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  • 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 -quantile q of the distribution of Y f pXq, for agiven in p0,1q,

    q inf tq : PpY qq u . f is an unknown, expensive-to-evaluate real-valued function X is a random vector having a known distribution on a compactsubset A Rd .

    We aim at estimating q 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 tpuq and covariancekt pu, vq are given by

    t puq ktpuq1K1t yt ,kt pu, vq kpu, vq kt puq1K1t ktpvq ,

    where ktpuq rkpx1,uq . . . kpxt ,uqs1, 1 denotes the matrixtransposition, Kt rkpxi , xjqs1i ,jt , 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}222 ,u, v P A , 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 q,m defined by

    q,m inf#q : 1m mi11tf pxi qqu + ,

    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 xt1 to add to the set of t observations:

    xt1 P arg maxxPA t pxq . Propose methodologies more adapted to our quantile estimation

    issue to realize the exploration-exploitation trade-off

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  • GPS Let Ut pxq t pxq ?tt pxq and Lt pxq t pxq ?tt pxqwith t 2 ln2t26 2 ln m where m is the cardinal of A Let qU,t and qL,t be the estimators of the -quantile of Ut and Lt

    qU,t inf#q : 1m mi11tUt pxi qqu +

    qL,t inf#q : 1m mi11tLt pxi qqu +

    PropositionFor all in p0,1q, for all t 1, with probability greater than p1 q,

    q,m P rqL,t , qU,t s .6/16

  • GPS (cont.)

    Let U,t and L,t be the following sets :

    U,t !x P A : Ut pxq qL,t) and L,t !x P A : Lt pxq qU,t) , t 1.PropositionWith probability greater than p1 q, for all t 1,|q,t q,m| at sup

    xPU,t t pxq .Criterionxt1 to add to the set of t observations is such that:

    xt1 P arg maxxPU,t t pxq .

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  • Illustration : 1D Gaussian Process sample path

    0 0.2 0.4 0.6 0.8 16

    4

    2

    0

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    6

    0 0.2 0.4 0.6 0.8 18

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    0 0.2 0.4 0.6 0.8 16

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    t 3 t 4 t 60 0.2 0.4 0.6 0.8 1

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    t 8 t 10 t 128/16

  • GPS+ Let S,t A be the compact subset such thatS,t di1rx piqmin,t , x piqmax,t s rx pdqmin, x pdqmaxs. Here x piqmin,t andx piqmax,t denote the smallest (resp. the largest) i th component ofthe points in U,t where U,t !x P S,t1 : Ut pxq qL,t) where S,t txt,1, . . . , xt,mt u Y U,t , where txt,1, . . . , xt,mt u are mt points randomly chosen in S,t By convention S,0 A.

    Criterionxt1 to add to the set of t observations is such that:

    xt1 P arg maxxPU,t t pxq .

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

    use t 2 ln2t26 2 ln |S,t1| .9/16

  • Illustration : 2D Gaussian Process sample path

    0 0.2 0.4 0.6 0.8 10

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    1

    0 0.2 0.4 0.6 0.8 10

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    t 2 t 620 0.2 0.4 0.6 0.8 1

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    0 0.2 0.4 0.6 0.8 10

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    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 .kg1)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 sourceFetus exposure Very few models are available The simulations are expensive in terms of computational load The preparation of the simulations is very complexWe 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

    /4

    /2

    3/4

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

    1

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    x 104

    ElevationAzimuth

    SA

    R in

    the

    feta

    l bra

    in (

    W/k

    g)

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    3D-plot Contour plot and observations

    0 10 20 30 40 50 60 700

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    0 10 20 30 40 50 60 703

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    8x 10

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    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

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    x 105

    t

    95%

    qua

    ntile

    of t

    he S

    AR

    (W

    /kg)

    20 30 40 50 60 700

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    1.2

    t

    Quantile convergence 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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