382 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 14, 2015

Automated Reduced Model Order SelectionMichał Rewieński, Member, IEEE, Grzegorz Fotyga, Adam Lamecki, Member, IEEE, and

Michał Mrozowski, Fellow, IEEE

Abstract—This letter proposes to automate generation ofreduced-order models used for accelerated -parameter com-putation by applying a posteriori model error estimators. So far,a posteriori error estimators were used in Reduced Basis Method(RBM) and Proper Orthogonal Decomposition (POD) to selectfrequency points at which basis vectors are generated. This lettershows how a posteriori error estimators can be applied to automat-ically select the order of the reduced model in second-order ModelOrderReduction (MOR)methods.Three different error estimatorsare investigated and compared in order to arrive at a new MORscheme that is fast, reliable, and fully automated. The effectivenessof the proposed approach is verified by very high accuracy of thecomputed scattering parameters ( -parameters) for an exampleof a waveguide filter over a prescribed frequency band.

Index Terms—a posteriori error estimator, model orderreduction, -parameter computation.

I. INTRODUCTION

M ODEL order reduction (MOR) is a powerful techniqueof reducing the numerical cost of spectral analysis of

linear time-invariant (LTI) dynamical systems. It uses projec-tion of the original large set of state-space equations onto amuch smaller one. In the moment-matching MOR, the projec-tion space is constructed so that the value and derivatives (mo-ments) up to a certain order of the transfer function derived fromthe resultant projected system and the original one are matchedat a certain frequency (expansion point). MOR has been shownto yield substantial speed gains in RF circuit design for inter-connects analysis [1] and also in computational electromag-netics for the finite element method (FEM) [2] and the finite-dif-ference time-domain (FDTD) method [3]. MOR has also beenapplied for selected regions in mesh-based techniques (FDFD,FDTD, FEM) [4]–[6] to reduce the numerical cost of problemsinvolving very fine local meshes.Spectral analysis (frequency sweep) of large dynamical

systems can also be accelerated using the reduced basis method(RBM) and proper orthogonal decomposition (POD). In boththese techniques, solution snapshots taken at different fre-quencies are found, and a projection subspace is constructedfrom these snapshots. RBM and POD do not have the mo-ment-matching feature of MOR, but have been shown to pro-vide excellent speedups in wideband FEM simulations [7]–[9].

Manuscript received August 22, 2014; accepted October 16, 2014. Date ofpublication October 23, 2014; date of current version February 05, 2015. Thiswork was supported by the under Polish National Science Centre under Agree-ment #2012/07/B/ST7/01241.The authors are with the Department of Electronics, Telecommunications and

Informatics, Gdańsk University of Technology, 80-233 Gdańsk, Poland (e-mail:mrewiens@eti.pg.gda.pl).Color versions of one or more of the figures in this letter are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/LAWP.2014.2364849

One advantage of MOR over RBM and POD is that not onlyis a certain number of moments matched, but also projectionspace can be found with a single matrix factorization. On theother hand, one of the key and long-standing problems of MORbased on moment-matching is finding a reliable criterion forselecting the order of the reduced model that would guaranteedesired accuracy of the model. Automated model order se-lection has been available only for certain specific problemssuch as circuits (modeled by first-order systems) andused conservative a priori model error estimates or heuristicconvergence detection [10]. However, no practical model orderselectors have been proposed for moment-matching-basedMOR of second-order systems used to accelerate computationof -parameters. Yet, for macromodeling methods based onRBM and POD, a few a posteriori model error estimatorshave been developed for second-order electromagnetic systemsdiscretized using FEM [7]–[9], [11]. Their key advantage isthat they all inexpensively compute the model error estima-tors by exploiting the knowledge of the approximate solutionfound using the macromodel. So far, those estimators wereused only in the context of RBM and POD to automate oroptimize selection of frequency points used to generate theprojection basis vectors. However, the discussed estimatorswere never applied in moment-matching-based second-orderMOR methods such as SAPOR [12]. The goal of this letter is toinvestigate how the available a posteriori error estimators canbe used to automate second-order MOR, and establish whichof the estimators is most effective for the problem of gener-ating reduced-order models used for high-accuracy accelerated-parameter computations.

