IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 11, NOVEMBER 2004 2593

A Space-Mapping Interpolating Surrogate Algorithm

for Highly Optimized EM-Based Design of

Microwave Devices

John W. Bandler, Fellow, IEEE, Daniel M. Hailu, Student Member, IEEE, Kaj Madsen, and Frank Pedersen

Abstract—We justify and elaborate in detail on a powerful

new optimization algorithm that combines space mapping (SM)

with a novel output SM. In a handful of ﬁne-model evaluations,

it delivers for the ﬁrst time the accuracy expected from classical

direct optimization using sequential linear programming. Our new

method employs a space-mapping-based interpolating surrogate

(SMIS) framework that aims at locally matching the surrogate

with the ﬁne model. Accuracy and convergence properties are

demonstrated using a seven-section capacitively loaded impedance

transformer. In comparing our algorithm with major minimax

optimization algorithms, the SMIS algorithm yields the same

minimax solution within an error of 10

15

as the Hald–Madsen

algorithm. A highly optimized six-section

-plane waveguide

ﬁlter design emerges after only four HFSS electromagnetic sim-

ulations, excluding necessary Jacobian estimations, using our

algorithm with sparse frequency sweeps.

Index Terms—Computer-aided design (CAD) algorithms,

electromagnetics, ﬁlter design, interpolating surrogate, microwave

modeling, optimization, output space mapping (OSM), space

mapping (SM), surrogate modeling.

I. INTRODUCTION

E

LECTROMAGNETIC (EM) simulators, long used by

engineers for design veriﬁcation, need to be exploited in the

optimization process. However, the higher the ﬁdelity (accuracy)

of the EM simulations, the more expensive direct optimization

becomes. For complex problems, EM direct optimization may

be prohibitive. Space mapping (SM) [1] aims to combine the

speed and maturity of circuit simulators with the accuracy

of EM solvers. The SM concept exploits “coarse” models

(usually computationally fast circuit-based models) to construct

a surrogate that is faster than the “ﬁne” models (typically CPU-

intensive full-wave EM simulations) and at least as accurate

as the underlying “coarse” model [1]–[4]. The surrogate is

Manuscript received April 29, 2004; revised July 8, 2004. This work was

supported in part by the Natural Sciences and Engineering Research Council

of Canada under Grant OGP0007239 and Grant STPGP 269760, through the

Micronet Network of Centres of Excellence and Bandler Corporation.

J. W. Bandler is with the Simulation Optimization Systems Research

Laboratory, Department of Electrical and Computer Engineering, McMaster

University, Hamilton, ON, Canada L8S 4K1 and also with Bandler Corporation,

Dundas, ON, Canada L9H 5E7 (e-mail: bandler@mcmaster.ca).

D. M. Hailu is with the Simulation Optimization Systems Research

Laboratory, Department of Electrical and Computer Engineering, McMaster

University, Hamilton, ON, Canada L8S 4K1.

K. Madsen and F. Pedersen are with the Department of Informatics and

Mathematical Modelling, Technical University of Denmark, DK-2800, Lyngby,

Denmark.

Digital Object Identiﬁer 10.1109/TMTT.2004.837197

iteratively updated by the SM approach to better approximate

the corresponding ﬁne model.

From the mathematical motivation of SM [4], it was found

that SM-based surrogate models provide a good approximation

over a large region, whereas the ﬁrst-order Taylor model is better

close to the optimal ﬁne-model solution. Based on this ﬁnding

and an explanation of residual misalignment, Bandler

et al..

[5] proposed the novel output space mapping (OSM) to further

correct residual misalignment close to the optimal ﬁne-model

solution. OSM reduces the number of computationally expen-

sive ﬁne-model evaluations, while improving accuracy of the

SM-based surrogate.

This paper elaborates on a new SM algorithm. Highly accu-

rate space-mapping interpolating surrogate (SMIS) models are

built for use in gradient-based optimization [6]. The SMIS is re-

quired to match both the responses and derivatives of the ﬁne

model within a local region of interest. It employs an output

mapping to achieve this.

