A Comparison of Some Model Order Reduction Techniques

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A Comparison of Some ModelOrder Reduction TechniquesRodney D. Slone a , Jin-Fa Lee a & Robert Lee aa ElectroScience Laboratory Department ofElectrical Engineering The Ohio State UniversityColumbus, Ohio, USAPublished online: 10 Nov 2010.

To cite this article: Rodney D. Slone , Jin-Fa Lee & Robert Lee (2002) AComparison of Some Model Order Reduction Techniques, Electromagnetics, 22:4,275-289, DOI: 10.1080/02726340290083888

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Electromagnetics, 22:275–289, 2002Copyright © 2002 Taylor & Francis0272-6343 /02 $12.00 C .00DOI: 10.1080/0272634029008388 8

A Comparison of Some Model OrderReduction Techniques

RODNEY D. SLONEJIN-FA LEEROBERT LEE

ElectroScience LaboratoryDepartment of Electrical EngineeringThe Ohio State UniversityColumbus, Ohio, USA

In this paper, an analysis of some model order reduction (MORe) techniques is pre-sented. More precisely, this paper considers asymptotic waveform evaluation (AWE),Galerkin asymptotic waveform evaluation (GAWE) with a short-term vector recurrencerelation, multipoint Galerkin asymptotic waveform evaluation (MGAWE) also usinga short-term recurrence, and matrix-Padé via Lanczos (MPVL). These techniques areapplied to matrix equations resulting when the � nite element method (FEM) is used tomodel electromagnetic wave propagation problems. The reduced order model equa-tions can then be solved repeatedly to obtain a wideband frequency simulation witha reduction in total computation time. The analysis contained herein compares andcontrasts the MORe techniques by not only considering the nature of the individualalgorithms, but also solving several illustrative numerical examples. These examplesshow how, for a MORe technique, a radiation and scattering problem might haveto be treated very differently. In addition, it is noted that the unknown(s) desiredas output(s) from the FEM mesh can in� uence which MORe technique is more ef� -cient. The solutions obtained through the MORe techniques are compared to an LUdecomposition at each frequency point of interest to benchmark their accuracy andef� ciency.

Keywords AWE, GAWE, MGAWE, model order reduction, MPVL, PVL

Introduction

Model order reduction (MORe) has been used extensively in the circuit analysis commu-nity over the past several years. The need for MORe techniques was fed by the desire todecrease the simulation time required for using computer-generated models in analysisand/or design. Of course, these techniques had to not only decrease simulation time,but also retain accuracy. These requirements helped motivate the development of asymp-totic waveform evaluation (AWE), where a reduced-order approximation of the systemresponse is formulated (Pillage & Rohrer, 1990). In AWE, as the order of approximationincreases, the response of the reduced-order model begins to “asymptotically” approachthe response of the original system (Raghavan et al., 1993). Once the smaller, reduced-

The work in this paper was supported by DSO National Lab under contract DSO/C/99134/0.Address correspondence to J. F. Lee, Department of Electrical Engineering, Ohio State Uni-

versity, 1320 Kinnear Rd., Columbus, OH 43212. E-mail: [email protected]

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276 R. D. Slone et al.

order system is generated, it can then be solved inexpensively for many different valuesof frequency. In this way, a broadband response can be calculated more ef� ciently thanif the original, large-circuit matrices were solved at each frequency point of interest.

In AWE analysis, to reduce the original problem to a smaller model, the matrix thatdescribes the system is expanded into a Taylor series where each of the coef� cients ofexpansion is a matrix. Moment matching is then used to approximate a solution to themodel. This approximation to the solution is then converted into a Padé representation bymatching the � rst 2Q moments (i.e., matching terms up to and including order 2Q ¡ 1)to a lower Q-order Padé model (Bracken, Raghavan, & Rohrer, 1992; Pillage & Rohrer,1990; Tang & Nakhia, 1992). AWE then uses the Padé approximation to provide anestimation of the system response. However, AWE is known to have issues involvingaccuracy and/or ef� ciency. The Padé via Lanczos (PVL) algorithm (Feldmann & Freund,1995; Gallivan, Grimme, & Van Dooren, 1994) was introduced to address these issues.In PVL, moment matching is not performed explicitly. Rather, the approximation to thesolution is calculated from quantities generated by a PVL algorithm. Although PVL wasapplied traditionally to only single-input, single-output systems, Freund (1998) reporteda matrix-Padé via Lanczos (MPVL) algorithm for multiple starting vectors that also hasthe ability to simultaneously produce output for many different unknowns.

