Efficient Bayesian Multifidelity Approach in Metamodeling of an Electromagnetic Simulator Response

Transcription

1 Progress In Eectromagnetics Research M, Vo. 54, 47 54, 2017 Efficient Bayesian Mutifideity Approach in Metamodeing of an Eectromagnetic Simuator Response Tarek Bdour 1, 2, *, Christophe Guiffaut 1, and Aain Reineix 1 Abstract Severa computer codes with varying accuracy from rigorous fu-wave methods (highfideity modes) to ess accurate Transmission Line (TL) approaches (ow-fideity mode) have been proposed to sove EMC probems of interference between parasitic waves and wired communication systems. For soving engineering tasks, with a imited computationa budget, we need to buid surrogate modes of high-fideity (HF) computer codes. However, one of their main imitations is their expensive computationa time. Rather than using ony the computationay costy HF simuations, we appy another type of surrogate modes, caed Mutifideity (MF) metamode which efficienty combines, within a Bayesian framework, high and ow-fideity (LF) evauations to speed up the surrogate mode buiding. The numerica resuts of combination of an expensive EMC simuator and a cheap TL code to sove a pane wave iumination probem, show that, compared to Kriging, a reiabe Bayesian MF metamode of equivaent or higher predictivity can be obtained within ess simuation time. 1. INTRODUCTION The anaysis of the couping between high-frequency eectromagnetic fieds and conducting wires has become a primary issue in EMC domain, due to various sources of eectromagnetic fied that can impact the devices connected by power or signa transmission ines. The probem of eectromagnetic fied-to-ine couping can be soved by using different approaches that can be cassified into two main categories with respect to their compexity: the cassica transmission ine (TL) theory [1] and fu-wave methods. The fu-wave approach based on Method of Moments (MoM) and impemented in Numerica Eectromagnetic Code (NEC-2) [2] is one of the we adapted approaches to sove the fied-to-ine probem in both ow and high frequencies. However, when eectricay ong or high ines are invoved, NEC-2 simuator becomes very time consuming, which penaizes the numerica cost to buid a surrogate mode with a reasonabe accuracy. Mutifideity surrogate (MFS) modes have been deveoped to compensate for inadequate expensive high-fideity data with cheap ow-fideity data by modeing the connection between them. The ast two decades have seen a major increase in pubications on mutifideity methods for computationa design. These methods use two types of modes for the evauation of designs: the highfideity mode (HF) provides the reference in terms of accuracy and the ow-fideity (LF) modes produce ess accurate response vaues, but at significanty ower expense. Space Mapping (SM), initiated in [3], is a physica mutifideity method based on a inear transformation (mapping) that makes a coarse mode behave as a fine mode. In eectromagnetics, Kozie and Bander [4] have used the SM technique to mode microwave devices. Received 1 December 2016, Accepted 19 January 2017, Schedued 8 February 2017 * Corresponding author: Tarek Bdour (tarek.bdour@teecom-paristech.fr). 1 EMC and Diffraction Research Team, Research Institute XLIM, Limoges 87066, France. 2 Chaire C2M, COMELEC dept, Teecom- ParisTech, 46 rue Barraut, Paris Cedex, France.

