trust region-based optimization of pki neural models for … region based... · mohamed bakr,3...

Trust Region-Based Optimization of PKI NeuralModels for RF/Microwave Devices

Lakshman Mareddy,1 Mohammad Almalkawi,1 Srinivasa Vemuru,2

Mohamed Bakr,3 Vijay Devabhaktuni1

1 EECS Department, University of Toledo, MS 308, 2801 W. Bancroft St., Toledo, OH 436062 ECCS Department, Ohio Northern University, Biggs 245, 525 South Main St., Ada, OH 458103 Department of Electrical and Computer Engineering, McMaster University, Hamilton,ON L8S 4K1, Canada

Received 28 May 2012; accepted 13 August 2012

ABSTRACT: This article proposes a new trust region-based optimization technique for Ra-

dio Frequency (RF)/microwave devices. The proposed approach is apt for modeling scenar-

ios, where standard ANN multilayer perceptron (MLP) and Prior Knowledge Input (PKI)

models fail to deliver a satisfactory model. This approach feeds output of standard ANN

model as knowledge input to PKI model. The ANN model and the PKI model form a symbi-

otic pair to yield accurate results. In this paper, the dogleg routine is exploited in the pro-

cess of optimization to obtain valid trust region steps. The proposed method is compared

with sensitivity technique via several RF/microwave components. VC 2012 Wiley Periodicals,

Inc. Int J RF and Microwave CAE 23:559–569, 2013.

Keywords: artificial neural networks; device modeling; dogleg; optimization; PKI models;

trust region

I. INTRODUCTION

The design of microwave devices requires tools that are

accurate and fast. Analysis and optimization with detailed

EM models is an important step in the process of design,

but it is computationally expensive. So, Artificial Neural

Networks (ANN), which are computationally cheap and

relatively accurate, are introduced for microwave design

[1–6]. Neural Network models are developed employing

the measured or simulated microwave data by a process

called training. These trained neural models are used in

the process of RF/microwave design and optimization [5].

The training data is specific to the structure of RF and

microwave devices and purely depends on the user speci-

fications for e.g. physical parameters like dimensions of

the structure. However, the advantage of ANNs lies in the

fact that, once an ANN model once trained can be imple-

mented to estimate the output for any given input. Neural

networks have been implemented in the modeling of sev-

eral active and passive microwave components including

microstrip interconnects [7–9], spiral inductors [10], Field

Effect Transistor (FET) devices [11] and patch antennas

[12].

Full wave EM modeling of passive components is

accurate but computationally expensive and very slow.

Although approximate/empirical models are fast, their ac-

curacy is limited. Neural models are more accurate com-

pared to empirical models and faster compared to physics-

based models [5]. Thus the neural models form an effec-

tive alternative with tradeoff between accuracy and com-

putational cost. However, developing accurate neural

models require a lot of training data. Obtaining training

data either by simulations or actual measurements is tedi-

ous. Hence several alternatives such as knowledge-based

neural networks (KBNNs), where additional knowledge

about the device being modeled is embedded into neural

models are introduced. The variants of KBNNs include

the source difference method (DM) [13], prior-knowledge

input (PKI) network [6], and space mapped neural net-

works (SMNNs) [14–16].

In the DM, two neural networks are used. First neural

network, termed as source model is trained to predict the

EM simulation output. On the other hand, the second net-

work, termed as difference model is trained to predict the

difference between the EM simulations and output of the

Correspondence to: V. Devabhaktuni; e-mail: [email protected]

VC 2012 Wiley Periodicals, Inc.

DOI 10.1002/mmce.20690Published online 5 October 2012 in Wiley Online Library

(wileyonlinelibrary.com).

559

source model. Finally, the outputs of both source and dif-

ference models are added to obtain the final estimate of

the desired output. In the PKI method, the source model

output is taken as input in addition to the other inputs.

The PKI method yields more accurate results compared

with the source difference method [6]. Although we

embed knowledge in the process of modeling, there are

issues with the accuracy of the neural models owing to

the complexity of the structure.

