Sequential Design of Computer Experiments forNumerical Dosimetry

Marjorie Jala(Telecom ParisTech/Orange Labs)

Ph.D. supervisors : Celine Levy-Leduc, Eric Moulines, EmmanuelleConil and Joe Wiart

MASCOT-SAMO 2013 * Ph.D. Students Day * July 1st 2013

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MotivationSequential Design of Computer Experiments ...Estimation of the α-quantile qα of the distribution of Y � f pXq, for agiven α in p0,1q,

qα � inf tq : PpY ¤ qq ¡ αu .� f is an unknown, expensive-to-evaluate real-valued function� X is a random vector having a known distribution on a compactsubset A � R

d .

We aim at estimating qα by using as few evaluations of f as possible

... for Numerical DosimetryAt wich level are fetuses exposed to RadioFrequency Electromagnetic Fields ?

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Background: Gaussian Process ModellingAssume that f is a sample of a zero-mean Gaussian process (GP)having a covariance function k : GPp0, kp., .qqConditionally to yt � py1, . . . , yt q1, the mean µtpuq and covariancekt pu, vq are given by

µt puq � ktpuq1K�1t yt ,

kt pu, vq � kpu, vq � kt puq1K�1t ktpvq ,

where ktpuq � rkpx1,uq . . . kpxt ,uqs1, 1 denotes the matrixtransposition, Kt � rkpxi , xjqs1¤i ,j¤t , u and v and the xi ’s are in A.

Covariance functionSince the SAR is supposed to be smooth, we shall use the squareexponential covariance function

kSEpu, vq � exp��}u � v}2

2ℓ2

,u, v P A , ℓ ¡ 0 ,

where }u} denotes the euclidean norm of u in Rd .

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Background: MethodologiesSequential strategiesBayesian optimization: find the maximum of f , optimizing anacquisition function� Expected Improvement [Vazquez et al., 2010]� Confidence Bound Criteria (GP-UCB [Srinivas et al., 2010],

Branch and Bounds [De Freitas et al., 2012])EI has been adapted for� Contour estimation [Ranjan et al., 2009]� Estimation of PpY ¥ sq where s is a given threshold (SUR) [Bect

et al., 2012]

Quantile estimation� Non sequential approach [Oakley, 2004]� Extension of the SUR criterion [Arnaud et al., 2010]We really need a sequential strategy, but improvement based criteriademand Monte Carlo samplings of the GP and the conditional GPs,which made them difficult to use for d ¡ 2

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Quantile estimationWe shall compare the quantile estimators with qα,m defined by

qα,m � inf

#q :

1m

m

i�1

1tf pxi q¤qu ¡ α

+,

where x1, . . . , xm are m fixed points in A.Let A � tx1, . . . , xmu � A .

Pure exploration criterion� Minimizes the global uncertainty on the estimation of f� New point xt�1 to add to the set of t observations:

xt�1 P arg maxxPA

σt pxq .� Propose methodologies more adapted to our quantile estimationissue to realize the exploration-exploitation trade-off

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GPS� Let µUt pxq � µt pxq �?

βtσt pxq and µLt pxq � µt pxq � ?

βtσt pxqwith βt � 2 ln

�π2t2

6

� 2 ln�

mδ

�where m is the cardinal of A� Let qU

α,t and qLα,t be the estimators of the α-quantile of µU

t and µLt

qUα,t � inf

#q :

1m

m

i�1

1tµUt pxi q¤qu ¡ α

+qLα,t � inf

#q :

1m

m

i�1

1tµLt pxi q¤qu ¡ α

+PropositionFor all δ in p0,1q, for all t ¥ 1, with probability greater than p1 � δq,

qα,m P rqLα,t , q

Uα,t s .

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GPS (cont.)

Let Uα,t and Lα,t be the following sets :

Uα,t � !x P A : µUt pxq ¥ qL

α,t

)and Lα,t � !x P A : µL

t pxq ¤ qUα,t

), t ¥ 1.