2 48 Bdour, Guiffaut, and Reineix Reying on the success of the geostatistica technique caed kriging [5, 12], the first MFS mode based on Gaussian Process (GP) [6] was introduced by Kennedy and O Hagan [7]. The direct Co- Kriging, introduced by Forrester et a. [8], provides a deterministic formuation for fitting the GP parameters with a non-informative prior. This Co-Kriging approach has been appied in [9] to design metamateria circuits for optimization purposes. However, the variance of the surrogate response coud be underestimated [7]. The Bayesian Co-Kriging mode, deveoped in [10], aows the incorporation of prior information on GP hyperparameters and is based on joint estimation between these hyperparameters. It is important to highight that this method benefits from good computationa characteristics of Bayesian approaches such as having information about the uncertainties of the estimation parameters (through their prior and updated distributions). In [13], the authors have appied a mutifideity Bayesian support vector regression to surrogate the input of panar antenna. To the authors knowedge, this work is the first contribution that appies a Co-Kriging based Bayesian mutifideity framework to eectromagnetics. In this artice, the direct and Bayesian approaches of MF Co-Kriging wi be used to buid surrogates modes for the EMC computer code, and their numerica performances wi aso be compared and discussed. This paper is organized as foows. Section 2 describes the derivation of the transmission ine mode (LF mode). Section 3 gives a brief theoretica background of the mutifideity modeing approach. In Section 4, the adopted mutifideity method is appied to a typica EMC probem, and its efficiency is compared to other metamodeing techniques. Some concuding remarks are provided in Section LOW-FIDELITY MODEL The aim of this section is to derive a simpe Transmission Line mode (ow-fideity mode) to investigate the radiated susceptibiity of a perfect eectric conductor (PEC) cabe iuminated by an externa eectromagnetic fied. The geometry of the studied EMC probem is iustrated in Fig. 1. A wire of ength L and radius a is ocated over an infinite and perfecty conducting ground at height h in the presence of a transverse eectric (TE) pane wave of ampitude E 0 and incidence ange θ. The termina resistances between the wire and the ground are R 1 and R 2. E θ k L I 1 I 2 R 1 R 2 h z Ground Figure 1. Geometry of a PEC ine iuminated by a TE pane wave. y x Using Baum-Liu-Tesche (BLT) equation [1], the induced currents at the ine terminas I 1 and I 2 can be written in a matrix form [ ] as: I1 = 1 [ ][ ] 1 ρ1 0 ρ1 e γl 1 [ ] S1 I 2 Z c 0 1 ρ 2 e γl (1) ρ 2 S 2 where S 1 and S 2 are the votage excitation source terms; γ = jk is the propagation constant at the frequency of interest; Z c is the characteristic impedance of the ine; ρ 1 and ρ 2 are the refection coefficients at resistances R 1 and R 2, respectivey.

3 Progress In Eectromagnetics Research M, Vo. 54, When the height of the cabe above ground h is no onger eectricay sma, the fied variation aong the vertica risers shoud be taken into account. Then, the excited umped sources into both vertica risers have to be incorporated in BLT equation. To simpify this atter, the three transmission ines (cabe and two risers) are supposed to have the same characteristic impedance Z c,sothewave propagation can be considered in a singe ine of effective ength L +2h. The resuting votage excitation sources are given by [11]: [ ] ( S1 = E ( 0 e S jkzh e jkzh) e (γ jk x)l 1 ) ( ) A γ jk x B jk z 2 2 e γl ( e (γ+jkx)l 1 ) ( ) A (γ+jk + (2) B x) jk z with A = cos θ, B =sinθ, k x = k sin θ and k z = k cos θ. 3. MULTIFIDELITY METAMODELING: BAYESIAN CO-KRIGING In this section, the theoretica background of Kriging [12] and Bayesian Co-Kriging [10] approaches are described. Ony the main properties and equations are provided. For further detais, the reader is referred to the mentioned references. Let us suppose that we want to surrogate an expensive computer code output z s (x) with x =[x 1,...,x d ] T a vector of d design variabes. We assume aso that ow fideity versions of this code (z (x)) =1,...,s 1 are avaiabe. These codes are sorted by order of fideity from the ess accurate z 1 (x) to the most accurate z s 1 (x). Our prior beief is that a code eves z (x) canbemodeedas reaizations of Gaussian processes (Z (x)) =1,...,s. For each eve =1,...,s, et define the experimenta design D = {x 1,...,x n } with n sampes, such that D s D s 1... D 1 i.e., n s n s 1... n 1 ), the vectors of true and approximated responses, respectivey z =[z (x 1 ),...,z (x n )] T and Z =[Z (x 1 ),...,Z (x n )] T and the iterativeyformed sets z () =(z 1,...,z )andz () =(Z 1,...,Z ) Kriging In this