This research explores new approaches to ANN model-

ing wherein the accurate PKI model and the not-so-accurate

original model form a synergetic pair which accurately

models the original problem. The proposed approach uti-

lizes a root-finding algorithm which is solved as an optimi-

zation problem. A trust region routine with a dogleg algo-

rithm is used to obtain valid steps in the optimization

process. The proposed approach is compared with sensitiv-

ity technique [17] via two RF/microwave device examples.

The paper is organized as follows: Section II describes

the overview of ANN modeling, Dogleg and Sensitivity

techniques. In section III, the proposed method is intro-

duced. In Section IV, presents the results from applying

the proposed modeling approach to RF/microwave exam-

ples. Finally, the paper is concluded in Section V.

II. OVERVIEW

A. ANN ModelingIn general, the relationship between the input and output

parameters of an RF/microwave device is multidimen-

sional and nonlinear. Such a relationship can be modeled

using a neural network that typically consists of a collec-

tion of interconnected neurons, e.g., Multi Layer Percep-

trons (MLP) networks [18] as shown in Figure 1.

Consider an RF/microwave modeling problem. Let

x[Rn is the input vector and y[Rm is the vector of out-

puts of the model. They are related through the model by

y ¼ f ðxÞ: (1)

In (1), f gives the RF/microwave relationship between

x and y. To illustrate the concept, without loss of general-

ity, we will consider the case where m ¼ 1. In such a

case, y is a scalar and hence nonbold. In general, a 3-layer

MLP or MLP3 network can be used to derive a represen-

tative ANN model fann that closely approximates f. The out-

put of standard ANN model is y ¼ fann(x, w), where w is the

ANN weight vector. The structure of PKI model consists of

an MLP3 network. The difference between the standard

ANN modeling and PKI modeling is that the PKI models

have additional input in the form of prior knowledge as

shown in Figure 2. The output of PKI model is given by

y ¼ fPKIðx; k;wÞ (2)

In (2), k represents prior knowledge input and fPKI rep-

resents the functional relationship between x, k, and y and

w is the ANN weight vector. In general, the source model is

either an approximate/empirical model or physical measure-

ments and its output is employed as the knowledge input.

Contrary to this, in the proposed approach, the output of the

standard ANN is fed as knowledge input of PKI model.

The error/quality measure of both standard and PKI

models is evaluated using

E ¼ 1

p

Xp

i¼1

yðiÞ � f ðxðiÞ;wÞyðiÞ

� ��100: (3)

In (3), p denotes number of training data (x(i), y(i)),

where y(i) is the desired output which is obtained from ei-

ther EM simulations or measurements and f (x(i), w) is the

neural model output when the input is x(i).

B. Dogleg TechniqueThe problem of learning an input–output relation in neural

networks can be seen as minimization of a nonlinear

function. The optimization techniques that use the

Figure 1 Block diagram of standard ANN modeling.

Figure 2 Block diagram of the PKI modeling approach.

Figure 3 The Dogleg Path from yt to C and to N.

560 Mareddy et al.

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 23, No. 5, September 2013

second order derivative information of the cost function

(i.e., quality measure E) provide a quick convergence

towards the solution [19, 20]. The main idea of these

methods is to minimize the cost function E, by a quad-

ratic form given by

EðDytÞ ¼ EðytÞ þ rEðDytÞTDyt þ1

2DyT

t r2EðDytÞDyt;

(4)

subject to ||Dyt|| < dt, and dt is the radius of trust region.

The solution to the above problem is given by

Dyt ¼ � Bt þ ltI½ ��1rEðytÞ: (5)

In (4) and (5), !E (yt) and !2E (yt) are gradient vec-

tor (gt) and Hessian matrix (Bt) of the cost function. Bt is

obtained using the Broyden–Fletcher–Goldfarb–Shanno

(BFGS) update given by

Figure 4 Illustration of proposed modeling approach.

Figure 5 Flowchart of the proposed approach.