After some successful applications of MORe techniques to circuit analysis, the com-putational electromagnetics (CEM) community became interested in MORe. MORe algo-rithms have been developed and/or applied in the CEM community to decrease the timerequired for simulations. In particular, MORe has been used to solve matrices generatedwhen the � nite element method (FEM) is used to model the computational domain. InZhang and Jin (1998), AWE was applied to solve scattering problems modeled usingFEM. The spectral Lanczos decomposition method (SLDM) was used in Zunoubi et al.(1998) to solve cavity problems modeled with FEM. The SLDM is limited in applicabil-ity, however, because open region problems cannot be treated. Since this paper focuses onradiation and scattering problems, the SLDM is not included in this study. In Slone andLee (2000), open domain radiation problems are treated using MPVL on a FEM mesh.However, to conform to the linear frequency requirement of PVL, the procedure in Sloneand Lee (2000) requires doubling the number of unknowns. To overcome some of thesecomputational dif� culties, a Galerkin asymptotic waveform evaluation (GAWE) was pre-sented in Slone, Lee, and Lee (2000). Unlike PVL, GAWE can solve a matrix equationthat has a polynomial frequency variation. To obtain even more accuracy, GAWE was alsoextended in Slone, Lee, and Lee (2000) to a multipoint Galerkin asymptotic waveformevaluation (MGAWE) in which more than one expansion point is considered simultane-ously. In this paper, short-term instead of long-term vector recurrences for GAWE andMGAWE are described and used. Using a short-term recurrence decreases the requirednumber of � oating point operations.

AWE, GAWE, MGAWE, and MPVL are compared and contrasted in this paper,which is organized as follows. In the next section, each of the MORe techniques isinvestigated to see what type of problem the technique can solve, how the ef� ciencyof the method depends on the number of inputs required to formulate the problem, thenumber of unknowns desired as outputs, and/or the number of frequency points requiredin the bandwidth of interest. In addition, the type of operations that are found in thealgorithms is analyzed. Then, in the next section, numerical simulations are shown toillustrate how accurate each method is, how the accuracy of the reduced order modelcreated by the method depends on its order, and how a radiation and scattering problemcan differ when MORe is applied. Finally, a summary and conclusion are given.

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Model Order Reduction 277

Some MORe Algorithms

Consider the matrix equation

a1

SiD0

.¾ iAi/X.¾/ Db1

SkD0

¾ kBk; (1)

where ¾ is related to the frequency (f ) distance from the expansion point (s0) by therelationship

¾ D j2¼f ¡ s0; (2)

each Ai is an N ´ N complex constant matrix, each Bk is a complex constant N ´ p

(where p is the number of independent inputs that can inject excitations into the FEMmesh) right-hand side matrix, and X.¾ / is the N ´ p solutions matrix. Assume it isdesired to calculate (1) for fnum different values of ¾ . In addition, assume there areo · N unknowns of interest that are desired as outputs. Let an N ´ o matrix L selectthe o outputs of interest from the unknowns. Finally, let H.¾ / denote the o ´ p solutionmatrix of the desired unknowns, where

H.¾/ D LT X.¾/: (3)

In the following subsections, it will be shown how each of the MORe algorithms solvesthe above problem.

AWE

AWE is applicable to any problem that can be cast into the form shown in (1). In fact,in AWE each of the right-hand side matrices Bk can usually be reduced to a right-handside vector, that is, p D 1. Therefore, in the following discussion, Bk will be denoted asbk , X.¾ / will be denoted as x.¾ /, and H.¾ / will be denoted as h.¾ /.

