application of design-of-experiment methods and …bilicz/research/phd/phd_bilicz_pres.pdf ·...

Motivation,framework andobjectives

Eddy-Current Testing

Surrogate Modeling &Design-of-Experiments

Research goals

Optimization-basedinversion

Expected Improvement

Simple illustration

ENDE examples

Conclusions

Surrogate modelingby functional kriging

Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Output space-filling

Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Application of

Design-of-Experiment Methods and Surrogate Models in

Electromagnetic Nondestructive Evaluation

PhD Defense, 30 May 2011

Sándor BILICZ

“Co-tutelle” thesis

Advisors:

Marc LAMBERT and Szabolcs GYIMÓTHY

Département de Recherche en ÉlectromagnétismeLaboratoire des Signaux et Systèmes

UMR8506 (CNRS-SUPELEC-Univ Paris Sud 11)

Department of Broadband Infocommunications and Electromagnetic TheoryBudapest University of Technology and Economics

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Outline

Introduction & framework

Electromagnetic Nondestructive Evaluation – illustrativeexample

Surrogate Modeling and Design-of-Experiments – some generaltools

Research goals

The backbone of the work: three independent approaches

Self-consistent entities

Same illustrative examples

Perspectives

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Eddy-Current Nondestructive Testing (ECT)

Probe coil

SpecimenDefect

Magnetic field

Eddy-current paths

Conductive specimen

Enclosed defect

Magnetic field→ eddy-currents

Defect→ impedance variation

Forward problem:defect→ response

Model/simulation

Usually straightforward

Computationally demanding

Inverse problem:response→ defect

Ill-posedness

A priori assumptions ?

Forward solver needed

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Our nondestructive evaluation framework

Parametric defect descriptionp = [p1, p2, . . . , pN]

T

︸︷︷︸input vector

∈ P ⊂ RN

︸︷︷︸input space

(geometry, constitutive properties, . . . )

Functional responseF p︸︷︷︸

forwardoperator

= q(t), t ∈ T︸︷︷︸output data

(measurement over a volume and/or time. . . )

Black-box model ofF

No need to know the inner structure Mathematical model: Maxwell’s equations Simulator:expensive-to-run(FEM, FD, integral eq.+MoM, . . . )

Simulator at hand

Thin crack(s)→ plane surface(s), current-dipole layer

Surface integral equation + MoM→ linear system of equations

Computer code: J. Pávó, CEA & BUTE project4 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Single-crack configuration

1) 2-parameter example (A = B = 0)

1 mm≤ L ≤ 10 mm5 %≤ D ≤ 90 %

2) 4-parameter example

−2 mm≤ A ≤ 2 mm−3 mm≤ B ≤ 3 mm

1 mm≤ L ≤ 10 mm5 %≤ D ≤ 90 %

r1

r2

ba

x

yy

z

L L

DA

B

d

coilcoilcrack

scanned area

|∆Z| (mΩ)

x (mm)

y (

mm

)

−2 −1 0 1 2−8

−6

−4

−2

0

2

4

6

8

5

10

15

20

25

30

35

40

45

arg ∆Z (rad)

x (mm)

y (

mm

)

−2 −1 0 1 2−8

−6

−4

−2

0

2

4

6

8

−0.5

0

0.5

1

1.5

2

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Double-crack configuration

3) 6-parameter example (D1 = D2 = 40 %)

−1 mm≤ A ≤ 1 mm0.25 mm≤ h ≤ 2 mm−3 mm≤ B1,B2 ≤ 3 mm

1 mm≤ L1, L2 ≤ 10 mm

4) 8-parameter example

−1 mm≤ A ≤ 1 mm0.25 mm≤ h ≤ 2 mm−3 mm≤ B1,B2 ≤ 3 mm

1 mm≤ L1, L2 ≤ 10 mm5 %≤ D1,D2 ≤ 80 %

r1

r2

ba

x

yy

zA

d

L1L1

D1

h

B1

L2L2

D2

B2

coilcoilcrack

scanned area

|∆Z| (mΩ)

x (mm)

y (

mm

)