section, we mode each computer code z (x) as a draw from an independent Gaussian process (GP) distribution [12]. Given the experimenta design of n sampes D and its corresponding response vector z, the Kriging mode Z (x) approximates the true function z (x) as ( ) Z (x) GP μ n (x),s (x) = μ n (x)+s (x) (3) where the deterministic function μ n (x) =f (x) T β is the mean vaue of the Gaussian process; T stands for the transpose; β =[β 0,...,β p ] T is a vector of p regression parameters; f T (x) =[f 1 (x),...,f p (x)] is a vector of p basis functions; the stochastic mode S (x), corresponding to the deviation from μ n at x, is defined as the Gaussian random variabe N (0,σ 2 ), with variance σ2 and a stationary covariance function C (x, x )=Corr[S (x),s (x )] = σ 2 R (x, x, ϕ ). The autocorreation function R describes the correation between two sampes x and x of the d-dimensiona input space, and depends on the vector of hyperparameters ϕ =[ϕ 1,...,ϕ d ]T. It is defined as R (x, x,ϕ )= d k=1 r k ( ) x k,x k,ϕ k where rk (x k,x k,ϕ k ) is a 1D correation function aong the k-th component x k of sampe x. One of the popuar choices for the correation function is the Gaussian function, reated to the weighed distance between the two corresponding points x and x as r k ( x k,x k,ϕ k ) =exp [ ϕ k xk x 2] ( ) k ϕ k 0 where the hyperparameter ϕ k is the weight for the distance aong each design variabe x k(k =1,...,d). (4) (5)

4 50 Bdour, Guiffaut, and Reineix Based on the best inear unbiased prediction (BLUP) [12], the Kriging predicted response at an unobserved point x is a Gaussian variabe Z (x )withmeanμ n (x )andvarianceŝ 2 (x ), defined as μ n (x )=E[Z (x ) z ]=f T ˆβ ( ) + r T R 1 z F ˆβ (6) ŝ 2 (x )=V[Z (x 2 ) z ]= ˆσ [ 1 r T R 1 T r + u ( F T R 1 ) ] F u (7) where the optima Kriging variance ˆσ 2 and the generaized east square regression weights ˆβ are respectivey given by: ˆσ 2 = 1 ( ) TR ) 1 z F ˆβ (z F ˆβ (8) n ˆβ = ( F T ) 1F R 1 T F R 1 z (9) where, for i, j =1,...,n and k =1,...,p, F =[F,ik ]=[fk (xi )] is the design matrix; f is a vector whose k-th eement is fk (x ); R =[R,ij ]=[R (x i,x j, ˆϕ )] is the correation matrix of the experimenta design D ; r is a vector whose i-th eement is r i = R (x i,x, ˆϕ ); the vector of optima hyperparameters ˆϕ is obtained through maximum-ikeihood estimation. In summary, to buid the mode Z s, surrogate of the HF code z s, we start from generating a random experimenta design D s, cacuate the vector of its corresponding responses z s by appying the computer code z s and assume an a priori spatia correation between design variabes (here a Gaussian correation function R s is chosen). After agebraic operations in Eqs. (6) and (7), we can get a Krigingbased prediction μ s n s (x )onanuntriedpointx, and even more we can get an idea about prediction uncertainty at x with the mean squared error ŝ 2 s (x ). This is the key advantage of Kriging, in opposite to other metamodes that cannot provide an inherent error measure Co-Kriging Framework Instead of metamodeing the computer code z s ony from its expensive HF evauations, we manage to combine it, within a unified Co-Kriging framework, to a of its LF s 1versionsz (=1,...,s 1). Since we have hierarchy of s responses, we can use the foowing autoregressive mode proposed by Kennedy and O Hagan [7]: Z (x) =ρ 1 Z 1 (x)+δ (x) Z 1 (x) δ (x) Z 1 (x) [ Z 1 (x) Z ( 1) = z ( 1), β 1, ρ 1, σ 2 1, ϕ 1] (10) where ρ 1 is the adjustment coefficient between two successive eves and 1; denotes the independence reationship; δ (x) modes the discrepancy between the eve and the adjusted eve 1. By convention, Z 1 (x) has the same distribution as δ 1 (x) (ρ 0 =0). For =1,...,s, conditioning on vectors of regression parameters β =[β1 T,...,βT ]T, adjustment coefficients ρ = [ρ 1,...,ρ ] T, variances σ 2 = [σ1 2,...,σ2 ]T and correation parameters ϕ = [ϕ T 1,...,ϕT ]T, the discrepancy function δ (x) is modeed as a Gaussian process, starting from the vector δ (D ) containing the vaues of δ (x) atthepointsind as: δ (x) GP ( f T (x)β,σ 2 r ( x, x )), ϕ (11) From the assumption of conditiona independence between δ (x) andz 1 (x),...,z 1 (x) in Eq. (10), we can separatey estimate the Co-Kriging hyperparameters {(β 1, ϕ 1,σ1 2), (β 2, ρ 1, ϕ 2,σ2 2),...,(β s, ρ s 1, ϕ s, σs)} 2 to buid the s-eve mode of interest Z s (x), surrogate of the most accurate code z s (x), given a the HF and LF observations Z (s) = z (s). There are two main approaches to estimate the Co-Kriging hyperparameters: Forrester s Co-Kriging [8] that uses deterministic parameter estimation with the Maximum Likeihood Estimate (MLE) method and Le Gratiet s Co-kriging [10] which is based on Bayesian approach that integrates prior information in the parameter estimation.