Expand the unknown solution vector x.¾/ into a Taylor series as

x.¾ / D1

SnD0

¾ nmn; (4)

where each of the mn is an N -vector moment. To obtain a Padé approximant of orderQ, there must by 2Q moments generated, which means that moments up to and includ-ing order 2Q ¡ 1 must be generated. Substituting (4) into (1) and performing momentmatching gives

m0 D A¡10 b0; (5)

m1 D A¡10 .b1 ¡ A1m0/;

m2 D A¡10 .b2 ¡ A2m0 ¡ A1m1/;

:::

mq¡1 D A¡10

0

@bq¡1 ¡q¡1

SiD1

Aq¡imi¡1

1

A ;

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where q D 2Q, bk D 0, for k > b1, and Ai D 0 for i > a1. Then let

hn D LT mn (6)

and denote the rth entry of the o-vector hn as ´rn. Finally, for each of the o unknowns

desired as outputs, let r D 1; 2; : : : ; o and form the Padé approximant by � nding crt for

t D 0; 1; : : : ; Q ¡ 1 and dru for u D 1; 2; : : : ; Q from

Q¡1

StD0

¾ t crt

1 CQ

SuD1

¾ udru

Dq¡1

SnD0

¾ n´rn; (7)

which requires solving the system

2

666664

´r0 ´r

1 ´r2 × × × ´r

Q¡1´r

1 ´r2 ´r

3 × × × ´rQ

´r2 ´r

3 ´r4 × × × ´r

QC1:::

::::::

: : ::::

´rQ¡1 ´r

Q ´rQC1 × × × ´r

2Q¡2

3

777775

2

666664

drQ

drQ¡1

drQ¡2:::

dr1

3

777775D ¡

2

666664

´Q

´QC1

´QC2:::

´2Q¡1

3

777775(8)

and

cr0 D ´r

0; (9)

cr1 D ´r

1 C dr1´r

0;

cr2 D ´r

2 C dr2´r

0 C dr1´r

1;

:::

crQ¡1 D ´r

Q¡1 CQ¡1

SiD1

drQ¡i´

ri¡1:

Then the r th desired output of the o-vector h.¾ /, denoted by hr .¾/, is given by

hr .¾ / ¼

Q¡1

StD0

¾ t crt

1 CQ

SuD1

¾ udru

: (10)

One important point to note in the above process is that the majority of the timeis spent in computing A¡1

0 , which must be done only once unless s0 changes. OnceA¡1

0 is computed, it must be applied q times. Of less importance is to note that as q

and/or a1 is increased, more matrix-vector multiplications must be performed in (5). Inaddition, as a1 increases, not only is more storage required for the Ai matrices, but also

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for the moment N -vectors computed in (5). Also note that the number of N -vector dotproducts performed in (6) is equal to the number of outputs times the number of momentsgenerated, which is oq . In addition, a higher value of q means that the order of the matrixin (8) and the number of equations in (9) increase. Also note that the number of times(8) and (9) must be solved in independent of the number of frequency points as which itis desired to calculate the solution h.¾ /, but rather depends on o. Finally, note that thenumber of times that (10) must be solved is equal to fnum times o.

GAWE

GAWE is similar to AWE in that it is applicable to any problem that can be cast into theform shown in (1), and the right-hand side can usually be represented as a vector series,so again p D 1. As before, Bk will be denoted as bk , X.¾ / as x.¾ /, and H.¾ / as h.¾ /.

