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

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Estimation of the mehanial p rop erties of a solid elasti medium in ontat with a uid medium doua rd Canot Jo elyne Erhel Nabil Nassif Samih Zein SA GE team INRIA/IRISA (Rennes, F rane) 1 Plan F o

slide-1
SLIDE 1 Estimation
  • f
the me hani al p rop erties
  • f
a solid elasti medium in
  • nta 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
slide-2
SLIDE 2 Plan
  • F
  • rw
a rd and inverse p roblems
  • Ba
y esian inferen e
  • Choi e
  • f
the estimato rs
  • The
MCMC metho d
  • The
SPSA metho d
  • Sensitivit
y analysis 2
slide-3
SLIDE 3 The fo rw a rd p roblem
  • Ωf

Γ Fluid

s

Ω Solid

3
slide-4
SLIDE 4 The fo rw a rd p roblem The
  • ntinuous
fo rmulation:

                                          

∂p ∂t + c2

fρf div vf = 0 (Ωf)

(1.1) ρf ∂vf ∂t + ∇p = 0 (Ωf) (1.2) A∂σ ∂t − ǫ(vs) = 0 (Ωs) (1.3) ρs ∂vs ∂t − div σ = 0 (Ωs) (1.4) vs.n = vf.n (Γ) (1.5) σ.n = −p.n (Γ) (1.6)

(1) 4
slide-5
SLIDE 5 The fo rw a rd p roblem The dis rete fo rmulation:

                                              

Mp P n+1

2 − P n−1 2

∆t + DfV n

f = 0

(2.1) Mf V n+1

f

− V n

f

∆t − Dt

fP n+1

2 + BΣn+1 + Σn

2 = 0 (2.2) MΣ Σn+1 − Σn ∆t + Dt

sV n+1

2

s

− BtV n+1

f

+ V n

f

2 = 0 (2.3) Ms V

n+1

2

s

− V

n−1

2

s

∆t + DsΣn = 0 (2.4)

(2) (J. Diaz and P . Joly 2005) 5
slide-6
SLIDE 6 The fo rw a rd p roblem Simulations results:

50 100 150 200 250 300 350 400 −5 5 10 x 10

−6

1 layer solid medium 2 layers solid medium 3 layers solid medium

6
slide-7
SLIDE 7 The inverse p roblem
  • θ
: the
  • e ients
to re over (λ, µ, ρ)
  • Pressure
measures

y = u + ǫ

where u := {uij = u(xi, tj)} ,ǫ := {ǫij} ∼ N(0, s2) . 7
slide-8
SLIDE 8 Estimato rs
  • f θ
1. the exp e tation with resp e t to the p
  • sterio
r p robabilit y:

E(θ|y) =

  • θ p(θ|y) dθ
2. the maximum a p
  • sterio
ri:

θ∗ = arg max

θ∈Dθ

p(θ|y),

8
slide-9
SLIDE 9 The Ba y esian mo del and the inverse p roblem F rom Ba y es's fo rmula:

p(θ|y) = 1 p(y)p(y|θ)p(θ)

W e
  • nsider:

p(θ) ∝

  

1

si ∀ i , θi ∈ [θmin, θmax] elsewhere

p(y|θ) ∝ exp

  • − 1

2

  • i

yi − u(xi, T, θ)

s

2

p(θ|y) ∝

          

exp

  • − 1

2

  • i

yi−u(xi,T,θ)

s

2

si ∀ i , θi ∈ [θmin, θmax] elsewhere 9
slide-10
SLIDE 10 Ma rk
  • v
Chain Monte Ca rlo Estimation
  • f θ
:

E[θ|y] =

  • θp(θ|y)dθ
This integral is app roa hed b y:

E[θ|y] ≈ 1 n

n

  • k=1

θk

with θ ∼ p(θ|y) the limiting distribution
  • f
a Ma rk
  • v
hain (Ha rold Niederreiter, SIAM, 1992). 10
slide-11
SLIDE 11 The Metrop
  • lis-Hasting
algo rithm The a elerated version
  • f
M-H algo rithm with p∗(.|y) a linea r interp
  • lation
  • f p(.|y)
: 1- A t θk generate a p rop
  • sal C
from q(·|θk) . 2- With p robabilit y αpred(C, θk) = min

p∗(C|y)

p∗(θk|y), 1

  • p
romote C to b e a andidate to the standa rd M-H algo rithm. Otherwise, p
  • se θk+1 = θk
. 3- With p robabilit y α(C, θ) = min

p(C|y)

p(θk|y), 1

  • a ept

θk+1 = C

; Otherwise reje t C , θk+1 = θk . (J. Andre`s Christen and C. F
  • x,
2005.) 11
slide-12
SLIDE 12 Ma rk
  • v
Chain Monte Ca rlo The va rian e
  • f
the estimato r given b y MCMC:

var(θMC) = τ var(θ) n

where τ is the integrated auto
  • va
rian e time (IA CT).

τ = 1 + 2

M

  • s=1

ρ(s).