II. FORMULATION OF ELECTROMAGNETICS PROBLEMS INTERMS OF SECOND-ORDER SYSTEMS

To introduce MOR, RBM, and POD algorithms, let us con-sider FEM applied to the vector Helmholtz equation. Dielectri-cally loaded cavity including ports that excite the electromag-netic field may be described using the following weak E-fieldformulation [13], [14]:

(1)

where is a volume enclosing the modeled system, isa vector testing function, is the electric field, and arerelative permittivity and permeability, respectively, is thewavenumber, is the angular frequency, is the number ofports, is the surface of the th port, is the number ofmodes excited at the th port, is the amplitude of the thmode (excited at port ), is a vector normal to surface , and

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REWIEŃSKI et al.: AUTOMATED REDUCED MODEL ORDER SELECTION 383

is the tangential magnetic field for the th mode at port .If finite-element (FE) discretization is applied, both andare expanded as

(2)

where are FEM vector interpolation functions. Substitutingthe above expansion to (1) yields the second-order system

(3)

where and are system matrices,is the vector of weights that specify the electric field in thecavity, is the input vector of amplitudes for excitation modesat ports, and is the ( ) port selectionmatrix, whose columns are expressed as

(4)

If one defines the complex frequency parameter, then system (3) transforms into

(5)

Applying normalization and ,where is the port impedance (at some given frequency ),and specifying the output at the ports in terms of amplitudes ofthe voltage waves yields

(6)

III. A POSTERIORI ESTIMATORS FOR REDUCED-ORDERMODEL ERRORS

Applying MOR, RBM, or POD to (6) yields a reduced-ordersystem

(7)

where , and are thereduced-order systemmatrices, is the projectionmatrix,and is the reduced-order internal state vector of order .Although the projected system looks identical for all three

approaches, what sets these methods apart is the way the pro-jection bases are constructed. In MOR, the projection vectorsare constructed so that the moments of the original and pro-jected systems at a selected frequency point are matched. Onthe other hand, themoment-matching property, which deempha-sizes the importance of the internal states in favor of the systeminput–output behavior, is absent in RBM and POD. As proposedin [7]–[9] and [11], the projection basis in the latter methodsis generated from snapshot solutions found at several frequencypoints selected by using the computed a posteriori model errorestimators. The starting point for all the considered error esti-mators is a residual error function defined as

(8)

where is the approximate electric field com-puted using the reduced-order model (7). (The weights are

related to the reduced-order vector through the projectionmatrix : .) Assuming unit input waveamplitudes, the discrete analogue of (8) is the residual errormatrix

(9)

where matrix , and theresidual vector corresponds to the excitation of the th modeat the th port.In RBM and POD, the frequencies for projection basis con-

struction are selected based on a greedy algorithm that requiresrepetitive evaluation of error estimators. The error estimatorcan be related to the residual (9) in several ways, but sincethe direct computation of (9) at many frequency points is time-consuming, the resulting expressions cannot involve high-orderspace computation.In [8], the error estimator is a norm of the residual error func-

tion (8), which can be computed (in the dual space) as

(10)

where and is a positive-definite matrix withelements . Reference [8]describes in detail how the above estimator can be computedefficiently. A simplified version of the above estimator has beenproposed in [11] and computes Euclidean norm of the residualerror

(11)

Finally, the third approach proposed in [7] defines a goalfunction, geared toward -parameter error estimation, based oncomputing the residual current interaction with the port modalfields. This function that describes the impact of the residualcurrent (excited by the th mode at the th port) on the thmode of the th port is specified as

where is the residual field due to excitation of mode atport , is the electric field for the th mode for the th port,and is a port impedance normalization for port . Expandingthe residual field in terms of the vector basis functions [cf. (2)and (4)] yields

(13)

Using the above gives a matrix formula for the goal functioninvolving all the ports and all the modes

(14)

where is a diagonal portimpedance normalization matrix. Finally, the error estimator iscomputed as the normalized goal function

(15)

384 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 14, 2015

Fig. 1. MOR with automated order selector.

Fig. 2. ErrEst: fast error estimator.

It can be shown that the above estimator can be evaluated in-expensively by performing computations solely in the reduced-order state space.

IV. MOR WITH AUTOMATED ORDER SELECTION

The a posteriori error estimators presented in Section III havebeen originally proposed to guide an algorithm for selectingfrequency points for solution snapshots. They can, however,be used also as stopping criteria for the model order reductionprocess by computing the error estimates over a specified fre-quency range and requiring that those estimates donot exceed a prespecified tolerance.The entire scheme with automated order selection based on

an arbitrary moment-matching MOR technique and one of thethree fast error estimators outlined in Section III is summarizedin pseudocodes in Figs. 1 and 2.