GAWE assumes that there is a collection of q linearly independent N -vectors vn

and q scalars °n.¾/ for n D 1; 2; : : : ; q. De� ne an N ´ q matrix Vq such that the nthcolumn of Vq is vn and also de� ne a q-vector gq.¾/ such that the nth component ingq.¾/ is °n.¾/. The quantities vn and °n.¾/ are chosen such that the approximation

x.¾ / ¼ Vqgq.¾/ Dq

SnD1

vn°n.¾/ (11)

minimizes the residual

rq.¾/ Da1

SiD0

.¾ iAi/

q

SnD1

vn°n.¾ / ¡b1

SkD0

¾ kbk (12)

in the sense that if rq.¾ / is expressed in a Taylor series as

rq.¾ / D1

SlD0

¾ lrlq ; (13)

then

rlq D 0 for l D 0 : : : q ¡ 1 (14)

and

rq.r/ ? vq (15)

The moments mn from AWE in (5) can be used to construct Vq if they are orthonormal-ized. Although this is not the way Vq is found in Slone, Lee, and Lee (2000), the spacethat is spanned is the same. However, note that this new way of constructing Vq gives ashort- (instead of long-) term recurrence. Therefore, fewer � oating point operations arerequired.

The difference between GAWE and AWE is that instead of performing a Padé ap-proximation, gq.¾/ is made to satisfy (15), and is found from

gq.¾/ D³

a1

SiD0

¾ iVHq AiVq

!¡1 ³b1

SkD0

¾ kVHq bk

!: (16)

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Then

h.¾ / ¼ LT Vqgq.¾/: (17)

Just as in AWE, the majority of the time spent in executing GAWE is in computingA¡1

0 , which must be done only once unless s0 changes; once A¡10 is computed, it must

be applied q times; q and a1 affect the number of N ´ N matrix N -vector multiplicationsperformed. However, in GAWE q dictates the number of N -vectors that must be stored;also, the q N -vectors must be orthonormalized. As a1 and/or q increases, more operationsmust be performed to compute VH

q AiVq for i D 0; 1; : : : ; a1; the same holds true forb1 and q in VH

q bk . Note that q affects the order of the matrix that must be factored andsolved in (14); the number of times that (16) must be factored and solved is independentof the number of desired outputs o, but rather depends on fnum. Note that LT Vq mustbe calculated once, which costs oq N -vector dot products. Finally, note that (17) mustbe computed fnum times.

MGAWE

The difference between MGAWE and GAWE is that in MGAWE more than one expansionpoint can be considered simultaneously. To see how MGAWE constructs the basis Vq ,consider the GAWE process. Assume num_pts expansion points are desired. Then q DS num_pts

vD1 qv , where qv is the number of moments to be generated at the vth expansionpoint. For the � rst expansion point, the � rst q1 vectors are exactly the same vectorsthat would be generated in GAWE. When it is desired to start generating vectors fromthe second expansion point, it is desired that the MGAWE solution still satisfy therequirement that rl

q1D 0 for l D 0 : : : q1 ¡ 1 at the previous expansion point. In addition,

it is desired that at the new expansion point, rlq2

D 0 for l D 0 : : : q2 ¡ 1. After newvectors are generated at the new expansion point, orthonormalize these new vectors notonly against themselves but also against all vectors generated at all previous expansionpoints. The above procedure is followed for all vectors generated at all num_pts expansionpoints. After Vq is � nally constructed, gq.¾ / is found from (16) (the same way as inGAWE, where ¾ corresponding to the last expansion point considered can be used ifa1 and b1 are chosen large enough so no signi� cant higher order term is truncated).Although the above description is not exactly the same as the one given in Slone, Lee,and Lee (2000), the space spanned is the same. Constructing Vq as described aboverequires fewer � oating point operations.

The computational difference between using GAWE at several expansion pointsthat are considered individually and MGAWE is that more orthogonalizations must beperformed in MGAWE. This is because in individually considered GAWE, each set ofvectors for a particular expansion point must be orthogonalized only against the vectorsthat are in that particular set. In MGAWE, however, a vector must be orthogonalized notonly against the vectors that are in its expansion point set, but also against the vectorsthat are in all the other expansion point sets. In addition, there is a slight increase inthe size of the reduced order model matrices that must be factored and solved aftergenerating Vq (i.e., a matrix of size S num_pts

vD1 qv instead of size ¼ qv). The gain ofMGAWE over GAWE is that by considering several expansion points simultaneously,more accuracy can be obtained for the same number of vectors. Therefore, for the sameaccuracy, MGAWE requires fewer total vectors to be generated and/or fewer expansionpoints to be considered than GAWE.