5 Progress In Eectromagnetics Research M, Vo. 54, Bayesian Estimation of Co-Kriging Hyperparameters Using Bayes theorem, the probabiities of Co-Kriging hyperparameters, at eve, are updated to a posterior estimation given the probabiity distributions of their priors and the observation sets Z () = z () [10]. Within this Bayesian framework, not ony the Co-Kriging parameters are estimated with a deterministic mean vaue, but aso their estimation uncertainties are cacuated. In the foowing, we give the main formuas of posterior hyperparameter estimates using the Bayesian approach. For a =1,...,s and thanks to the nested property D D 1, the joint posterior Bayesian distribution of parameters ρ 1 and β has the foowing cosed form: ( ) ( ρ 1 (H T N β R 1 ) 1 H H T R 1 z,σ 2 ( H T R 1 ) 1 ) H (12) with H = z 1 (D )F. For each eve =1,...,s, the variance parameter σ 2 is estimated with a restricted maximum ikeihood method. Its estimate is given by ˆσ 2 =(z H ( ˆρ 1 ˆβ )) T R 1 (z H ( ˆρ 1 ))/(n ˆβ p 1) where ( ˆρ 1 ) is the mean estimate of ( ˆβ ρ 1 ) in Eq. (12). β The predictive distribution Zs (x ) is not Gaussian. Nevertheess, we can obtain cosed form expressions for its mean μ s n s (x )andvarianceŝ 2 s(x ) (expression given in [10]). Finay, the Bayesian Co-kriging-based prediction μ s n s, resuting from the combination of a HF code z s and its LF versions (z ) =1,...,s 1, at an unobserved point x is given by the foowing iterative reationship for a =1,...,s: μ n (x )=ˆρ 1 μ 1 n 1 (x )+μ δ (x ) (13) where μ δ (x) =f T (x) ˆβ ( ) + r T (x)r 1 z F ˆβ ˆρ 1 z 1 (D ) (14) 4. NUMERICAL RESULTS In this section, a pane wave iumination exampe is used to demonstrate the characteristics of the Co- Kriging mutifideity surrogate mode deveoped in Section 3. With reference to the geometry described in Fig. 1, a ine of ength L = 8 m, radius a =0.01 m, and height above the ground h = 4 m is iuminated by a TE poarized pane wave with ampitude E 0 = 1 V/m, at frequency f =20MHz. Thetermina oads are respectivey R 1 =50ΩandR 2 =50Ω. Despite its simpicity, this EMC probem is chosen for didactic purposes. In fact, the corresponding high-fideity simuations can be conducted in a reasonabe amount of time in order to numericay and graphicay show the advantages of the Bayesian mutifideity technique. Here, the output of interest is the magnitude of the induced current response I 1 on the oad R 1.The simuation resuts are obtained by the proposed anaytic BLT mode in Eq. (1) (LF mode) and fu-wave commercia software NEC-2 (HF mode). The average CPU runtime of a singe NEC-2 simuation is 0.67 s on a 2.6 GHz I7 processor computer with 1 GB of RAM, whereas one BLT simuation takes ony 0.52 ms (one LF simuation is 1288 times faster than one HF run). Our purpose is to buid an efficient MF mode ˆM (with s = 2 eves) that repaces the expensive NEC-2 code M and gives accurate predictions for the whoe input space. According to the framework in Eq. (10), BLT and NEC-2 simuations are denoted respectivey by eves 1 and 2. Next, we assume uncertainty (due to randomness or engineering design requirements) in the pane wave incidence ange θ and the vaue of the termina resistance R One-Dimensiona Probem The design variabe is x = θ. The input space is defined by 85 θ 85. The next step is to derive a surrogate mode ˆM(θ) that repaces the intensive NEC-2 mode M(θ) =z2 (θ) =I 1 (θ) using some