−2 −1 0 1 2−8

−6

−4

−2

0

2

4

6

8

10

20

30

40

50

60

70

80

90

100

arg ∆Z (rad)

x (mm)

y (

mm

)

−2 −1 0 1 2−8

−6

−4

−2

0

2

4

6

8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Surrogate Modeling (SM) and Design-of-Experiments (DoE)

“Expensive” true modelF

SurrogateModeling

−−−−−−−−−→ Approximate modelF Reducing computational complexity

(CPU, memory, price of commercial code, . . . )

Often interpolation supported by some pre-computed results

Fp1, Fp2, . . . , Fpn −→ Fp

Criterion: “small error” using “few” samples

Choice ofp1, p2, . . . , pn −→ Design-of-Experiments Systematically performing observations for a specific purpose

n = ? Error = ?

“Real” experiments / computer simulations

Classical methods (pi )

Model-oriented methods (pi , F )

Adaptive methods (pi , F , Fpi )

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Sampling by means of classical Design-of-Experiments

Goals

filling “well” P by the samplesp1, p2, . . . , pn

representing “well” the effect of each coordinate ofP

. . .

Examples

Full-Factorial (FF) design:K levels inN dimensions:n = KN

Latin Hypercube (LH) design:n intervals along each axis→ n points

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Approximation of scalar and functional data by kriging

Scalar (classical,∼1960) Functional (recent,∼2004)Observations: Q(p1), . . . , Q(pn) qp1(t), . . . , qpn(t)

Model: Gaussian processξ(p) Functional Gaussian proc.ξp(t)

• Mean: m(p) m(p, t)

• Covariance: k(pi − pj) kT(pi − pj) =

∫

Tk(pi − pj , t)dt

Prediction: ξ(p) =n∑

i=1λi(p)ξ(pi) ξp(t) =

n∑i=1

λi(p)ξpi (t)

Properties: “Best” Linear Unbiased Predictor (BLUP)

Coefficients: linear system of∼n equations

Interpolation : Q(p) =n∑

i=1λi(p)Q(pi) qp(t) =

n∑i=1

λi(p)qpi (t)

“Uncertainty”: σ2(p) = E

[(ξ(p)− ξ(p)

)2]

σ2T(p) =

∫

TE

[(ξp(t)− ξp(t)

)2]

dt

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Choice of the covariance

Parametric covariance model: the Matérn function

k(h |σ20, ν) = σ

20(2√ν h)ν Kν(2

√ν h)

2ν−1Γ(ν), h =

√√√√ N∑k=1

(p(k)

i − p(k)j

ρi

)2

Variance:σ20

Smoothness:ν

Scaling factors:ρ1, ρ2, . . . , ρN →anisotropic distance

more. . .

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ν = 0.25

ν = 1

ν = 4

hρ (ρ = 0.25)

k(h)

Estimation of the parameters:

(Restricted) Maximum Likelihood

Cross-Validation

Numerical implementation (partly) by E. Vazquez, Supélec10 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Research goal and applied methodology

Efficiency improvement of ENDE forward and inverse problem solutions byinvolving design-of-experiment and surrogate modeling schemes:

1. An optimization-based inversion scheme

2. A generic surrogate model to reduce the interpolation error

3. A generic surrogate model with optimally distributed output dataq1(t), q2(t), . . . , qn(t) ( · · · → the inverse problem )

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives



1. An optimization-based inversion scheme Global minimization of a data-misfit function Kriging model Adaptive sampling (“expected improvement”)



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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Optimization-based inversion

Measured data:q(t)

Parametric model:q(t) = Fp Data-misfit→ objective function:Q(p) = ‖q(t)−Fp‖ The regularized inverse problem:

p⋆ = arg minp∈P

Q(p)

Simulator

Measurement Strategy

q(t)

Fpp

Q(p)

Challenges:Q(p) expensive-to-evaluate

might be multimodal

−→ “Efficient Global Optimization”

kriging model (Surrogate Model)

sequential sampling (DoE)

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

The “Expected Improvement” sampling criterion

Sequential sampling for the solution of

p⋆ = arg minp∈P

Q(p), p1, p2, p3, . . . p⋆

0. Initial observations:Q(p1), Q(p2), . . . , Q(pn)

1. Qmin = min[Q(p1), . . . , Q(pn)

]