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MPVL

The structure of MPVL is very different from the other methods considered in this paper.For MPVL to be applicable to a problem written in the form given in (1), it is requiredthat a1 D 1 and b1 D 0. This restriction in b1 means that to be able to handle scatteringproblems it is necessary for p > 1. For radiation-type problems, however, if the entireradiator can be excited in phase, then the problem can be posed so that p D 1.

Under the above constraints, (1) becomes

.A0 C ¾A1/X.¾ / D B0: (18)

Before applying MPVL, (18) is � rst premultiplied by A¡10 to obtain

.I C ¾ A¡10 A1/X.¾ / D A¡1

0 B0: (19)

If X.¾ / were known, then the o outputs for H.¾ / could be found as shown in (3).However, X.¾ / is not known. Therefore, to compute an approximation to H.¾ /, theMPVL algorithm given in Freund (1998) is run on the input matrices ¡A¡1

0 A1 andA¡1

0 B0 to form several reduced-order output matrices. If the algorithm is run for Q

iterations, then these reduced-order output matrices are a Q ´ Q matrix T, a Q ´ o

matrix h, and a Q ´ p matrix f. The resulting MPVL approximation is

H.¾ / ¼ hT .I ¡ ¾ T/¡1f: (20)

As in the methods discussed previously, the majority of the time spent in executingMPVL is in computing A¡1

0 . Once A¡10 is computed it must be applied 2Q C p times;

Q affects the number of N ´ N matrix N -vector multiplications performed; o and p

affect the number of N -vectors that must be stored. As Q increases, the order of T in(20) increases. The number of times that (20) must be solved depends on fnum. Foreach frequency point, a matrix of size Q ´ Q must be factored and then applied to p

right-hand sides, and op Q-vectors dot products must be evaluated.A key fact about the double-sided Padé approximation via the Lanczos process, on

which MPVL is based, is that during each iteration two vectors are generated and twomoments are matched. In contrast, the other methods discussed herein only generateone vector and match one moment per iteration. However, there is a huge differencein the moment matching characteristics of PVL and MPVL because in MPVL onlyapproximately 1=o C 1=p moments are matched per iteration. Therefore, the accuracy ofa reduced-order model approximation computed with MPVL decreases with increasinginputs and/or outputs.

Numerical Examples

A nodal-based FEM is used to solve some two-dimensional TMz .TEz/ problems forEz .Hz/. The resulting matrix equation from each example given below is solved usingthe methods from the previous section that are applicable for that particular simulation.All solutions are computed using MATLAB running on a 266 MHz Pentium II with256 Megabytes of RAM.

Example 1. The � rst numerical example is a two-dimensional TMz model of a hornantenna shown in Figure 1. The circular outer boundary is treated with an absorbing

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Figure 1. Geometry of the horn in Example 1.

boundary condition. In this example, it is desired to � nd the wave impedance insidethe horn from 500 MHz to 1.5 GHz. Table 1 compares the unknowns for the severalmethods considered in the previous section. It should be noted that the reason b1 is equalto zero instead of one is that even though the right-hand side contains a linear term in ¾ ,it does not contain a constant term. Therefore, the linear term can be considered to beconstant during the MORe procedure, and then the solution obtained can be multipliedby ¾ during postprocessing. In addition, note that MPVL requires a1 D 1 and thereforeneeds two times the number of unknowns that the other methods require (Slone & Lee,2000). Figure 2 shows the impedance for the structure shown in Figure 1. In Figure 2,the response computed using a LU decomposition is shown as a solid curve, the AWEsolution is shown as a dashed curve, the GAWE solution is shown as a dash-dotted curve,