6 52 Bdour, Guiffaut, and Reineix HF simuations z 2 (θ i ) i=1,...,n2 and a ot of LF simuations z 1 (θ i ) i=1,...,n1 obtained by the cheapest code (BLT equation). The goa of this section is to compare the numerica performances of metamodes buit using Kriging [6], direct Co-Kriging [8] and Bayesian Co-Kriging [10]. To investigate the accuracy of each metamode, the Mean Reative Square Error (MRSE) of a separate vaidation dataset is chosen here as a criterion to measure the prediction abiity of such metamodes at non-observed ocations. It is defined by MRSE = 1 N v N v i=1 ( M(x i) ˆM(x i ) M(x i ) ) 2,whereN v is the number of vaidation sampes. We choose n 2 = 6 HF training points (D 2 ) to buid the Kriging metamode of the induced current response. It is worth noting that the reative sma budget n 2 = 6 is deiberatey chosen since it is chaenging to any metamode surrogating the mutimoda NEC-2 response iustrated in Fig. 2. The Kriging method is impemented using a Matab code caed DACE [6]. The interpoated Kriging curve ˆM K (θ), using Eq. (6), is iustrated in Fig. 2. The n 1 = 69 LF training points (D 1 ) are generated using a uniform spatia step Δθ =2.5. We reca here the nested property (D 2 D 1 ) noted earier. To buid the MFS modes, the 6 HF points are combined to the 69 LF points to generate the Bayesian Co-Kriging mode ˆMBCoK according to the framework in Eq. (10) and the direct Co-Kriging mode ˆMDCoK according to Forrester s method [8]. Both Co-Kriging curves are potted in Fig. 2. The MRSE of each metamode is computed at N v = 18 random test sampes. The numerica vaues of RMSEs are given in Tabe 1. The resuting reative errors are MRSE BCoK = , MRSE K = and MRSE DCoK = This means that, with ony 6 HF training points, the Bayesian Co-Kriging captures the actua behavior of NEC-2 current response better than Forrester s method and significanty improves the Kriging metamode. The difference between BCoK and DCoK approaches can be expained by the fact that Bayesian approach takes into account the uncertainties of its estimation hyperparameters whie the Direct approach tends to underestimate them. Tabe 1. RMSEs of surrogate modes. Surrogate mode 1D 2D Kriging Bayesian Co-Kriging Direct Co-Kriging LF HF R1 ( Ω ) Figure 2. Metamodeing methods appied to the one-dimensiona probem M(θ) =I 1 (θ) θ ( ) Figure 3. Two-dimensiona uniformy generated random experimenta designs (20 HF and 116 LF sampes).

7 Progress In Eectromagnetics Research M, Vo. 54, In the next section, the Bayesian approach wi be retained as the main MF metamodeing technique Two-Dimensiona Probem In this section, we assume an additiona uncertainty in the vaue of the termina resistance R 1.Thus,the design variabes are x =[θr 1 ] T, and the design space is defined by 85 θ 85 and 50 R Ω. For training input data, n 2 = 20 HF points and n 1 = 116 LF points were randomy seected respecting the nested property (D 2 D 1 ). The corresponding experimenta designs D 2 and D 1 are depicted in Fig. 3. Figure 4 depicts the high-fideity output of the induced current I 1 computed by NEC-2 simuator. As shown in Fig. 5, the Kriging of HF sampes fais to accuratey predict the output of interest compared to the accurate response in Fig. 4. As shown in Tabe 1, the MRSE cacuated at a vaidation set of N v = 100 random sampes gives MRSE K =0.0847, which shows that the 20 HF observation points are not sufficient to buid a metamode with a good predictivity. However, the same set of accurate observation sampes, combined to the 116 ow-eve sampes in the Bayesian Co-Kriging framework in Eq. (10) yieds a reiabe MF metamode with a good mean reative error MRSE BCoK = Moreover, the goodness of fit of the obtained metamode can be graphicay checked by comparing Figure 4. High-fideity response surface using NEC-2 simuations. Figure 5. Response surface of the Kriging metamode using [6]. Figure 6. Response surface of the co-kriging metamode using [10].