V. NUMERICAL VERIFICATION OF THE ERROR ESTIMATORS

A series of numerical tests were performed that investigatethe quality of the three different estimators for reduced modelerrors presented above. The goals of the tests were to compareaccuracy of the error estimators and to determine the robustnessand usability of the error estimators as automated model orderselectors.To this end, reduced-order models were generated for a

fourth-order E-plane metal inset WR90 waveguide filter shownin Fig. 3. The inset lengths equaled 5, 15, 15, 15, and 5 mm. Theinset widths equaled 2 mm. The four waveguides (resonators)connecting the insets were each 15 mm long. The referencefilter response computed using FEM model is shown in Fig. 3.In this letter, (block) SAPOR [12] moment-matching MOR

algorithm has been used since it provides a numerically stablemethod for generating reduced-order models for second-order

Fig. 3. Values of and for the fourth waveguide filter example (shownin the inset).

Fig. 4. Estimated and actual absolute error for values computed using areduced model of order .

systems, and it also readily extends to second-order systemswith losses.Fig. 4 compares the three error estimates with the actual error

for values computed with a reduced-order model of orderover the specified frequency range. (The actual error has

been computed using the reference -parameter values com-puted with the full-order model.) First, note that the reducedmodel provides excellent accuracy over the frequency range,as confirmed by the values of the actual error. It is also ap-parent that the “goal-oriented” error estimator (15) providesa much more accurate approximation of the actual -param-eter error as compared to the two other estimators (10) and (11)based on residual norms over the entire frequency range.Further experiments investigated accuracy of the error

estimators for different sizes of the reduced-order models.Fig. 5 shows the maximum value of each error estimator overthe frequency range as a function of the model order . Also,the actual maximum error for the -parameters computedwith the reduced-order models is shown. It follows from the

REWIEŃSKI et al.: AUTOMATED REDUCED MODEL ORDER SELECTION 385

Fig. 5. Estimated and actual absolute maximum error for -parameters versusorder of the used reduced model.

TABLE IFINAL ORDER OF THE REDUCED MODEL SELECTED BY DIFFERENTSTOPPING CRITERIA. THE REFERENCE OPTIMAL ALGORITHMUSED EXACT -PARAMETER ERROR IN ITS STOPPING CRITERION

graph that while for lower orders of the reduced model, allthree estimates reasonably approximate the actual error, forhigher orders, the estimates (10) and (11) substantially divergefrom the actual error. Only the “goal-oriented” estimator (15)consistently follows the actual -parameter error.This result indicates that error estimates (10) and (11) unfor-

tunately cannot provide robust stopping criteria for the modelorder reduction process. Only the estimator (15) can be used toautomate the model order selection. This is confirmed by the re-sults fromTable I, which shows the orders of the reducedmodelsproduced by Algorithm 1 for different values of the target tol-erance (“tol” parameter) when different error estimates are ap-plied. As expected, only the “goal-oriented” estimator providesa usable stopping criterion for different tolerances, while theother two estimators lead to false nonconvergence of the MORprocess. The table also shows the order of the reduced modelselected using an optimal “model” MOR algorithm with stop-ping criterion based on actual, exact -parameter error. (Thisalgorithm clearly cannot be used in practice since it requires ana priori knowledge of the exact solution.) This result confirmsthat Algorithm 1 with “goal-oriented” error estimator yields re-duced models of orders that are close to the optimal orders forthe considered test structure and frequency range.

VI. CONCLUSION

This letter proposed a method for automated selection ofthe model order during MOR process, based on computinga posteriori error estimates. Three different error estimators, sofar used only in RBM and POD, were considered jointly withSAPOR reduction algorithm as stopping criteria. Numericaltests indicate that applying one of the estimators (the “goal-ori-ented” estimator) yields a reliable and fully automated modelorder selector. It is shown that the proposed automated MORapproach generates reduced models that very accurately ap-proximate -parameters of the initial system over a prescribedfrequency range. The described automated MOR algorithmcan also be readily extended to systems that include losses.Finally, the “goal-oriented” estimator can be easily adaptedto reduction algorithms based on the finite difference method(e.g., [5] and [6]) and applied in FDTD analysis.

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