Trust Region Based Opt. of PKI Models for RF Devices 561

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

Btþ1 ¼ Bt þBtDytDyT

t Bt

DyTt BtDyt

þ DgtDgTt

DgTt Dyt

: (6)

To solve problems of this nature, the two possible

globalization strategies are line search and trust region

method [20]. The latter approach is preferred since the

line search methods can be computationally expensive as

step length calculations typically requires 3–15 function

evaluations [19]. On the other hand, trust region methods

choose an initial step length and then utilize multidimen-

sional model to choose the step direction [19, 20]. The

trust region denotes a subset of domain of the objective

function to be optimized using the model function (often

a quadratic). If an adequate model of the objective func-

tion is found within the trust region then the region is

expanded; conversely, if the approximation is poor then

the region is contracted [20].

It is often difficult to obtain the solution to Eq. (4)

directly [21]. More specifically, when dt < ||Dyt||, there is

no finite method to determine lt > 0 such that ||Dyt|| ¼ dt.

Hence practical trust region methods attempt to find an

approximate solution only. One such approach is to ap-

proximate the variation of Dyt as a function of lt by a

path called dogleg path which connects piece-wise linear

yt, the Cauchy point (C) and the Newton point (N).

Cauchy point is the minimizer of quadratic model in the

steepest descent direction and Newton point is the iterate

point from DytN (as shown in Figure 3). The Newton step

DytN is given by

DyNt ¼ � Bt½ ��1�gt: (7)

The Cauchy step is given by

DyCt ¼

� gtk k2

gTt Btgt

; if DyCt

�� < dt

� dt

gtk k gt; otherwise

8><>: : (8)

In Figure 3, D is the Dogleg step and is in between

Cauchy step and Newton step. This Dogleg path has two

important properties [22]. One of them is that as we move

along the linear curve from yt to C and then to N, the dis-

tance from yt increases monotonically by construction.

Thus for any dt < ||DytN||, there will be a unique point

ytþ1 on the dogleg path such that ||ytþ1 – yt|| ¼ dt. The

other one is the cost function E (yt) decreases monotoni-

cally as Dyt goes from yt to C and finally to N.

Let DytTR be the trust region step, the possible cases

are presented below;

1. If dt = ||DytC||, then Dyt

TR ¼ DytC;

2. If dt > ||DytN||, then Dyt

TR ¼ DytN;

3. If ||DytC|| < dt < ||Dyt

N||, then Dogleg step is chosen

i.e. DytTR ¼ Dyt and given by

Dyt ¼ DyCt þ k DyN

t � DyCt

� �; k 2 ð0; 1Þ: (9)

And k can be obtained by solving the equation

DyCt þ k DyN

t � DyCt

� �� 2¼ d2t : (10)

C. Sensitivity TechniqueThe robustness of the dogleg technique is compared with the

sensitivity technique presented in [17]. For the benefit of the

readers, the sensitivity technique is presented in this article.

The sensitivity technique is more robust than the steepest-

descent method and combines the advantages of the New-

ton’s method and the steepest-descent method [19]. In this

method, the sensitivity information, i.e., partial derivatives of

the PKI model, to obtain both step-size and update direction.

In other words, derivatives of the PKI model fPKI with

respect to output y need to be computed. Applying the chain

rule of calculus to the PKI model fPKI, we obtain

@y

@y0¼Xm

k¼1

@y

@zk

@zk

@ck

@ck

@y0(11)

and

zk ¼1

1þ e�ck: (12)

In Eqs. (11) and (12), ck is the weighted sum of all inputs

to fPKI, m is the number of hidden neurons in fPKI, and zk

is the sigmoid activation function.

When the PKI model fails to achieve the accuracy

specified by user (i.e., E > Euser, after exhaustive train-

ing), our proposed approach is invoked as an alternative.

For a given input vector x, RF/microwave output y can be

computed using: (i) the not-so-accurate model fann; (ii) the

PKI model fPKI; and (iii) a typical root-finding algorithm.

Details of the procedure are provided in the following section.