Table 1Comparison of variables associated with Example 1

LU AWE GAWE MGAWE MPVL

a1 2 2 2 2 1b1 0 0 0 0 0

fnum 1001 1001 1001 1001 1001N 3438 3438 3438 3438 6876o 3 3 3 3 3p 1 1 1 1 1

q D 2Q N/A 30 30 20 C 20 200s0=j2¼106 500; 501; : : : ; 1500 1200 1200 700, 1200 1150Time (sec) 42,651 101 106 174 684

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Figure 2. Responses for the impedance of the structure shown in Figure 1. Solid: LU response;dashed: AWE response; dash-dotted: GAWE response; dotted: MGAWE response; solid: MPVLresponse (essentially indistinguishable from LU response except for a slight deviation in zoommode near f D 1:5 ´ 109).

the MGAWE solution is shown as a dotted curve, and the MPVL solution is also shownas a solid curve because it is indistinguishable from the LU response except for a veryslight deviation in zoom mode near f D 1:5 ´ 109. There are several issues to note fromTable 1 and Figure 2. First, not how GAWE gives a somewhat better response than AWEfor essentially the same computational cost. Second, note that MGAWE gives a betterresponse (indistinguishable from the LU response using the large scale) than GAWE.Although MGAWE takes more computation time than GAWE, if another expansion pointwere added for GAWE at j2¼700 ´ 106, with the number of moments added at thesecond expansion point being q2 D 20, the computation time for GAWE would increaseto 188 sec (which is more than the 174 sec required by MGAWE, as shown in Table 1).In addition, GAWE would still not be as accurate as MGAWE (not shown in the � gure).Finally, note that MPVL gives the most accurate response of any of the reduced-ordermodels shown here, and it does so with just one expansion point. However, note howlarge q is required to be and how much computation time the method takes. Furthermore,it will be shown in the other numerical examples that MPVL is not applicable to manyproblems that are of interest.

Example 2. The second numerical example is a radiation problem where a model ofa material cylinder is illuminated by a uniform electric line source. The outer boundaryis treated with perfectly matched layer (PML) backed by a perfect electric conductor(PEC). The solution is computed up to 500 MHz, which is the frequency where the edge

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length of a side of an element is about 1/20 of a wavelength. Table 2 shows the relevantquantities needed to compare the methods.

There are several important issues illustrated by Table 2. First, note that a1 D 4because anisotropic, dispersive PML is used to treat the outer boundary of the FEMmesh. Since a1 6D 1, PVL cannot be applied directly, although the method shown inSlone and Lee (2000) could be extended to the case where a1 D 4. However, doing sowould require quadrupling the size of the matrix that would have to be factored and solvedduring the execution of the PVL algorithm. This cost is prohibitive, and therefore MPVLis not used to solve this example. Second, note that o D 124 because the response iscomputed for all the nodes in the FEM mesh that are on an imaginary circle that enclosesthe cylinder and source, so a far � eld pattern can be constructed if desired. In Figure 3the magnitude of the response of Ez at one of these nodes is shown. Note that in thelow-frequency region, both GAWE and MGAWE match the LU solution well. However,AWE would require another expansion point in the low-frequency region to obtain a goodmatch to the LU solution. This extra expansion point would increase the computationtime required for AWE by 675 sec for 10 moments or 703 sec for 20 moments. Inthe high-frequency region, MGAWE matches the LU solution well, but both AWE andGAWE need at least two more expansion points each. These two new expansion pointswould at least increase the required computation time for AWE by an additional 1,472 secand GAWE by an additional 1,474 sec. So, in summary, MPVL is impractical for thisproblem. AWE requires a total of at least 4 expansion points with a total cost of at least2,877 sec to match the LU solution throughout the entire frequency band and is thereforethe most expensive MORe technique computed for this example. GAWE requires a totalof at least 3 expansion points with a total cost of at least 2,216 sec and is thereforeless expensive than AWE, and MGAWE is more computationally ef� cient than the othermethods and can match the entire desired frequency band using only 2 expansion pointswith a total cost of 1,682 sec.

