A Fully Adaptive Scheme for Model Order Reduction ... - IEEE Xplore

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A Fully Adaptive Scheme for Model Order Reduction Based on Moment Matching Lihong Feng, Jan G. Korvink, Member, IEEE, and Peter Benner Abstract— A fully adaptive model order reduction scheme based on moment matching is proposed to derive the reducedorder models of linear time-invariant (LTI) systems. According to the given error tolerance, the order of the reduced-order model as well as the expansion points for the transfer function is automatically determined on the fly during the process of model reduction. In this sense, the reduced-order model is automatically obtained without assigning the number of moments and expansion points in a priori, which is a prerequisite for the standard implementation of model reduction based on moment matching. The proposed adaptive scheme is found to be efficient when it is tested on various LTI systems. Index Terms— Adaptive algorithms, mathematical model, numerical simulation, reduced order systems.

I. I NTRODUCTION

M

ODEL ORDER reduction (MOR) for linear timeinvariant (LTI) systems has demonstrated great efficiency in the simulation of large-scale systems. Model reduction methods based on balanced truncation are very efficient for medium- to large-scale problems [1], [3]. It is well known that a global error bound for the reduced-order model makes the model reduction process automatic. Although model reduction methods based on the Krylov subspace and moment matching are much simpler to be implemented and also require much less computational complexity than methods based on balanced truncation, they cannot be performed adaptively. One has to always decide a priori on the number of moments to be matched and the position of the expansion points to be used for the transfer function. In this case, one runs the risk of performing model reduction repeatedly before Manuscript received February 25, 2015; revised August 3, 2015; accepted September 13, 2015. Date of publication November 3, 2015; date of current version December 11, 2015. This work was supported in part by the Deutsche Forschungsgemeinschaft under Grant BE 2174/7-1, in part by the Automatic, Parameter-Preserving Model Reduction for Applications in Microsystems Technology, in part by the Bundesministerium für Bildung und Forschung (BMBF)-Imaging of Neuro Disease Using high field MR And Contrastophores (INUMAC) Project, an operating grant of the University of Freiburg, and in part by the Collaborative Project entitled Nanoelectronic Coupled Problems Solutions through the European Union within the FP7-ICT-2013-11 Program under Grant 619166. Recommended for publication by Associate Editor F. G. Canavero upon evaluation of reviewers’ comments. L. Feng and P. Benner are with the Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg 39106, Germany (e-mail: feng@ mpi-magdeburg.mpg.de; [email protected]). J. G. Korvink was with the Department of Microsystems Engineering, Institute for Microsystems Technology, Freiburg Institute of Advanced Studies, University of Freiburg, Freiburg im Breisgau 79085, Germany. He is now with the Institute of Microstructure Technology, Karlsruhe Institute of Technology, Karlsruhe 76344, Germany (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCPMT.2015.2491341

a satisfactory solution is found. The worst case is that after several attempts, the reduced-order model remains inaccurate. The most difficult task is choosing suitable expansion points, and in many cases, using only one expansion point at zero is not sufficient for an accurate and small reduced-order model. However, how to choose proper nonzero expansion points and decide on the corresponding number of moments accordingly are still problems unsolved in general. The above problems are therefore the most tricky issues for model reduction based on moment matching. An early method called complex frequency hopping was proposed in [5] to illustrate a principle of choosing multiple expansion points of the transfer function. Using a binary search algorithm, the expansion points were chosen with respect to the common poles contained in both circles of the neighboring expansion points. However, the poles of the transfer function were computed based on explicit moment matching. This means that the moments needed to compute the poles were explicitly computed by recursive matrix-vector multiplications, in the same way as the asymptotic waveform evaluation method in [6]. Therefore, the poles computed were actually not accurate because of numerical instability, although they would represent the actual poles if computed with precise arithmetic. Moreover, in order to compute the actual poles according to the theorem of convergence (see [6, Th. 1]), higher order moments must be computed. However, explicit moment computation cannot guarantee that the higher order moments are accurately computed, again because of numerical instability. Usually, only the first few moments can be accurately determined. A transfer-function-based approach was proposed in [7] using a similar binary search algorithm as in [5]. However, one reduced-order model was constructed at each expansion point, implicating that multiple reduced-order models must be constructed for multiple expansion points. The reduced-order models were still based on explicitly calculating the moments. Later on, two approaches were proposed in [17] for sample point selection, which were stated to be applicable to MOR methods requiring appropriate sample (expansion) point selection. One of them was called the linear matrix inequality (LMI) approach and hence was too expensive for large-scale systems due to the complexity of current LMI solvers. A second approach based on a heuristic search scheme with lower cost was then presented. The basic principle followed the variance of the reduced-order models motivated by the ideas from statistics. The idea was that an initial set of search points was randomly defined over the search space (frequency space), then the variance was evaluated at

