse p roblem F rom Ba y es's fo rmula: W e onsider: 1 si ∀ i , θ i ∈ [ θ min , θ max ] p ( θ | y ) = p ( y ) p ( y | θ ) p ( θ ) elsewhere  1  p ( θ ) ∝ 0  si ∀ i , θ i ∈ [ θ min , θ max ] − 1 � y i − u ( x i , T, θ ) � 2 � � � p ( y | θ ) ∝ exp 2 s i elsewhere � 2 �  � � y i − u ( x i ,T,θ ) − 1 exp  �  i  2 s   9 p ( θ | y ) ∝    0  

Ma rk ov Chain Monte Ca rlo Estimation of θ : � E [ θ | y ] = θp ( θ | y ) dθ This integral is app roa hed b y: with θ ∼ p ( θ | y ) the limiting distribution of a Ma rk ov n E [ θ | y ] ≈ 1 � θ k hain (Ha rold Niederreiter, SIAM, 1992). n k =1 10

The Metrop olis-Hasting algo rithm The a elerated version of M-H algo rithm with p ∗ ( . | y ) a linea r interp olation of p ( . | y ) : 1- A t θ k generate a p rop osal C from q ( ·| θ k ) . 2- With p robabilit y α pred ( C, θ k ) = min p romote C to b e a andidate to the standa rd M-H � p ∗ ( C | y ) algo rithm. Otherwise, p ose θ k +1 = θ k . � p ∗ ( θ k | y ) , 1 3- With p robabilit y α ( C, θ ) = min a ept ; Otherwise reje t C , θ k +1 = θ k . (J. Andre`s Christen and C. F o x, 2005.) � p ( C | y ) � p ( θ k | y ) , 1 θ k +1 = C 11

Ma rk ov Chain Monte Ca rlo The va rian e of the estimato r given b y MCMC: where τ is the integrated auto ova rian e time (IA CT). var ( θ MC ) = τ var ( θ ) n (S. Mey er, N. Christensen and G. Ni holls, 2001) M τ = 1 + 2 � ρ ( s ) . s =1 12

Ma rk ov Chain Monte Ca rlo Autocorelation 1 0.5 Lambda 0 −0.5 0 50 100 150 200 250 300 350 400 450 500 1 0.5 Mu 0 −0.5 0 50 100 150 200 250 300 350 400 450 500 1 0.5 Rho 0 −0.5 12 0 50 100 150 200 250 300 350 400 450 500 lag s

Ma rk ov Chain Monte Ca rlo 19000 samples of the Ma rk ov hain with the a - elerated version of M-H algo rithm (6000 simu- lations and noise < 6% ): • Exa t V alue (SI) Conf. Interval of erro r 11.5 × 10 9 6 × 10 9 % θ 10 . 9 × 10 9 ± 2 . 6% λ 5 . 2% 6 . 5 × 10 9 ± 2% µ 8% ρ 1850 1867 ± 0 . 15% 0 . 9% 13

Ma rk ov Chain Monte Ca rlo 19000 samples of the Ma rk ov hain with the standa rd M-H algo ritm and di�erent sta rting p oints: • θ θ 0 = θ min θ 0 = θ max 11 . 1 × 10 9 ± 2 . 8% 10 . 8 × 10 9 ± 2 . 7% λ 5 . 82 × 10 9 ± 2 . 5% 6 . 14 × 10 9 ± 2 . 6% µ ρ 1827 ± 0 . 21% 1911 ± 0 . 4% 13

Ma rk ov Chain Monte Ca rlo Frequency 1500 Lambda 1000 500 0 0 2 4 6 8 10 12 14 16 18 9 x 10 1500 1000 Mu 500 0 0 1 2 3 4 5 6 7 8 9 10 9 x 10 6000 4000 Rho 2000 0 13 0 500 1000 1500 2000 2500

Ma rk ov Chain Monte Ca rlo 10 Convergence of the estimators x 10 1.15 1.1 Lambda 1.05 1 0.95 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4 x 10 9 x 10 8 7 Mu 6 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4 x 10 2000 1950 (C. Rob ert 1996) Rho 1900 1850 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4 x 10 13

