Lecture 1: Getting started with problematic inversions Denis - - PowerPoint PPT Presentation

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Lecture 1: Getting started with problematic inversions

Denis Maillet, Yvon Jarny, Daniel Petit, Olivier Fudym, Philippe Le Masson

LEMTA Nancy - LTN Nantes - Institut P’ Poitiers - Ecole Mines Albi - LIMATB Lorient

Example 1: Square system of linear equations Example 2: Different inverse problems fo steady state 1D heat transfer through a wall

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

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Example 1: Square system of linear equations

1 81 39 9 21 10

2 1 2 1

= − = − x x x x

      = = ⇒       = =       − − = 36 9 1 3 81 39 21 10

exact mo exact

x S y x x S Direct problem: input (known)

• utput (calculated)

scalar relationship → vector

Model : ymo = η η η η (x) Structure of model

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

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inverse problem: data (known) unknown

      = =       − − = 36 9 81 39 21 10

exact mo

x S y S

Solution with exact data ymo :       = = =

−

1 3

1 exact mo

x y S x No problem ! Solution with noisy data y :

      = + = 7 35 1 9 . .

mo

ε y y

      − = 3 1 . . ε ≈ 1 % of ymo 1 ≈ 1 % of ymo 2

noise

      =       − − 7 35 1 9 81 39 21 10 . , x       = + = 233 40 1 . . ˆ

exact x

e x x

53 % error for x1 77 % error for x2

estimate estimation error

Noise amplification !

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

4 absolute

61 5 316 774 1 ) (

1

. . . ka = = = =

−

ε e ε ε S ε

x

8 65 11 37 316 16 3 774 1 ) (

1 1

. . / . . / . / / / / k

mo exact mo mo r

= = = =

− −

y ε x e y ε y S ε S ε

x 2 1 2 1 2

: norm) (L2 distance Euclidian

/ j j

u         = ∑

=

u

coefficients of amplification of measurement error (leverage)

relative

exact

ˆ x x ex − =

estimation error

mo

y y ε − =

Measurement error (noise)

958 ) ( cond ) ( = ≤ S ε

r

k

maximum:

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

9 ) ( det = S

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Model(s) Heat flux Computed Temperatures Direct problem Measured temperatures Retrieved heat flux ? Inverse problem

i n v e r s e method

Example of inverse heat conduction problem (IHCP)

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

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Example 2: Different inverse problems fo steady state 1D heat transfer through a wall

Thermal model for exact output of sensor Ts

• homogeneous material

(conductivity λ)

• steady state
• stimulation q (x = 0)
• 1D heat transfer
• no internal source
• Fourier law

Physical system with sensors

• plane wall
• 2 temperature sensors:

1 on rear face (exact measurement Te) 1 inside (x = xs ; noisy measurement y) Possible objectives (types of inverse problems)

• flux q entering the wall (x = 0) ?
• front face temperature T0 ?
• internal temperature distribution T (x) ?
• conductivity λ ?

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

7 λ λ η / x q T T q x T y

x mo

− ≡ = =

1

) , , ; (

dependent variable

• r explained

variable parameters model structure

e e x x

T T q x T x T = = ∂ ∂ − = ∂ ∂

= =

and with

2 2

λ

State equations: T ? Assumption: λ known, no error for Te no error for e and xs Objective: find q, T0, Tx Output equation:

ε + =

s

T y

measurement exact unknown temperature noise

p.d.f: E ( ε ) = 0 E (ε2) = σ2 Random variable

σ ε

Measurement: Estimation = exact matching:

estimate Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011 e s s s s s

T e x T e x T x T q x T +       = ≡ =

2 1

• 1

) /e, ( ) , , ; ( η λ η       ≡

e s

T T e x , ,

2

η ) /e, (

2

T ˆ x T y

s s

η = =      

e s

T T ˆ e x , ,

2

η

8

Solution of inverse problem: estimation of T0

e / x x T x x y x T ˆ

s s e s s s

* * * *

= − − − = with 1 1 1

) 1 ( d a ) ( E ) 1 (

* *

s T s T

x / n e x / e − = = ⇒ − = σ σ ε

Estimation error:

e / x x T x x x y x x T ˆ T ˆ x x T

e s s s x

= − − + − = = =

* * * * * * *

with 1 1

• 1

) , ( ) (

2 recalc

η

Estimation of T (x) :