8 54 Bdour, Guiffaut, and Reineix Fig. 6 to Fig. 4. Numerica experiences show that we need to add more than 6 HF sampes for the Kriging framework [6] in order to obtain the same mean error as the Co-Kriging metamode, e.g., MRSE K = MRSE BCoK which costs additiona runtime of 4.02 s whie the overa runtime of 116 LF simuations is ony about 0.06 s (6600% runtime gain). This computationa gain highights the strength of the proposed Co-Kriging framework [10], since it aows us to make good ow-cost predictions of the expensive NEC-2 code in a reasonabe computationa time by efficienty combining both expensive and cheap computer code simuations. 5. CONCLUSION In this paper, a methodoogy to speed up the soution of couping between eectromagnetic fieds and terminated ines of finite ength is proposed by means of a metamodeing technique. In fact, a costeffective mutifideity surrogate mode based on a Bayesian Co-Kriging method is buit by combining two independent high- and ow-fideity sampes computed respectivey by NEC-2 and BLT codes. The resuts obtained so far with mutifideity metamodeing on fied-to-wire couping are encouraging. Further tests shoud be performed on more compex cases where we have more than two code eves or more than two design parameters. Aso, we have ony considered fixed designs of experiments. It woud be interesting to investigate sequentia designs to iterativey improve the accuracy of mutifideity metamodes. REFERENCES 1. Tesche, F., M. Ianoz, and T. Karsson, EMC Anaysis Methods and Computationa Modes, John Wiey & Sons, New York, Burke, G. J. and A. J. Poggio, Numerica Eectromagnetic Code (NEC)-method of moments, Technica Report, Nava Ocean Systems Center, San Diego, Bander, J., R. Biernacki, S. Chen, P. Grobeny, and R. Hemmers, Space mapping technique for eectromagnetic optimization, IEEE Transactions on Microwave Theory and Techniques, Vo. 42, No. 12, , Dec Kozie, S. and J. W. Bander, Space-mapping modeing of microwave devices using mutifideity eectromagnetic simuations, IET Microwaves, Antennas & Propagation, Vo. 5, No. 3, , Feb Krige, D. G., A statistica approach to some mine vauations probems at the Witwatersrand, Journa of the Chemica, Metaurgica and Mining Society of South Africa, Vo. 52, , Lophaven, S. N., H. B. Niesen, and J. Søndergaard, Aspects of the Matab Toobox DACE, Report IMM-REP , Informatics and Mathematica Modeing, DTU, 44 pages, Kennedy, M. C. and A. O Hagan, Predicting the output of a compex computer code when fast approximations are avaiabe, Biometrika, Vo. 87, No. 1, 1 13, Forrester, A. I. J., A. Sobester, and A. J. Keane, Mutifideity optimization via surrogate modeing, Journa of Mechanica Engineering Science, Vo. 463, , Bradey, P. J., A mutifideity based adaptive samping optimisation approach for the rapid design of doube-negative metamaterias, Progress In Eectromagnetics Research B, Vo. 55, , Le Gratiet, L., Bayesian anaysis of hierarchica mutifideity codes, SIAM/ASA J. Uncertainty Quantification, Vo. 1, No. 1, , Lugrin, G., N. Mora, F. Rachidi, and S. Tkachenko, Eectromagnetic fied couping to transmission ines: A mode for the risers, 2016 Asia-Pacific Internationa Symposium on Eectromagnetic Compatibiity (APEMC), , Shenzhen, Sacks, J., W. J. Wech, T. J. Mitche, and H. P. Wynn, Design and anaysis of computer experiments, Statistica Science, Vo. 4, No. 4, , Jacobs, J. P., S. Kozie, and S. Ogurtsov, Computationay efficient muti-fideity Bayesian support vector regression modeing of panar antenna input characteristics, IEEE Transactions on Antennas and Propagation, Vo. 61, No. 2, , Feb