2. Kriging prediction:ξ(p) ∈ N(

Q(p), σ2(p))

(normally distributed)

Improvement atp by evaluatingQ there:I(p) = Qmin − ξ(p)

“Expected Improvement” atp:

η(p) := E

[I(p)

∣∣∣ I(p) > 0]

closed-form expression available

3. The next evaluation:pn+1 = arg max

p∈Pη(p)

4. Update the model withQ(pn+1). UNTIL the stopping criterion is met,increasen := n+ 1 and GOTO step1.

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

A simple illustrationHigh expected improvement:

Low Q(p) : “promising” p

High σ2(p) : “unexplored”p ⇒ Balanced local/global search

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Numerical ENDE examples Normalized misfit function:Q(p) =

‖q(t)−Fp‖‖q(t)‖

Synthetic data ( ) Initial samples (N ): Latin Hypercube design Sequential sampling (• ) until n = nmax

N 2 4 6 8ninit 10 40 60 80nmax 40 160 240 320

A 2-parameter example (L = 2.5 mm,D = 75 %)

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Numerical ENDE examples

A 2-parameter case

0

0.05

0.1

0.15

0.2

Qm

in

0 5 10 15 20 25 30 35 40

−8

−6

−4

−2

0

Iterations

lg η

max

A 6-parameter case

0

0.05

0.1

0.15

Qm

in

0 50 100 150 200−20

−15

−10

−5

Iterations

lg η

max

A 4-parameter case

0

0.2

0.4

Qm

in

0 20 40 60 80 100 120 140 160

−4

−3

−2

−1

0

Iterations

lg η

max

A 8-parameter case

0.1

0.2

0.3

Qm

in

0 50 100 150 200 250 300

−3

−2

−1

Iterations

lg η

max

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Conclusions

Pros

Usually “few” iterations F is black-box→ generalization Gradient-free Implementation:

→ access to the precision ofdefect reconstruction

Cons

Not always few iterations Initial design ? Stopping criteria ? Gradient-free “Curse-of-Dimensionality”

Covariance parameters Exponential increase ofP

Perspectives

Gradient available: improved kriging model / stopping criterion

Combined stopping criteria (n, Qmin, ηmax)

Two-stage optimization

Related articleS. Bilicz, E. Vazquez, M. Lambert, Sz. Gyimóthy, and J. Pávó,“Characterization of a 3D defect using the Expected Improvement algorithm,”COMPEL, 28(4), pp. 851-864, 2009.

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives






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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives




2. A generic surrogate model to reduce the interpolation error Functional kriging model ofF Error estimation in a statistical framework Adaptive sampling


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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Interpolation-based generic surrogate modeling

A 2-stage approach

1. Database:Dn =(p1, q1(t)) , . . . , (pn, qn(t))

, qi(t) = Fpi

time-consuming, performed onlyonceby “experts”

2. Interpolation:Fp ≃ Fp, ∀p ∈ P and Fpi ≡ Fpi, ∀pi

“real-time”, performed by the“end-user”

Properties

Error: ε(p) = ‖Fp − Fp‖,

with a function norm, e.g.:‖·‖ ≡√

1|T|

∫

T

| · |2dt

General purpose→ general criteria forε(p) all overP :

εmax = maxp∈P

ε(p), εRMS =

√1|P|

∫

p∈P

ε2(p)dp, . . .

Reducedn, as far as possible

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

. . . how to choose the observations?−→ DoE

Classical sampling plans: Full-Factorial, Latin Hypercube, . . .

. . . are not adapted either toF or qi(t)

Adaptive sequential DoE

Initial samples F

Add sample

No

Stop? Final dataset

Small number of initial samples (Dninit )

Add samples one-by-one driven by asmart strategy

Stop when the stopping criterion is met

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Sequential sampling strategy:p1, p2, . . . , pn → pn+1 =???

Implicit formulas

pn+1 = arg minp∈P

[ε(n+1)max

], pn+1 = arg min

p∈P

[ε(n+1)RMS

], . . .