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FENG et al.: FULLY ADAPTIVE SCHEME FOR MOR BASED ON MOMENT MATCHING

each search point, and the point with maximal value of the variance was chosen as the next sample (expansion) point. At each iteration, the search points were resampled so that those points producing smallest variances were dropped from the search points, since the reduced-order model was already accurate at those points. The variance computation is independent of the original full system and possesses low computational cost. An adaptive scheme for automatically choosing the number of moments for each given expansion point was proposed in [10]. The set of expansion points was predetermined using engineering heuristics or experience. At each iteration, an expansion point causing the biggest moment error (the difference between the moment of the original transfer function and that of the reduced transfer function) was selected, and a higher order moment was matched at that point. Using a heuristic estimation of the moment error, an adaptive rule of selecting the expansion points was proposed in [2]. Combined with the method of selecting the moments in [10], it was able to give an adaptive scheme of selecting both the expansion points and the moments. How to determine the order of the reduced-order model was unclear, though. Recently, the issue of multipoint expansion for the transfer function has been readdressed in [8]. The proposed method selected interpolation points yielding a locally H2 -optimal reduced-order model, but the number of moments matched was fixed to two in each point for single-input single-output (SISO) systems. For more than one input and/or output, the method yielded no moment matching, but tangential interpolation conditions were satisfied. One disadvantage is that the method is actually not flexible in choosing the number of the interpolation points, as a fixed number of interpolation points must be prescribed. Adapting the order to a given tolerance requires rerunnig the whole algorithm, possibly several times with increased number of prescribed interpolation points, without the possibility of reusing the already computed information. The rational global Arnoldi algorithm proposed in [10] and the iterative rational Krylov algorithm in [8] were combined in [4] to be able to adaptively determine the expansion points, the number of moments, and the order of the reduced-order model. A similar limitation exists, i.e., to start the algorithm, a starting set of expansion points must be assigned. An adaptive scheme for computing the reduced-order models was explored in [9]. Therein, the expansion points and the corresponding moments can be automatically selected by computing and monitoring the norms of the residuals at samples sweeping through the frequency intervals of the selected expansion points. It may happen that the residualbased error monitor may overestimate the error of the reducedorder model, especially for those problems where the output response and the output error are of importance. As a result, the reduced-order model may not be as small as desired. Beyond the achievements of the reviewed methods above, we not only propose a fully adaptive scheme of choosing the expansion points but also show how to adaptively determine the corresponding number of moments matched at each chosen expansion point, and finally, we show that the order of the reduced-order model can also be automatically decided.

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The starting set of the expansion points is not required from the user. We review the standard moment matching model reduction method passive reduced order interconnect macromodeling algorithm (PRIMA) [12] in Section II. In Section III, the key ideas of our adaptive scheme are described, and the implementation details are discussed as well. The simulation results for several examples in integrated circuit design, including a package model, are presented in Section IV. II. R EVIEW OF MOR BASED ON M OMENT M ATCHING Among the various model reduction methods based on moment matching, the basic methods for linear systems include those described in [12] and [13]. Since the method PRIMA in [12] preserves the passivity of the original system, it is usually the method of choice for MOR of linear systems. Our adaptive scheme is based on PRIMA and aims to obtain the reduced-order model of the linear descriptor system d (1) C x(t) + Gx(t) = Bu(t), y(t) = L T x(t). dt n m Here, x(t) ∈ R is the state vector, u(t) ∈ R 1 is the input signal, and y(t) ∈ Rm 2 is the output response. We consider general multi-input and multioutput (MIMO) systems, i.e., m 1 ≥ 1 and m 2 ≥ 1. Many model reduction methods are based on the idea of projection, i.e., a basis V ∈ Rn×r of a subspace that approximates the manifold in which the state vector x(t) resides is first computed, and then the reduced-order model is obtained by the Petrov–Garlerkin projection by forcing the residual to be zero in the test subspace spanned by the basis W ∈ Rn×r . The reduced-order model is d W T C V z(t) + W T GV z(t) = W T Bu(t) dt yr (t) = L T V z(t). (2) The variable r ≤ n indicates the dimension of system (2), which is also called the order of the reduced-order model. Model reduction methods based on projection differ in the computation of the matrices W and V . In order to preserve the passivity of the original system, PRIMA uses W = V such that the reduced-order model is guaranteed to be passive if the original system is passive and the system matrices C, G, B, and L satisfy certain conditions [12]. The matrix V is constructed from the moments of the transfer function (matrix) H (s) = L T (sC + G)−1 B.