* *

1

• 1

with

s Tx Tx

x x K K K e − = = ⇒ = σ σ ε

Estimation error:

Good estimation of T0 for shallow measurement

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

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* *

1

• 1

wit

s Tx

x x K h K − = = σ σ Two regions for estimation of Tx

• in between measurements points

interpolation = attenation of error well-posed problem (Hadamard, 1902):

• solution exists
• it is unique
• it depends continuously of the data

] [ e , x x

s

∈ 1 ≤ K

• outside measurements points interval

extrapolation = attenation of error ill-posed problem (Hadamard, 1902)

[ [

s

x , x ∈ 1 > K

e x K e xs ≠ ∀ ∞ → ⇒ →

Very bad design !

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

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SNR x q / x e x e e x e T y q ˆ

* s q s q s q s e

1 1 1 − = ⇒ − = ⇒ − = − − = σ σ λ σ ε λ λ

Estimation of flux q : Estimation error: σ )/ ( T T SNR

e −

=

signal over noise ratio

e = 0.2 m - λ = 1 W.m-1.K-1 - T0 – Te = 30° C xs = 0.18 m - σ = 0.3° C

% 10 and 100 = = ⇒ q / SNR

q

σ % 2 m 10 2 = ⇒ = = q / . / e x

q s

σ

Numerical application: mid-slab measurement:

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

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Errors for parameters "assumed to be known"

Assumption: λ known, no error for Te no error for e error for xs Objective: find q, T0, Tx

δ + =

s nom s

x x

nominal (« a priori ») location of sensor (deterministic) exact location (random) location error (random)

p.d.f: E ( δ ) = 0 E (δ 2) =

σpos δ

2 pos

σ

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

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( )

( )

pos pos 2 pos 2 2 2 pos 2 2 2

with 1 e / ) ( ) ( var σ σ σ σ ε σ / e R R / SNR T T ' '

e

= + = − + = =

ε δ ε ε η ε η + − = + = + = e T T ' ' T T , e / x T T , e / x y

e e nom s s

)/ ( with ) , ( ) , (

2 e 2

e = 0.2 m - λ = 1 W.m-1.K-1 - Te – T0 = 30° C xs = 0.18 m - σ = 0.3° C - σpos = 2 mm

100 = ⇒ SNR

Numerical application: 100 2 200

pos pos

= = = / / e R σ

% 1 14. q /

q

= ⇒ σ

Signal and model : Equivalent temperature noise :

' SNR x q /

* s q

1 1 1 − = σ

' T T ' SNR

e

σ )/ ( − = Estimation error:

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

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Assumption: no error for Te no error for e no error for xs error for λ λ λ λ Objective: find q, T0, Tx

λ

λ λ e

exact nom

+ =

nominal (« a priori ») conductivity (deterministic) exact conductivity (random) conductivity error (random)

p.d.f: E ( eλ ) = 0 E (eλ

2) =

σλ δ

2 λ

σ

( )

        − +       + − − = + − − + = − − =

e s exact s e s exact e s s exact s e nom

T T e x e T T T T x e e x e T y q ˆ ε λ λ ε λ λ

λ λ

1 1 ) (

exact

/ e λ

λ

assumptions: - small

• large SNR

σ ε λ ε λ

λ λ

SNR e q e T T e q e q

exact exact q e s exact exact q exact

1 1 + = ⇒         − + + = +

( )

2 1 2 2 2

1

/ exact exact q

SNR q         + ≈ λ σ σ

λ

Estimation error: Estimation of flux q :

e = 0.2 m - λ = 1 W.m-1.K-1 - Te – T0 = 30° C xs = 0.18 m - σ = 0.3° C - σpos = 0 mm σλ = 0.1 W.m-1.K-1

Numerical application:

% 1 10. q /

q

= ⇒ σ

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

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Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

Thank you for your attention !