Towards realizable explicit approximations

pn+1 = arg maxp∈P

[ε(n)(p)

]still computationally intractable. . .

Variance-driven sampling: Kriging (trace) varianceσ2

T(p) ∼ interpolation errorε(p)

Use the “jackknife-variance”σ2T,Jack(p)

more. . .

Naive sampling:

pn+1 = arg maxp∈P

[σ2

T,Jack(p)]

Two-factor product (heuristic):

pn+1 = arg maxp∈P

[σ2

T,Jack(p)︸︷︷︸“adaptivity”

factor

× mini=1,2,...,n

‖p − pi‖

︸︷︷︸input space-filling

]

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

A simple illustration:R 7→ R polynomial function (not functional kriging)

Auxiliary function: Q(p) = σ2T,Jack(p)× min

i=1,2,...,n‖p− pi‖

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Numerical ENDE examples

Initial samples: a classical input space-filling (Fractional-Factorial)

Stopping criterion:n = nmax

Comparisons:

Adaptive sampling +functional kriging

1 2 3 4 5 6 7 8 9 105

20

40

55

75

90

L (mm)

D(%

)

Full-factorial sampling +multilinear interpolator

1 2 3 4 5 6 7 8 9 105

20

40

55

75

90

L (mm)

D(%

)Full-factorial sampling +

functional kriging

1 2 3 4 5 6 7 8 9 105

20

40

55

75

90

2

4

6

8

10

L (mm)D

(%)

ε(p) is evaluated at a finite number of test points

εmax andεRMS are approximately computed

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

The 2-parameter example

9 16 25 360

5

10

15

20

25

30

Samples

εm

ax (

mΩ

)

Kriging full−fact.

Kriging adaptive

Multilinear

9 16 25 360

1

2

3

4

5

6

7

8

9

Samples

εR

MS (

mΩ

)


Kriging adaptive

Multilinear

The 4-parameter example

81 144 2565

10

15

20

25

30

35

Samples

εm

ax (

mΩ

)


Kriging adaptive I

Kriging adaptive II

Multilinear

81 144 2562

3

4

5

6

7

8

9

10

Samples

εR

MS (

mΩ

)


Kriging adaptive I

Kriging adaptive II

Multilinear

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Conclusions

Pros

Two stages→ “off-line” simulator

Fully adaptive

F is a black-box→ generalization

Cons

Does the interpolation always make sense?

Heuristic generation rule

Reliable, error-based stopping rule?

Rapidly growing computational cost(jackknifing)

Perspectives

Constant mean function→ more complicated mean functions series expansion of the data + kriging on the residual

Improvement when gradient available?

Applications: inversion, uncertainty propagation, . . .. . . and non-ENDE setups, e.g., forest characterizationmore. . .

Related articleS. Bilicz, E. Vazquez, Sz. Gyimóthy, J. Pávó, and M. Lambert“Kriging for eddy-current testing problems,”IEEE Transactions on Magnetics, 46(8), pp. 3165-3168, 2010.

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives






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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives






Measure of sampling uniformity in the output domain Adaptive sampling Kriging → numerically tractable algorithm Characterization of inverse problems

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Spaces, sampling, space-filling

RN ⊃ P︸︷︷︸inputspace

∋ p 7→ Fp = q(t) ∈ Q︸︷︷︸outputspace

⊂ L2(T)

SamplingDn =

(p1, q1(t)) , (p2, q2(t)) , . . . , (pn, qn(t))

, qi(t) = Fpi

a discrete representation ofF

Repartition of samples inP andQ → properties of representation

Input space-filling: classical DoE (Full-factorial, Latin hypercube, . .. )

Output space-filling: adaptive DoE Criteria forq1(t), q2(t), . . . , qn(t)

Inherently more difficult (F : P 7→ Q )

Iterative / incremental realization

Our proposal: a combined use of two distance-based criteria

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Output space-filling. . . How and why?