Example 3. The third numerical example is a TEz scattering problem where a modelof a material cylinder is illuminated by a uniform plane wave. The outer boundary istreated with an absorbing boundary condition. The solution is computed up to 500 MHz,which is again the frequency where the edge length of an element is about 1/20 of awavelength. Table 3 shows the relevant quantities needed to compare the methods. Notethat b1 D 10. This is because in the scattering problem, the right-hand side is not just



a1 4 4 4 4 4 6D 1b1 3 3 3 3 N/A

fnum 201 201 201 201 N/AN 4734 4734 4734 4734 N/Ao 124 124 124 124 N/Ap 1 1 1 1 N/A

q D 2Q N/A 30 30 30 C 30 N/As0=j2¼106 0; 2:5; : : : ; 500 200 200 200, 400 N/ATime (sec) 130,718 730 742 1682 N/A

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Figure 3. LU, AWE, GAWE, and MGAWE responses for the magnitude of the response in Exam-ple 2. Solid: LU response; dashed: AWE response; dash-dotted: GAWE response; dotted: MGAWEresponse.



a1 2 2 2 2 N/Ab1 10 10 10 10 N/A

fnum 201 201 201 201 N/AN 1276 1276 1276 1276 N/Ao 100 100 100 100 100 À 1p 1 1 1 1 100 À 1

The following data is for Simulation 1, shown in Figure 4.

q D 2Q N/A 10 C 10 10 C 10 see Figure 5 N/As0=j2¼106 0; 2:5; : : : ; 500 100, 400 100, 400 see Figure 5 N/ATime (sec) 241 24.55 5.95 see Figure 5 N/A

The following data is for Simulation 2, shown in Figure 5.

q D 2Q N/A 20 C 20 20 C 20 10 C 10 N/As0=j2¼106 0; 2:5; : : : ; 500 100, 400 100, 400 100, 400 N/ATime (sec) 241 40.42 9.55 5.97 N/A

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a constant or a linear function of ¾ . In fact, the right-hand side contains an exponentialterm of ¾ that is handled by expanding it into a Taylor series around s0. The form of anentry in the right-hand side looks like

.sC1 C C2/esCe ; (21)

where

C1 D p¹².1 ¡ cos.Á//; (22)

C2 D1

2½; (23)

C2 D ¡p

¹²½ cos.Á/; (24)

Á is the angle difference between the location of the � nite element under considerationand the incoming uniform plane wave, and ½ is the distance of the boundary from thecenter of the mesh. The Taylor series for (19) is

1

SkD0

¾ k

k![kC1Ck¡1

3 es0C3 C .s0C1 C C2/Ck3 es0C3]: (25)

Figure 4. LU, AWE, and GAWE responses for the magnitude of the response in Example 3,Simulation 1. Solid: LU response; dashed: AWE response; dash-dotted: GAWE response.

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It is found that this Taylor series expansion introduces insigni� cant error into the solutionif b1 D 10 (that is, 11 terms are used with powers ranging from ¾ 0 to ¾ 10). Since b1 6D 0,MPVL cannot handle this problem without treating the exponential in the right-hand sideduring postprocessing. However, this would require each of the 100 outer boundary nodesto be an independent input, so p D 100. Since all the outer boundary nodes are alsodesired as outputs, o D 100 for this mesh, and this results in the moment matching powerof MPVL being approximately 1=100C1=100 per iteration, which is much too small to beef� cient. Actually, an MPVL simulation for q D 100 took 263.6 sec (which is more timethan LU required) and the results were not even accurate enough to show. In Figure 4 theresults for the � rst simulation are shown, where the solid curve is the response computedusing an LU decomposition at each frequency point of interest, the dashed curve is theAWE solution computed using two independent expansion points at j2¼100 ´ 106 andj2¼400 ´ 106 with q1 D q2 D 10, and the dash-dotted curve is the GAWE solutioncomputed with the same expansion points and number of moments as AWE. As canbe seen in Figure 4, neither GAWE nor AWE converged using the number of momentsgiven above. Simulation 2, shown in Figure 5, shows the results for AWE and GAWEwhen q1 D q2 D 20 (for a total of 40 moments). Also shown in Figure 5 is the result forMGAWE (dotted curve) when the expansion points at j2¼100 ´ 106 and j2¼400 ´ 106

with q1 D q2 D 10 (for a total of 20 moments) are considered simultaneously. Notefrom Figure 5 that GAWE and MGAWE have converged, but AWE still needs more