Simultaneous P erturbation Sto hasti App ro ximation Optimize the log of the p osterio r p robabilit y: In our ase, it is a least squa res p roblem L ( θ ) = − log p ( θ | y ) The fo rm of the algo ritm is as follo ws: with θ k +1 = ˆ ˆ g k (ˆ θ k − a k ˆ θ k ) (J.C. Spall, 2003). θ k ) = L (ˆ θ k + c k ∆ k ) − L (ˆ θ k − c k ∆ k ) [∆ − 1 k 1 , ∆ − 1 k 2 , . . . , ∆ − 1 kp ] T g k (ˆ ˆ 14 2 c k

Simultaneous P erturbation Sto hasti App ro ximation Asymptoti no rmalit y: dist (J.C. Spall, 2003). It is p ossible to have a on�den e interval fo r θ ∗ b y k β/ 2 (ˆ θ k − θ ∗ ) → N (0 , Σ) − − − − − − − − − − applying the Monte Ca rlo metho d: N 1 k β/ 2 (ˆ θ k − θ ∗ ) = 0 � N k =1 � N k =1 k β/ 2 (ˆ θ k ) ⇒ θ ∗ = = � N k =1 k β/ 2 N Σ = 1 15 k β (ˆ θ k − θ ∗ )(ˆ θ k − θ ∗ ) T � N k =1

Simultaneous P erturbation Sto hasti App ro ximation Exa t V alues (SI) Con�den e Intervals Erro rs 11.5 × 10 9 6 × 10 9 1850 θ Inje ted noise < 1% and 600 simulations 11 . 83 × 10 9 ± 1 . 4% 2 . 8% λ Exa t V alues (SI) Con�den e Intervals Erro rs 5 . 75 × 10 9 ± 1 . 6% 4 . 1% µ 11.5 × 10 9 ρ 1856 ± 0 . 12% 0 . 03% 6 × 10 9 1850 θ Inje ted noise < 6% and 700 simulations 12 . 2 × 10 9 ± 6 . 12% 6 . 6% λ 5 . 4 × 10 9 ± 7 . 2% 9 . 5% µ ρ 1868 ± 0 . 83% 1% 16

Simultaneous P erturbation Sto hasti App ro ximation Exa t V alues (SI) Estimated V alues Erro rs 11.5 × 10 9 11.43 × 10 9 6 × 10 9 6.01 × 10 9 1700 1702 θ 9 × 10 9 8.93 × 10 9 λ 1 0 . 6% 7 × 10 9 7.02 × 10 9 µ 1 0 . 1% 2000 2003 ρ 1 0 . 1% 11.5 × 10 9 11.4 × 10 9 λ 2 0 . 7% 6 × 10 9 6.02 × 10 9 µ 2 0 . 2% 2400 2405 ρ 2 0 . 1% Inje ted noise < 1% and 6000 simulations λ 3 2 . 1% µ 3 1 . 8% ρ 3 0 . 1% 17

Sensitivit y analysis Consider the singula r value de omp osition (SVD) of the Ja obian matrix ( F ′ ( θ 0 )) . ǫ ≈ F ′ ( θ 0 ) δθ One easily veri�es that: F ′ ( θ 0 ) = USV T 18 δθ k = ǫ k , ∀ k = 1 , ..., p. s k

Sensitivit y analysis A �rst o rder analysis yields: Thus 0 = F (0) = F ( θ 0 ) − F ′ ( θ 0 ) θ 0 � u 0 � = � F ( θ 0 ) � ≃ � F ′ θ 0 � < s 1 � θ 0 � Hen e, if the a ura ies on θ and u , resp e tively σ θ ∀ k, | δθ ∗ � θ 0 � ≤ s 1 k | | ǫ k | � u 0 � ≤ σ u and σ u , verify the inequalit y: s k 18 s k ≥ σ u ∀ k, s 1 σ θ