Our criteria forq1(t), q2(t), . . . , qn(t)

1. Fill the wholeQ: maxq(t)∈Q

[min

i‖q(t)− qi(t)‖

]< ∆

2. With evenly spaced samples:max

i

[minj 6=i

‖qi(t)− qj(t)‖]

mini

[minj 6=i

‖qi(t)− qj(t)‖] = 1

Expected benefits of such output space-filling databases

Addressing the inverse problem:

Higher precision due to the better describedQ

“Meta-information” from the structure of such database

→ Characterization of the inverse problem

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

How does the difference ofP- andQ-filling look like?

F : a simpleR2 → R2 analytical function

q1 = coshπp1

2cos

πp2

2,

q2 = sinhπp1

2sin

πp2

2.

Input space-filling(Full-factorial design inP)

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 30

0.5

1

1.5

2

2.5

p1

p 2

q1

q 2

Output space-filling(Adaptive sampling)

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 30

0.5

1

1.5

2

2.5

p1

p 2

q1

q 2

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Generation of output space-filling databases I.

Governing tool: the distance functions

Qi(p) = ‖Fp − qi(t)‖, i = 1, 2, . . . , n

Adaptive sampling (in theory)

0. Initialization: choosep1 ∈ P

D1 =(p1, q1(t))

n := 1

1. Sample insertion:pn+1 = arg maxp∈P

[min

iQi(p)

]

Dn+1 = Dn ∪ (pn+1, qn+1(t)) n := n+ 1

2. Until a stopping rule is met, GOTO step1.

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

A simple illustration:R27→ R2 analytical function (as before)

Input spaceP Output spaceQ

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Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Generation of output space-filling databases II.

Sample insertion: numerically intractable algorithm

pn+1 = arg maxp∈P

[min

iQi(p)

]Kriging model

of Qi(p)−−−−−−−−−−−−→ pn+1 = arg maxp∈P

[min

iQi(p)

]

Adaptive sampling (realizable)

0. Initialization: choosep1, p2, . . . , pm ∈ P

Dk =(p1, q1(t)) , . . . , (pm, qm(t))

n := m

1. Sample insertion:pn+1 = arg maxp∈P

[min

iQi(p)

]

Dn+1 = Dn ∪ (pn+1, qn+1(t)) n := n+ 1

2. Sample removal (if needed):k = arg mini

[minj 6=i

‖qi(t)− qj(t)‖]

Dn−1 = Dn \ (pk, qk(t)) n := n− 1

3. Until a stopping rule is met, GOTO step1.

32 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

The sampling algorithm

Kriging prediction

Initial observationsStart

N

New observation

Remove?Stop?End

N

Remove sample

Initial observations: classical input space-filling

Sample removal: in every second cycle (other rules can also be applied)

Stopping rule:n = nmax

33 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Illustration via ENDE examples I. :Visualization of the output data inL2(T)

Multidimensional scaling (MDS):q1(t), . . . , qn(t) → π1, . . . , πn ∈ R2,

such that‖qi(t)− qj(t)‖ ≃ ‖πi − πj‖, ∀i, j

Input samples

L (mm)

D (

%)

1 2 3 4 5 6 7 8 9 105

20

40

55

75

90

L (mm)

D (

%)

1 2 3 4 5 6 7 8 9 105

20

40

55

75

90

Output samples by MDS

−20 0 20 40 60 80−15

−10

−5

0

5

10

15

π1 (mΩ)

π2 (

mΩ

)

−40 −20 0 20 40 60−15

−10

−5

0

5

10

π1 (mΩ)

π2 (

mΩ

)

34 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

What can an output space-filling database be used for?Inverse mapping of a noise levelδ

inverse problem:q(t) → p “noise level” :‖q(t)−Fp‖ ≤ δ −→ “noise patch” inP

quantitative characterization of the ill-posedness

P

Q

δδ

δ

δ

δ

p1

p2

F

inverse mapping: computationally expensive

at theith sample:‖qi(t)−Fp‖ = Qi(p) distance function

Qi(p) ≤ δkriging−−−−−→ Qi(p) ≤ δ numerically tractable approximation

35 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Illustration via ENDE examples II. : Noise level mappingThe 2-parameter example, output space-filling databases

9 samples,δ = 1 mΩ

L (mm)

D (

%)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 105

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90


L (mm)

D (

%)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 105

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90


L (mm)

D (

%)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 105

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90


L (mm)