The SMIS framework is formulated in Section IV. An

algorithm based on it is outlined in Section V. Convergence

is compared with two classical minimax algorithms, and a

hybrid aggressive space-mapping (HASM) surrogate-based

optimization algorithm using a seven-section capacitively

loaded impedance transformer. Finally, the SMIS algorithm is

implemented on a six-section

-plane waveguide ﬁlter [7].

II. D

ESIGN PROBLEM

A. Design Problem

The original design problem is

(1)

Here,

is the ﬁne-model response vector, e.g.,

at selected frequency points is the

number of response sample points, and the ﬁne-model point is

denoted

, where is the number of design parameters.

is a suitable objective function, and is

the optimal design.

III. OSM

OSM addresses residual misalignment between the optimal

coarse-model response and the true ﬁne-model optimum re-

sponse

. In the original SM [1], an exact match between

0018-9480/04$20.00 © 2004 IEEE

2594 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 11, NOVEMBER 2004

Fig. 1. Error plots for a two-section capacitively loaded impedance

transformer [4] exhibiting the quasi-global effectiveness of SM (light grid)

versus a classical Taylor approximation (dark grid). See text.

the ﬁne model and mapped coarse model is unlikely. For ex-

ample, a coarse model such as

will never match the

ﬁne model

around its minimum with any mapping

.An“output” or response mapping can

overcome this deﬁciency by introducing a transformation of the

coarse-model response based on a Taylor approximation [8].

The results of Bakr et al. [4] indicate that “input” SM-based

surrogates are good approximations to the ﬁne model over a

large region, which makes them useful in the early stages of an

optimization process. The residual misalignment between the

corresponding mapped coarse model(s) and the ﬁne model ren-

ders an exact match between them unlikely. Consequently, con-

vergence to

should not be expected.

Fig. 1 depicts model effectiveness plots [4] for a two-sec-

tion capacitively loaded impedance transformer at the ﬁnal it-

erate

, approximately . Centered at ,

the light grid shows

. This

represents the deviation of the mapped coarse model (using the

Taylor approximation

to the mapping, i.e., a

linearized mapping) from the ﬁne model. The dark grid shows

. This is the deviation of the ﬁne

model from its classical Taylor approximation

.

The gradient of the two-section capacitively loaded impedance

transformer, used in the Taylor approximation, was obtained an-

alytically using the adjoint network method [9]. The light grid

surface passing over the dark grid surface near the center of

Fig. 1 veriﬁes that the Taylor approximation is most accurate

close to

, whereas the mapped coarse model is best over a

larger region. The reason that the Taylor approximation is best

in the vicinity of

is that the Taylor approximation inter-

polates at the development point, whereas the mapped coarse

model does not.

Based on the above ﬁnding, Bakr et al. [10] use a surrogate

that is a convex combination of a mapped coarse model and a

ﬁrst-order Taylor approximation of the ﬁne model. Madsen and

Søndergaard [11] prove convergence of such HASM algorithms.

Fig. 2. Error plots for a two-section capacitively loaded impedance

transformer [4] exhibiting the quasi-global effectiveness of SM-based

interpolating surrogate, which exploits OSM (light grid) versus a classical

Taylor approximation (dark grid). See text.

In this paper, we introduce a novel method to ensure con-

vergence of the SM technique. OSM is incorporated into SMIS

to ensure that we obtain the same solution as classical direct

gradient-based optimization. Fig. 2 depicts model effectiveness

plots for the two-section capacitively loaded impedance trans-

former corresponding to Fig. 1. Centered at

, the light grid

shows

. This represents the de-

viation of the SMIS surrogate from the ﬁne model. The dark

grid shows the deviation of the ﬁne model from its classical

Taylor approximation as in Fig. 1. Thus, Fig. 2 demonstrates

that the SMIS surrogate, because of its interpolating properties,

performs better than the ﬁrst-order Taylor approximation even

close to

.

IV. SMIS F

RAMEWORK

A. Surrogate

The SM-based interpolating surrogate

is

deﬁned as a transformation of a coarse model

through input and output mappings at each sampled re-

sponse. Fig. 3 illustrates the SMIS framework. Here,

, where ,

[1], [2] is an input mapping for the

th coarse response ,

and

[8] is an output mapping applied to the

coarse response. Using the function

with individually adjusted coarse responses, deﬁned as

, where

, the surrogate can be expressed as

a composed mapping

.