III. PROPOSED MODELING APPROACH

In this research work, we propose a new modeling

approach which uses both standard ANN model, PKI

model along with a trust region method for optimization

of RF/microwave devices. First the standard ANN model

is trained using the data obtained from either physical

measurements or EM simulations or both. The output of

standard ANN serves as the knowledge input to the PKI

model. Then the PKI model is trained with inputs and out-

put of standard model as inputs (as shown in Figure 4).

Training of both standard ANN and PKI model concludes

the model-development phase.

In what we term as the model-utilization phase, for a

given input x not-seen during training, the not-so-accu-

rate desired/original model fann helps initialize the

desired output y’ (an approximate of y). This initial esti-

mate is fed as the knowledge input to the PKI model

fPKI. From this point forward, y’ is iteratively adjusted,

until E < Euser is satisfied. Such iterations involve

562 Mareddy et al.


typical optimization routine(s) providing both update

direction and value. As mentioned earlier, Dogleg tech-

nique is implemented and compared with sensitivity

technique [17]. The proposed modeling approach is

depicted in Figure 5. It is worth mentioning that the

flowchart for both Dogleg and Sensitivity techniques is

same and is shown in Figure 5. The proposed approach

is further explained using pseudocode.

Pseudocode (Dogleg Case)

Step 1: For the given data (x1, x2, x3, …, xn), use the pre-

trained less-accurate MLP model i.e. fann to evaluate the

approximate output y’;

Step 2: Feed inputs (x1, x2, x3, ..., xn, y’) to the PKI

model fPKI resulting in y, and y’ is an approximate of y;

Step 3: Evaluate the objective function (Eobj) using eq. 3;

Step 4: If Eobj < <Euser, then go to Step 7 and return y.

Else, evaluate Dy using eq. 4 through 10.

Step 5: Once Dy is obtained, set y’ ¼ y’ – Dy;

Step 6: Go to Step 2 and replace y’ with the adjusted or

modified y’;

Step 7: Return y and STOP.

IV. EXAMPLES

The proposed approach can be applied in the design stage of

diverse RF/microwave components. For example in the

design stage of microwave antennas, filters, etc. where the

geometrical dimensions of the component is used as an out-

put and electrical parameters from the design specifications

such as S-parameters, center frequency can be given as inputs

to yield practical implementations. Two examples are used

to demonstrate the robustness of the proposed approach.

A. Spiral Inductor ModelingInductors are critical components in the design of analog/

mixed signal and RF circuits [23]. Although the structure of

spiral inductor is simple its optimization is complicated [24].

The compact ANN model of spiral induction is given by

din ¼ fannðW;L; freq;wÞ: (13)

In (13), din and W represent the inner diameter and line

width, L represents the inductance, freq represents the fre-

quency and w is the ANN weight vector. The structure of

spiral inductor is shown in Figure 6.

Training and validation datasets are generated using an

electromagnetic (EM) solver, namely, Ansys HFSS, after

fixing the number of turns N ¼ 4.5 and spacing between

the lines S ¼ 5 lm and varying: W (from 1 to 15 lm), din

Figure 6 Geometry of square spiral inductor.

Figure 7 Comparison of outputs from EM simulations versus Standard ANN and PKI models for spiral the inductor example.



Figure 8 Comparison of outputs from EM simulations versus Dogleg and Sensitivity techniques for the spiral inductor example.

Figure 9 Comparison of quality measure E of Dogleg and sensitivity techniques for the spiral inductor example. [Color figure can be

viewed in the online issue, which is available at wileyonlinelibrary.com.]