D (

%)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 105

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

36 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Conclusions

Database generation

New adaptive DoE scheme: output space-filling

Samplingdistance functions−−−−−−−−−−−−−→ optimization task

Numerical burdenkriging−−−−−→ tractable algorithm

Inverse mapping

Tools involved in the database generation

Ill-posedness quantitatively accessible

Limitation: “Curse of Dimensionality”Perspectives

Other norms inQ

“Density estimation”→ inhomogeneous distance measure inP

Related articleS. Bilicz, M. Lambert and Sz. Gyimóthy“Kriging-based generation of optimal databases as forward and inverse surro-gate models,”Inverse Problems, 26(7), p. 074012, 2010.

37 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Perspectives

Uncertainty propagation, sensitivity analysis

Uncertain configuration parameters?−→ performance of inversion

Defect parameters: which are the “most important”?

Interpolation-based inversion:

Fp1, Fp2, . . . , Fpn −→ F−1q(t)

p =

n∑

i=1

λi(q(t)) pi

Extension to other nondestructive tests and remote sensing

Ultrasonic NDT

Pulsed eddy current NDT

Forest characterization by means of radar observations

38 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Choice of the covariance back

Parametric covariance model

Matérn:k(h |σ20, ν) = σ

20(2√ν h)ν Kν(2

√ν h)

2ν−1Γ(ν), h2 =

N∑k=1

(p(k)

i − p(k)j

ρi

)2

Estimation ofσ20, ν, ρ1, ρ2, . . . , ρN :

(Restricted) Maximum Likelihood, Cross-Validation, . . .

Illustration: a polynomial test function (black) and the kriginginterpolators (red) based on 5 samples:

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

p

Q(p)

σ20 = 100, ν = 4, ρ = 0.1

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

p

Q(p)

σ20 = 82.3, ν = 2.74, ρ = 0.28

(maximum likelihood estimates)39 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Jackknife variance estimator back

Well-known statistical tool Similar to Cross-Validation (a “goodness of fit”) Directly depends on the observations

(contrarily to the classical kriging variance)

0

0.5

1

1.5

2

f(p)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

p

σ(p) Jackknife

Unknown function

Classical

Kriging interpolation

Controversy: not zero at the sampled points Drawback: computational cost

40 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Forest characterization by radar measurementsback

Inverse problems: e.g., biomass retrieval, mean height and age retrieval(allometric equations: biological properties→ EM parameters)

EM scattering models→ time-consuming simulations→ need of surrogate modeling

An illustrative problem statement: Input: age (a), ground moisture (m) Output: polarimetric backscattering coefficients (σHH , σVV, σHV) Configuration parameters: frequency (f ), incident angle (θ)

σuv = Ff ,Θa,m

2-level adaptive sampling ofF

1. Ff ,Θa,m =n∑

i=1λi(a,m)FS

f ,Θai ,mi upper: functional kriging

2. FSf ,Θa,m =

k∑j=1

λj(f ,Θ)Ffj ,Θja,m lower: scalar kriging

[MSc. Thesis ofAndrásVASKÓ, L2S-SONDRA-BUTE, 2011.]

41 / 38




Research goals



Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions


Adaptive sampling

Simple illustration

ENDE examples

Conclusions

Perspectives

Age

Mois

ture

10 20 30 40 500.1

0.2

0.3

0.4

0.5

0.6

0.7

VV (σ = −28.2 dB)

Age

Mois

ture

10 20 30 40 500.1

0.2

0.3

0.4

0.5

0.6

0.7

VH (σ = −57.0 dB)

Age

Mois

ture

10 20 30 40 500.1

0.2

0.3

0.4

0.5

0.6

0.7

HH (σ = −20.6 dB)

Age

Mois

ture

10 20 30 40 500.1

0.2

0.3

0.4

0.5

0.6

0.7

Combined polarizations

Error maps for(a = 40, mv = 0.25), frequency bandwidth=[1000, 2000] MHz and incidenceangle range=[59, 61]. Black area: error less than 1 dB, dark grey: error between 1 dB and 2 dB,

light grey: between 2 dB and 3 dB. back 42 / 38

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