(3)

The Laplace domain variable s is related to frequency f by √ s = 2πj f , with j = −1. If we expand H (s) around an expansion point s0 as H (s) = L T [(s − s0 )C + (s0 C + G)]−1 B ∞  = L T [−(s0 C + G)−1 C]i (s0 C + G)−1 B (s − s0 )i    i=0

:=m i (s0 )

then m i (s0 ), i = 0, 1, 2, . . ., are the i th-order moments of the transfer function H (s). The columns of V span the subspace ˜ 0 )−1 C) B(s ˜ 0 ), . . . , ˜ 0 ), (G(s range{V } = span{ B(s −1 q ˜ 0 ) C) B(s ˜ 0 )} (G(s (4)

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˜ 0 ) = s0 C + G and B(s ˜ 0 ) = (s0 C + G)−1 B. In this where G(s −1 i ˜ ˜ paper, we call (G(s0 ) C) B(s0 ), i = 0, 1, . . . , the i th-order moment vectors. It is proved in [12] that the first q + 1 moments of the (reduced) transfer function of the reduced-order model match those corresponding moments of the original transfer function H (s) Hr (s) − H (s) = o(s − s0 )q .

(5)

Here, Hr (s) is the reduced transfer function. It implies that for a fixed s0 , a larger q produces a more accurate Hr (s). From (5), one can see that the accuracy of Hr also depends on s0 . For a fixed q, if s  is far away from s0 , the error of Hr at s  is usually larger than for those variables that are close to s0 . This implies that if s0 is chosen as zero, then Hr (s) is probably not accurate at high frequencies. One may also use multiexpansion points to overcome the large error caused at high frequencies. However, it is not immediately clear as to how to choose suitable expansion points or indeed how many corresponding moments should be taken. For the current model reduction methods based on moment matching, the two variables s0 and q are heuristically given a priori, because there is not a general rule for properly choosing them. III. A DAPTIVE S CHEME A technique of automatically choosing the number of moments as well as the expansion points is proposed in this section. The motivation is that using a single expansion point may not get a reduced-order model satisfying the preassigned accuracy and the acceptable model order r . Usually multiple expansion points must be used to cover the accuracy over the whole frequency interval: [ fl , f h ]. A brief summary of the scheme is as follows. A single expansion points is initially decided (usually sl = 2πj fl ). Then keep matching the moments at sl . Once the order r exceeds a preassigned order rmax , a second expansion point is selected as the one farthest from sl . The moments for each expansion point are iteratively added to the subspace of range{V }. By observing the error estimated at a point, which causes the biggest error in the frequency interval between two neighboring expansion points, one can decide on the number of moments necessary for each expansion point. If at some stage, the order r exceeds rmax again, new expansion points must be selected. They are selected as the points at which the biggest error is estimated. The above process can be repeated until certain criteria are satisfied. The motivation of our approach regarding the adaptive selection of expansion points in some sense resembles the greedy sampling procedures used in reduced basis (RB) methods for model reduction of parameterized partial differential equations (see [15], [16], and the references therein). In [15] and [16], the parameter samples that cause the largest error (according to certain error estimation) are selected as the candidates for generating the snapshots. Our adaptive moment selection process has no direct analogy in greedy sampling/RB methods. To make the procedure clear, we first explain the adaptive scheme of matching moments for a given single expansion

point in Section III-A. Then in the following section, the adaptive scheme of matching moments is applied to the scheme of adaptively choosing both the expansion points and the corresponding number of moments, as well as the proper order r of the reduced-order model. A. Adaptively Choosing Moments Adaptive Scheme of Choosing Moments: M1: As input, we specify the acceptable accuracy of the reduced-order model tol. tol here means the acceptable maximal error maxs∈[sl , sh ] ||H (s) − Hr (s)||2 /||H (s)||2 between the reduced transfer function and the original transfer function. Note that the absolute error maxs∈[sl , sh ] ||H (s) − Hr (s)||2 should be used if appropriate, though we take the relative error as an example here. ˜ 0 ) is chosen, and the M2: The first moment vector B(s projection matrix V is computed such that range{V } = ˜ 0 )}. Assume that s ∗ ∈ [sl , sh ] is the variable span{ B(s that is farthest away from s0 (s ∗ can be taken as sh if s0 = sl ). Compute the reduced transfer function Hr (s ∗ ) and the error ε(s ∗ ) = ||H (s ∗) − Hr (s ∗ )||/||H (s ∗)||. M3: For i = 1, 2, . . . , while ε > tol, add new moment ˜ 0 ) to V such that vectors (G(s0 )−1 C(s0 ))i B(s ˜ 0 ), (G(s ˜ 0 )−1 C) range{V } = span{( B(s ˜ 0 )−1 C)i B(s ˜ 0 ), . . . , (G(s ˜ 0 )} × B(s and compute Hr (s ∗ ) and ε until ε < tol. From the above steps, we see that at each iteration step i , only the error at the point s ∗ is checked, and usually this is sufficient. Although in many situations, especially when q is small, the largest error over [sl , sh ] is not exactly at s ∗ , it has a similar magnitude as the error at s ∗ , i.e., the largest error is O(error(s ∗ )). Once H (s ∗) is computed in M2, it will be reused in M3. The computation of the reduced transfer function Hr (s ∗ ) costs even less, because V usually has very few columns. This means that only little additional computation is necessary in order to check the error of the reduced-order model. B. Adaptive Scheme for Choosing Expansion Points, Moments, and the Order In the following, the adaptive scheme of both choosing expansion points and deciding the number of moments is delineated. 1) Adaptive Scheme for Choosing Expansion Points, Moments, and the Order r : E1: At the start, one should choose an acceptable dimension of the reduced-order model, say rmax (depending on one’s experiences on the problem under consideration), as well as the acceptable accuracy tol. rmax will be adjusted to a proper number during the adaptive scheme if it was selected too small. E2: The first expansion point is chosen as s0 = sl and the projection matrix V is computed such that ˜ 0 )}. Compute ε = ||H (sh ) − range{V } = span{ B(s Hr (sh ))||/||H (sh )||.