TABLE I Accuracy of the Proposed Approach as Compared with Conventional ANN, PKI, and Sensitivity Techniquesof Spiral Inductor

Freq (GHz) W (lm) L (mH)

Inner Diameter din (lm)

Actual

Value

Conventional

ANN Model PKI Model

Sensitivity

Technique

Dogleg

Technique

17.2 5 7.502 70 74.191 71.529 71.894 70.529

1 7 5.721 110 105.593 107.899 109.297 109.254

1.4 9 2.483 100 106.350 101.452 101.932 100.452

10.4 3 2.922 40 45.267 43.667 42.478 41.077

1.8 11 10.733 80 84.183 81.165 81.097 80.042

11.6 9 3.413 90 95.081 93.712 92.668 90.257

7.5 7 6.025 30 27.443 24.208 28.401 30.404

12.7 15 10.344 120 111.951 109.087 121.724 119.114

2.8 3 2.892 20 24.275 23.279 21.644 20.725

18 13 9.983 100 106.543 103.472 104.707 101.511

564 Mareddy et al.


(from 20 to 120 lm), and frequency (from 0.1 to 20

GHz), resulting in 2200 simulation data. Data is random-

ized and 1760 data points are employed for training a

standard ANN and 540 for verification. Quality measure Eof the standard ANN and PKI models turned out to be an

unacceptable 10.40 and 6.40%. In such cases, we advocate

to implement the proposed approach. In the proposed

approach for a given input vector x, the accurate model is

developed employing the pseudocode.

Figure 7 presents EM simulations versus standard

ANN and PKI models. To make the figures simpler 120

randomly selected data are used for comparing the inner di-

ameter (din) estimation from the proposed approach with

that from EM simulations. The results yielded from the

Dogleg technique and sensitivity technique are presented in

Figure 8. It is clearly seen that the EM simulation results

are well matched with that of proposed approach. Error

comparison of PKI model and Dogleg technique is pre-

sented in Figure 9. At around sample 46 in Figure 9, the

quality measure (E) is maximum and has a value of 4.26%.

Randomly selected 10 sample data points are presented in

Table I to illustrate the performance of the Dogleg tech-

nique. Dogleg technique yielded major improvement over

the conventional straightforward ANN and PKI approaches

(refer to Table I and spiral inductor example in Table III.

B. Microstrip Array Antenna with 4 Elements ModelingThe antenna structure contains four elements (patches)

and the input is fed by using microstrip transmission line

as shown in Figure 10. The physical parameters of mean-

der line such as the length and width play a dominant role

on resonant frequency in higher mode of operation as the

current path length changes with change in the physical

parameters [25]. The structure is symmetric about the cen-

ter and feed point can be placed anywhere on the bridge

such that the steering produced is zero. The compact

model is given by

L ¼ fannðfreq; S11; d;wÞ: (14)

In (14), L represents the length of the patch, d represents

the width of the meander line, freq represents the fre-

quency and S11 represents the reflection co-efficient of the

antenna.

Training and validation datasets for the structure

shown in Figure 10 are generated using an electromag-

netic (EM) solver, namely, Ansys HFSS, varying: L (from

28 to 32 lm), d (from 9 to 13 lm), and frequency (from

Figure 10 Geometry of four patch array antenna with meander

lines. [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

Figure 11 Comparison of outputs from EM simulations versus Standard ANN and PKI models for antenna example. [Color figure can

be viewed in the online issue, which is available at wileyonlinelibrary.com.]



Figure 13 Comparison of quality measure E of Dogleg and sensitivity techniques for antenna example. [Color figure can be viewed in

the online issue, which is available at wileyonlinelibrary.com.]

Figure 12 Comparison of outputs from EM simulations versus Dogleg and sensitivity techniques for antenna example. [Color figure can

be viewed in the online issue, which is available at wileyonlinelibrary.com.]

TABLE II Accuracy of the Proposed Approach as Compared with Conventional ANN, PKI, and Sensititivity Techniquesof Antenna

Freq (GHz) d (lm) S11 (dB)

Length of the Patch L(lm)

