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.
![Fig. 10. Six-section H-plane waveguide filter [7]. (a) Physical structure. (b) Coarse model as implemented in MATLAB .](/figures/fig-10-six-section-h-plane-waveguide-filter-7-a-physical-1bz72gq2.webp)
![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.](/figures/fig-1-error-plots-for-a-two-section-capacitively-loaded-3dqus87x.webp)
![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.](/figures/fig-2-error-plots-for-a-two-section-capacitively-loaded-2uki65k7.webp)