(S. Mey er, N. Christensen and G. Ni holls, 2001) 12
slide-13
SLIDE 13 Ma rk
  • v
Chain Monte Ca rlo

50 100 150 200 250 300 350 400 450 500 −0.5 0.5 1 Lambda Autocorelation 50 100 150 200 250 300 350 400 450 500 −0.5 0.5 1 Mu 50 100 150 200 250 300 350 400 450 500 −0.5 0.5 1 Rho lag s

12
slide-14
SLIDE 14 Ma rk
  • v
Chain Monte Ca rlo
  • 19000
samples
  • f
the Ma rk
  • v
hain with the a - elerated version
  • f
M-H algo rithm (6000 simu- lations and noise < 6% ):

θ

Exa t V alue (SI) Conf. Interval

%

  • f
erro r

λ

11.5×109

10.9×109±2.6% 5.2% µ

6×109

6.5×109±2% 8% ρ 1850 1867 ± 0.15% 0.9%

13
slide-15
SLIDE 15 Ma rk
  • v
Chain Monte Ca rlo
  • 19000
samples
  • f
the Ma rk
  • v
hain with the standa rd M-H algo ritm and dierent sta rting p
  • ints:

θ θ0 = θmin θ0 = θmax λ 11.1×109±2.8% 10.8×109±2.7% µ 5.82×109±2.5% 6.14×109±2.6% ρ 1827 ± 0.21% 1911 ± 0.4%

13
slide-16
SLIDE 16 Ma rk
  • v
Chain Monte Ca rlo

2 4 6 8 10 12 14 16 18 x 10

9

500 1000 1500 Lambda Frequency 1 2 3 4 5 6 7 8 9 10 x 10

9

500 1000 1500 Mu 500 1000 1500 2000 2500 2000 4000 6000 Rho

13
slide-17
SLIDE 17 Ma rk
  • v
Chain Monte Ca rlo

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10

4

0.95 1 1.05 1.1 1.15 x 10

10

Lambda Convergence of the estimators 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10

4

5 6 7 8 x 10

9

Mu 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10

4

1850 1900 1950 2000 Rho

(C. Rob ert 1996) 13
slide-18
SLIDE 18 Simultaneous P erturbation Sto hasti App ro ximation Optimize the log
  • f
the p
  • sterio
r p robabilit y:

L(θ) = − log p(θ|y)

In
  • ur
ase, it is a least squa res p roblem The fo rm
  • f
the algo ritm is as follo ws:

ˆ θk+1 = ˆ θk − akˆ gk(ˆ θk)

with

ˆ gk(ˆ θk) = L(ˆ θk + ck∆k) − L(ˆ θk − ck∆k) 2ck [∆−1

k1 , ∆−1 k2 , . . . , ∆−1 kp ]T

(J.C. Spall, 2003). 14
slide-19
SLIDE 19 Simultaneous P erturbation Sto hasti App ro ximation Asymptoti no rmalit y:

kβ/2(ˆ θk − θ∗)

dist

− − − − − − − − − − →N(0, Σ)

(J.C. Spall, 2003). It is p
  • ssible
to have a
  • nden e
interval fo r θ∗ b y applying the Monte Ca rlo metho d:

1 N

N

  • k=1

kβ/2(ˆ θk − θ∗) = 0 = ⇒ θ∗ =

N

k=1 kβ/2(ˆ

θk)

N

k=1 kβ/2

Σ = 1 N

N

  • k=1

kβ(ˆ θk − θ∗)(ˆ θk − θ∗)T

15
slide-20
SLIDE 20 Simultaneous P erturbation Sto hasti App ro ximation

θ

Exa t V alues (SI) Conden e Intervals Erro rs

λ

11.5×109

11.83×109±1.4% 2.8% µ

6×109

5.75×109±1.6% 4.1% ρ

1850

1856 ± 0.12% 0.03%

Inje ted noise < 1% and 600 simulations

θ

Exa t V alues (SI) Conden e Intervals Erro rs

λ

11.5×109

12.2×109±6.12% 6.6% µ

6×109

5.4×109±7.2% 9.5% ρ

1850

1868 ± 0.83% 1%

Inje ted noise < 6% and 700 simulations 16
slide-21
SLIDE 21 Simultaneous P erturbation Sto hasti App ro ximation

θ

Exa t V alues (SI) Estimated V alues Erro rs

λ1

11.5×109 11.43 ×109

0.6% µ1

6×109 6.01 ×109

0.1% ρ1

1700 1702

0.1% λ2

9×109 8.93×109

0.7% µ2

7×109 7.02×109

0.2% ρ2

2000 2003

0.1% λ3

11.5×109 11.4×109

2.1% µ3

6×109 6.02×109

1.8% ρ3

2400 2405

0.1%

Inje ted noise < 1% and 6000 simulations 17
slide-22
SLIDE 22 Sensitivit y analysis

ǫ ≈ F ′(θ0)δθ

Consider the singula r value de omp
  • sition
(SVD)
  • f
the Ja obian matrix (F ′(θ0)) .

F ′(θ0) = USV T

One easily veries that:

δθk = ǫk sk , ∀k = 1, ..., p.

18
slide-23
SLIDE 23 Sensitivit y analysis A rst
  • rder
analysis yields:

0 = F(0) = F(θ0) − F ′(θ0)θ0

Thus

u0 = F(θ0) ≃ F ′θ0 < s1θ0 ∀k, |δθ∗

k|

θ0 ≤ s1 sk |ǫk| u0 ≤ σu

Hen e, if the a ura ies
  • n θ
and u, resp e tively σθ and σu , verify the inequalit y:

sk s1 ≥ σu σθ ∀k,

18
slide-24
SLIDE 24 Sensitivit y analysis then:

|δθ∗

k|

θ0 ≤ σθ ∀k

whenever

|ǫk| u0 ≤ σu.

(P . Al Khoury, G. Chavant, F. Clment and P . Herv, 2002). 18
slide-25
SLIDE 25 Sensitivit y analysis

1 2 3 4 5 6 7 8 9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ratio of the sigular values singular value’s index

18
slide-26
SLIDE 26 Con lusion
  • MCMC
is mo re a urate than SPSA but it is ha rder to implement and to have
  • rre t
results.
  • SPSA
is mu h less exp ensive in
  • mputations
then MCMC .
  • In
the ase
  • f
multiple la y ers, SPSA is a mo re app rop riate metho d. 19