Actual

value

Conventional

ANN Model PKI Model

Sensitivity

Technique

Dogleg

Technique

2.59 13 �5.517 29 29.409 29.338 28.507 29.280

2.35 10 �1.519 29 31.248 28.891 30.529 29.025

2.68 9.5 �3.678 28 26.475 28.267 28.390 28.419

2.53 11 �3.521 29.5 33.254 29.583 31.859 29.691

2.5 12 �3.569 28 30.713 29.838 28.630 28.931

2.32 11.5 �2.711 29.5 27.550 34.476 29.941 29.662

2.53 9.5 �5.173 31 32.158 30.158 30.158 30.158

2.5 9.5 �5.064 31.5 34.061 30.330 30.330 31.330

2.26 9.5 �5.83 31.5 27.411 31.581 30.622 31.609

2.26 9 �0.563 28 27.348 25.445 28.142 28.575

566 Mareddy et al.


2.2 to 2.7 GHz), resulting in 4,131 simulation data. Of

these, 3305 are used for training and 826 samples for test-

ing. Quality measures of standard ANN and PKI models

turned out to be unacceptable 12.10 and 7.51%, respec-

tively. At this juncture, we advise to implement the pro-

posed modeling approach to yield accurate results. For a

given input sample x, the inputs are fed to standard ANN

model then the inputs along with the output of the stand-

ard ANN are fed to PKI model and dogleg technique is

implemented via pseudocode.

The quality measures of standard ANN model, PKI

model and proposed approach are presented in Table III.

To make the plots simpler Figure 11, presents EM simula-

tion versus standard ANN and PKI models. The compari-

son of EM simulations and proposed approach is presented

in Figure 12. The Errors of PKI model and proposed

approach are compared in Figure 13. Randomly selected 10

data samples are shown in Table II to demonstrate the per-

formance of the Dogleg technique. The proposed approach

yielded accurate results compared to PKI model and sensi-

tivity technique as shown in Tables II and III.

Comparison of elapsed time for model utilization phase

of proposed models for entire data set (i.e., 540 samples

for Spiral Inductor and 826 samples for Antenna exam-

ples) based on two optimization routines is presented in

Table IV. It is noteworthy that the runtimes presented in

Table IV does not include the time taken for the training

of ANN and PKI models. The total elapsed time for each

of spiral inductor and antenna examples is presented in

Figure 14. As it can be seen, Dogleg technique led to

faster convergence. Tests are performed on a PC running

on Intel CoreTM2 Duo E8600.

V. CONCLUSIONS

This article introduced a trust region based PKI modeling

approach for efficient CAD modeling applicable in RF/

microwave context. In a situation where standard ANN and

PKI models fail, the proposed approach offers an accurate

and fast model with its structural simplicity duly preserved.

Initializing root-search with the empirical model (i.e., the

standard ANN model) offers a reasonable trust. The Dogleg

technique implemented offers trust in terms of conver-

gence. The proposed approach offers an effective solution

for handling intense computational requirements in the field

of RF/microwave devices. Implementation of this approach

will require generation of data using full-wave EM solvers

and/or high-frequency measurements by varying design pa-

rameters, which could be time-consuming. In the modeling

cases where standard ANN and PKI models fail to deliver

satisfactory results, the proposed approach is a simpler al-

ternative. Output of the less accurate standard ANN model

is used as the initial guess in the proposed pseudocode. As

such, the error between EM/physics simulations and the

estimated value from standard ANN model is small result-

ing in faster rate of convergence. It being the first study in

this direction, the proposed approach covers modeling

cases with only one output. As such, extension to multiple

outputs becomes a natural alternative for future work.

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PKI

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BIOGRAPHIES

Lakshman Mareddy received his

Bachelor of Technology (B.Tech.) in

Electrical and Electronics Engineer-

ing from Jawaharlal Nehru Techno-

logical University, India in May

2010 and Master of Science (M.S) in

Electrical Engineering from the Uni-

versity of Toledo in August 2012.

He received Outstanding Teaching Assistant award from

College of Engineering, University of Toledo. His current

research interests include Optimization methods, com-

puter-aided design and modeling of RF/Microwave cir-

cuits and neural networks applications.