We wish to consider individual mappings of each coarse re-

sponse

. These (nonlinear) mappings will be

approximated by a sequence of local linear mappings. The

th

linearized input mapping at the

th iteration is assumed to be of

the form

(2)

BANDLER et al.: SMIS ALGORITHM 2595

Fig. 3. Illustration of the SMIS concept. The aim is to calibrate the mapped

coarse model (the surrogate) to match the ﬁne model using different input and

output mappings for each sampled response.

where the matrix and vector . The th output

mapping is deﬁned as

(3)

where

are the th components of . is deﬁned

as

, where is a constant vector. Deﬁning

similarly, the th component of the surrogate becomes

(4)

We now discuss how to determine the constants

deﬁning the linear mappings

and . Assume we have reached the th iterate in the

iterative search for a solution. At

, the surrogate must

agree with the ﬁne response [12]

(5)

We also aim to align the surrogate with the ﬁne-model re-

sponse at the previous points in the iteration, as well as aim to

have agreement between the Jacobians at the current point, i.e.,

(6)

where

and are the Jacobians of the surrogate

and ﬁne model at

, respectively.

The constants

are determined

in such a way that the alignment (5) holds and the requirements

in (6) are satisﬁed as well as possible (in some sense to be spec-

iﬁed). The alignment (5) is satisﬁed by choosing

and ap-

propriately. If we let

, then (5) only depends on the

choice of

.

Thus, the

th surrogate of response number is

and (7)

where

(8)

In the ﬁrst iteration, the mapping parameters

and are used, which

ensure that

.For , the parameter

is utilized, which ensures (5).

In summary, the surrogate used in the

th iteration is given by

(9)

In each iteration, the surrogate is optimized to ﬁnd the next it-

erate by solving

(10)

B. Surface Fitting Approach for Parameter Extraction (PE)

PE is a crucial step in any SM algorithm. In this paper,

we employ a surface ﬁtting approach for PE, which involves

the minimization of residuals between the surrogate and ﬁne

models, and extracting the parameters , and

.

Assume

has been found. We now wish to ﬁnd the

th set of mapping parameters . Since

(5) is automatically satisﬁed by using (7), the aim is to ensure

the matching (6). Thus, for ﬁnding ,we

aim to minimize the following set of residuals in some sense [6]:

.

.

.

(11)

where

and are the th columns of and , respec-

tively. The residual (11) is used during the PE optimization

process

(12)

which extracts the mapping parameters for the

th response, and

for iteration

. Hence, we have the complete set of mapping

parameters after

PE optimizations.

V. P

ROPOSED SMIS ALGORITHM

Our proposed algorithm begins with the coarse model as the

initial surrogate. The algorithm incorporates explicit SM [1] and

OSM [5] to speed up the convergence to the optimal solution.

Step 1)

Select a coarse and ﬁne model.

Step 2)

Set

, and initialize .

2596 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 11, NOVEMBER 2004

Fig. 4. Seven-section capacitively loaded impedance transformer: “Fine”

model.

Fig. 5. Seven-section capacitively loaded impedance transformer: “Coarse”

model.

TABLE I

F

INE MODEL

CAPACITANCES, AND THE

CHARACTERISTIC

IMPEDANCES FOR THE

SEVEN-SECTION

CAPACITIVELY

LOADED IMPEDANCE

TRANSFORMER

Step 3)

Optimize the surrogate (9) to ﬁnd the next iterate

by solving (10).

Step 4)

Evaluate

.

Step 5)

Terminate if the stopping criteria are satisﬁed.

Step 6)

Update the input and output mapping parameters

through PE

by solving (12).

Step 7)

Set

, and go to Step 3.

As stopping criteria for the algorithm in Step 5, the relative

change in the solution vector, or the relative change in the ob-

jective function, could be employed.

VI. E

XAMPLES

A. Seven-Section Capacitively Loaded

Impedance Transformer

We consider the benchmark synthetic example of a seven-sec-

tion capacitively loaded impedance transformer [4]. We apply

the proposed SMIS algorithm to that example. The objective

function is given by

. We consider a

“coarse” model as an ideal seven-section transmission line (TL),

where the “ﬁne” model is a capacitively loaded TL with capac-

itors

pF. The ﬁne and coarse models are shown

in Figs. 4 and 5, respectively. Design parameters are normalized

lengths

with respect to the

quarter-wave length

at the center frequency of 4.35 GHz.