PropositionWith probability greater than p1 � δq, for all t ¥ 1,|qα,t � qα,m| ¤aβt sup

xPUα,t

σt pxq .Criterionxt�1 to add to the set of t observations is such that:

xt�1 P arg maxxPUα,t

σt pxq .7/16

Illustration : 1D Gaussian Process sample path

0 0.2 0.4 0.6 0.8 1−6

−4

−2

0

2

4

6

0 0.2 0.4 0.6 0.8 1−8

−6

−4

−2

0

2

4

6

0 0.2 0.4 0.6 0.8 1−6

−4

−2

0

2

4

6

t � 3 t � 4 t � 6

0 0.2 0.4 0.6 0.8 1−5

0

5

0 0.2 0.4 0.6 0.8 1−5

−4

−3

−2

−1

0

1

2

0 0.2 0.4 0.6 0.8 1−4

−3

−2

−1

0

1

2

t � 8 t � 10 t � 12

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GPS+� Let Sα,t � A be the compact subset such that

Sα,t �±di�1rx piqmin,t , x

piqmax,t s � � � � � rx pdqmin, x

pdqmaxs. Here x piqmin,t and

x piqmax,t denote the smallest (resp. the largest) i th component ofthe points in Uα,t� where Uα,t � !x P Sα,t�1 : µU

t pxq ¥ qLα,t

)� where Sα,t � txt,1, . . . , xt,mt u Y Uα,t ,� where txt,1, . . . , xt,mt u are mt points randomly chosen in Sα,t� By convention Sα,0 � A.

Criterionxt�1 to add to the set of t observations is such that:

xt�1 P arg maxxPUα,t

σt pxq .Note: Since the size of the grid varies at each iteration of the process, we

use βt � 2 ln�

π2t2

6

� 2 ln� |Sα,t�1|

δ

.

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Illustration : 2D Gaussian Process sample path

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

t � 2 t � 62

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

t � 102 t � 162 10/16

Numerical Dosimetry ?In generalVirtually expose human 3D-models to one source of EMF in order toevaluate the Specific Absorption Rate (the SAR, in W .kg�1)SAR computation in our case is done through Finite Difference inTime Domain (FDTD) methodThe SAR depends on� the geometry of the models� the dielectric properties of the tissues� the type and position of the EMF source

Fetus exposure� Very few models are available� The simulations are expensive in terms of computational load� The preparation of the simulations is very complex

We focus on the fetal brain exposure

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Application I: GPS, Japanese model and plane wave� Plane wave exposure:far field sources (basestations antennas,WiFi boxes)� 900 MHz verticallypolarizedelectromagnetic planewaves with a 1 Volt permeter amplitude� Start by performing 5randomly chosenevaluations of the SARin order to have anestimation of l

0

/2 3 /2

π/4

π/2

3π/4

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Application I: GPS, Japanese model and plane wave(cont.)

1

1.5

2

0

2

4

6

2

3

4

5

6

7

x 10−4

ElevationAzimuth

SA

R in

the

feta

l bra

in (

W/k

g)

1

2

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2223

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3132

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5657

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6263

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6566

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7172

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75

1 1.5 20

1

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6

3D-plot Contour plot and observations

0 10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0 10 20 30 40 50 60 703

4

5

6

7

8x 10

−4

Relative errors Quantile convergence 13/16

Application II: GPS+, Victoria and Samsung GalaxyTab� Model Victoria is sitting

working on herSamsung Galaxy Tabat 3G frequency (1940MHz)� 3 parameters: height,nearness and slope ofthe tablet� Start by performing 20evaluations of the SARfrom a LHS in order tohave an estimation of l

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Application II: GPS+, Victoria and Samsung GalaxyTab (cont.)

20 30 40 50 60 70

1.2

1.4

1.6

1.8

2

x 10−5

t

95%

qua

ntile

of t

he S

AR

(W

/kg)

20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

1.2

t

ℓQuantile convergence ℓ convergence

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Conclusion

� We propose two novel sequential approaches for quantileestimation� Successfully applied to real data coming from numericaldosimetry

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