Mohammad Almalkawi received

B.Sc. degree in Telecommunications

Engineering (with honors) from Yar-

mouk University, Irbid, Jordan, in

2007. In 2009, he received M.Sc.

degree in Electrical Engineering from

South Dakota School of Mines &

Technology, SD, USA. In 2011, he received Ph.D. degree

in Engineering from the University of Toledo, OH, USA. In

the summer of 2011, he was an RF Modeling Engineer in-

tern at Freescale Semiconductor, Inc., Tempe, AZ, working

on modeling RF/Microwave components for the develop-

ment of product-level models. In January 2012, he joined

the EECS Department at the University of Toledo, as a

Post-Doctoral Research Associate and he is currently a

Research Assistant Professor in the EECS Department. His

current research interests include RF/Microwave circuit

design and modeling, UWB antennas, computational elec-

tromagnetics, and electromagnetic compatibility.

Srinivasa Vemuru received his BS

and MS degrees in electrical engi-

neering from Indian Institute of

Technology, Madras in 1984 and

1986, respectively. He received his

PhD from the University of Toledo

in 1991. He is currently a Professor

of Electrical and Computer

568 Mareddy et al.


Engineering at Ohio Northern University. Prior to that he

was a faculty member in the Department of Electrical En-

gineering at the City College of the City University of

New York. His research interests are in analog and digital

electronic systems, RF circuit design, embedded systems,

and nanoelectronics.

Mohamed Bakr (S’98, M’00, Sr’

2010) received a B.Sc. degree in

Electronics and Communications En-

gineering from Cairo University,

Egypt in 1992 with distinction (hon-

ors). In June 1996, he received a

Master’s degree in Engineering

Mathematics from Cairo University.

In 1997, he was a student intern with Optimization Sys-

tems Associates (OSA), Inc. From 1998 to 2000, he

worked as a research assistant with the Simulation Opti-

mization Systems (SOS) research laboratory, McMaster

University, Hamilton, Ontario, Canada. He earned the

Ph.D. degree in September 2000 from the Department of

Electrical and Computer Engineering, McMaster Univer-

sity. In November 2000, he joined the Computational

Electromagnetics Research Laboratory (CERL), University

of Victoria, Victoria, Canada as an NSERC Post Doctoral

Fellow. Dr. Bakr received a Premier’s Research Excel-

lence Award (PREA) from the province of Ontario, Can-

ada, in 2003. He also received an NSERC Discovery Ac-

celerator Supplement (DAS) award in 2011. His research

areas of interest include optimization methods, computer-

aided design and modeling of microwave circuits and pho-

tonic devices, nanotechnology, neural network applica-

tions, smart analysis of high frequency structures and effi-

cient optimization using time/frequency domain methods.

He is currently working as an associate professor with the

Department of Electrical and Computer Engineering,

McMaster University.

Vijay Devabhaktuni received the

B.Eng. degree in EEE and the M.Sc.

degree in physics from BITS, Pilani,

in 1996, and the Ph.D. in electronics

from Carleton University, Canada, in

2003. He was awarded the Carleton

University’s senate medal at the PhD

level. During 2003-2004, he held the

NSERC (Natural Sciences and Engg. Research Council)

postdoctoral fellowship and spent the tenure at the Univer-

sity of Calgary. In spring of 2005, he taught at Penn State

Erie, PA. During 2005-2008, he held the prestigious Canada

Research Chair in Computer-Aided High-Frequency Mod-

eling and Design at Concordia University, Montreal. Since

2008, he is an Associate Professor in the EECS Department

at the University of Toledo. Dr. Devabhaktuni’s R&D inter-ests include applied electromagnetics, biomedical applica-tions of wireless sensor networks, computer aided design,device modeling, signal/image processing, infrastructuremonitoring, neural networks, optimization methods, RF/microwave devices, and virtual reality applications. In theseareas, he secured external funding close to $4M (sponsoringagencies include AFOSR, CFI, ODOT, NASA, NSERC,NSF, as well as industry). He co-authored 100 peer-reviewed papers, and is advising 17 MS/PhD students and 3PDFs. He is recipient several teaching excellence awards inCanada and USA. Dr. Devabhaktuni currently serves as theAssociate Editor of the International Journal of RF andMicrowave Computer-Aided Engineering. He is a regis-tered member of the Association of Professional Engineers,Geologists and Geophysicists of Alberta and a Senior Mem-ber of the IEEE.



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