Design speciﬁcations are

for 1 GHz GHz (13)

with 68 points per frequency sweep. The characteristic imped-

ances for the transformer are ﬁxed as shown in Table I. The

Fig. 6. Seven-section capacitively loaded impedance transformer: optimal

coarse-model response

(

--

)

, the optimal minimax ﬁne-model response (—),

and the ﬁne-model response at the initial solution or at the optimal coarse-model

solution

(

)

.

TABLE II

O

PTIMIZABLE PARAMETER

VALUES OF THE SEVEN-SECTION

CAPACITIVELY

LOADED IMPEDANCE TRANSFORMER

Fig. 7. Seven-section capacitively loaded impedance transformer: optimal

coarse-model response

(

--

)

, the optimal minimax ﬁne-model response (—),

and the ﬁne-model response at the SMIS algorithm solution obtained after ﬁve

iterations (six ﬁne-model evaluations)

(

)

.

Jacobians of both the coarse and ﬁne models were obtained

analytically using the adjoint network method [9]. We solve

BANDLER et al.: SMIS ALGORITHM 2597

Fig. 8. (a) First 25 iterations of the difference between the ﬁne-model

objective function

U

obtained using the SMIS algorithm

(

)

and the

ﬁne-model objective function at the ﬁne-model minimax solution

U

obtained

by the Hald–Madsen algorithm

( )

, the HASM surrogate optimization

algorithm using exact gradients

(

r

)

, and the HASM surrogate optimization

algorithm using the Broyden update

(1)

. (b) The corresponding difference

between the designs.

the PE problem using the Levenberg–Marquardt algorithm for

nonlinear least squares optimization available in the M

ATLAB

Optimization Toolbox.

1

Optimizing the ﬁne model directly using the gradient-based

minimax method of Madsen [13], and Hald and Madsen [14]

conﬁrms that the problem has numerous local solutions. Starting

from the optimal coarse-model solution (the initial solution for

the SMIS method), the Hald–Madsen minimax algorithm needs

13 iterations, or 13 ﬁne-model evaluations, to converge to the

ﬁne-model minimax solution. Note that both the direct opti-

mization method of Hald and Madsen and the SMIS approach

employ exact gradients.

The ﬁne-model response at the optimal coarse-model solu-

tion is shown in Fig. 6. Table II shows the lengths for solutions

obtained using the SMIS algorithm and the ﬁne-model direct

minimax optimization solution [13], [14]. Our SMIS algorithm

1

MATLAB, ver. 6.1, MathWorks Inc., Natick, MA, 2001.

Fig. 9. (a) Difference between the ﬁne-model objective function

U

obtained

using the SMIS algorithm

(

)

and the ﬁne-model objective function at the

ﬁne-model minimax solution

U

obtained by the Hald–Madsen algorithm

(

)

,

the HASM surrogate optimization algorithm using exact gradients

(

r

)

, and

the HASM surrogate optimization algorithm using the Broyden update

(1)

.

(b) The corresponding difference between the designs.

took six ﬁne-model evaluations or ﬁve iterations to reach the

same accurate solution as the Hald–Madsen direct minimax op-

timization algorithm.

Fig. 7 shows the ﬁne-model response at the SMIS algorithm

solution. The difference between the minimax objective func-

tion at the optimal minimax ﬁne-model response and the re-

sponse obtained using the SMIS algorithm is shown in Figs. 8

and 9.

Corresponding results reached by the Hald–Madsen method

are shown in Figs. 8 and 9. In these ﬁgures, we show the

HASM surrogate exploiting exact gradients. The minimax

objective function and solution reached by the HASM surro-

gate optimization approach using the Broyden update [10] are

also shown. The four methods converged to the same highly

accurate solution.

The optimization methods used for solving (1) and a compar-

ison is shown in Table III. Using the adjoint technique, the SMIS

algorithm was able to obtain the same optimum solution as the

Hald–Madsen algorithm within an error of 10

after only ﬁve

iterations.