Figure 5. LU, AWE, GAWE, and MGAWE responses for the magnitude of the response in Ex-ample 3, Simulation 2. Solid: LU response; dashed: AWE response; dash-dotted: GAWE response;dotted: MGAWE response.

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moments. In addition, from Table 3 note that the time taken by AWE is much morethan the time taken by GAWE. This could be because o is a larger fraction of N thanwhat it was in the previous examples, and as pointed out earlier, the steps following themoment generation in AWE are a function of the number of outputs. Finally, note fromthe computation times given in Table 3, and from the � gures, that although GAWE forq1 D q2 D 10 is slightly quicker than MGAWE for the same number of moments, GAWEis not accurate and MGAWE is. When GAWE is computed for q1 D q2 D 20, althoughthe response is accurate, the computation time for GAWE is then longer than MGAWEfor q1 D q2 D 10. Therefore, using MGAWE is again the best choice for solving theproblem.

Summary and Conclusions

This paper investigates several MORe techniques that are used to solve matrix equationsresulting when the FEM is applied to model electromagnetic wave propagation problems.The methods considered herein include AWE, GAWE with a short-term vector recurrencerelation, MGAWE also using a short-term recurrence, and MPVL. The methods arecompared and contrasted from several perspectives.

The � rst perspective is a detailed analysis of the algorithms which shows whattype of computations are performed during program execution. This indicates how muchcomputation time the algorithm requires and shows what requirement are placed on thematrix equations before the method can be applied. It is found that these requirementsare especially severe for MPVL. As a results, MPVL is extremely limited in applica-bility.

The second perspective from which the methods are compared and contrasted is thenumerical examples that are considered. It is shown how the methods can change depend-ing on whether a radiation or a scattering problem is considered. In particular, althoughMPVL can give an accurate solution for a radiation problem as shown in Example 1, it isnot feasible to apply MPVL to scattering problems as shown in Example 3. In addition,the numerical examples show how the ef� ciency of a method can depend on the numberof outputs required from the FEM mesh. This is the case in Example 3 for AWE, wherethe solution takes much longer than GAWE. Overall, it is found in this paper that themost computationally ef� cient method that gives an accurate response throughout thebandwidth of interest is MGAWE, followed by GAWE, and then by AWE. This result isfound by comparing the solutions generated by the methods to an LU decomposition ateach frequency point of interest.

References

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Freund, R. W. 1998. Reduced-order modeling techniques based on Krylov subspaces and their usein circuit simulation. Numer. Anal. Manuscr., 98-3-02, Bell Labs, Murray Hill, NJ.

Gallivan, K., E. Grimme, & P. Van Dooren. 1994. Asymptotic waveform evaluation via a Lanczosmethod. Appl. Math. Lett. 7(5):75–80.

Pillage, L. T., & R. A. Rohrer. 1990. Asymptotic waveform evaluation for timing analysis. IEEETrans. Computer Aided-Design 9(4):352–366.

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Tang, T. K., & M. S. Nakhla. 1992. Analysis of high-speed VLSI interconnects using the asymptoticwaveform evaluation technique. IEEE Trans. Computer-Aided Design 11(3):341–352.

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Zunoubi, M. R., K. C. Donepudi, J. M. Jin, & W. C. Chew. 1998. Ef� cient time-domain andfrequency-domain � nite-element solution of Maxwell’s equations using spectral Lanczos de-composition method. IEEE Trans. Microwave Theory Tech. 46(8):1141–1149.

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