FENG et al.: FULLY ADAPTIVE SCHEME FOR MOR BASED ON MOMENT MATCHING

E3: Let col be the number of columns in V . For i = 1, 2, . . . , while ε > tol and col < rmax , add new moment ˜ 0 )−1 C)i B(s ˜ 0 ) to V , such that range{V } = vectors (G(s −1 ˜ 0 ), (G(s ˜ 0 ) C) B(s ˜ 0 ), . . . , (G(s ˜ 0 )−1 C)i B(s ˜ 0 )}. span{ B(s Compute ε = ||H (sh ) − Hr (sh )||/||H (sh )||. E4: Inside the for loop in E3, if col > rmax , i.e., if the dimension of the reduced-order model that will be produced by V is larger than the acceptable dimension, then add the second expansion point s1 = sh and go to E5. Otherwise, if col < rmax and ε < tol, compute the reduced-order model and stop. E5: If s1 is added as the second expansion point, then compute a new matrix V based on the two expansion points s0 = sl and s1 = sh , such that it includes the 0th-order moment vectors of both expansion ˜ 1 )}. Because ˜ 0 ), B(s points, i.e., range{V } = span{ B(s ˜ B(s0 ) has been computed, it is simply picked out from the previously computed matrix V . One only has ˜ 1 ). Now only check the error at s2 : to compute B(s ε = ||H (s2 ) − Hr (s2 )||/||H (s2)||. Here, s2 = 2πj f 2 is the midpoint between s0 and s1 with f 2 = f 0 + ( f 1 − f 0 )/2. E6: While ε > tol and col < rmax , add the i th (i > 0)-order moment vectors of both expansion points, such that ˜ 0 ), B(s ˜ 1 ), (G(s ˜ 0 )−1 C) B(s ˜ 0 ), range{V } = span{ B(s −1 ˜ ˜ ˜ (G(s1 ) C) B(s1 ), . . . , (G(s0 )−1 C)i ˜ 1 )}. ˜ 0 ), (G(s ˜ 1 )−1 C)i B(s × B(s (6) ˜ 0 )−1 C)i B(s ˜ 0 ) can be simply picked out Likewise, (G(s ˜ 1 ) is ˜ 1 )−1 C)i B(s from the previous V so that only (G(s computed. E7: If ε > tol and col > rmax , then take s2 as the new expansion point and go to E8. Otherwise, if col < rmax and ε < tol, then compute the reduced-order model and stop. E8: Consider a general case of m expansion points. Similarly, the 0th-order moments of all expansion points ˜ 0 ), are added to V , such that range{V } = span{ B(s ˜ ˜ B(s1 ), . . . , B(sm )}. For i = 1, 2, . . . , check the error at the midpoint of each pair of neighboring expansion points. If none of them is smaller than tol, continue adding the i th-order moments. Otherwise, if any of them, say s j 0 is smaller than tol, stop adding higher order moments of the two expansion points that generate s j0 . Also stop checking the error corresponding to s j0 and only check the midpoints with errors larger than tol. Keep adding higher moments of the corresponding expansion points until all the errors are smaller than tol or col > rmax . Midpoints like s j0 may arise at different steps of i , and they will be deleted gradually with increasing order i . E9: Final check: a) If all of the errors are smaller than tol and col < rmax , then stop. The projection matrix V of the reduced-order model is obtained. b) If all of the errors are smaller than tol and col > rmax , and only the 0th-order moments are included in range(V ), then it means that it

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is impossible to obtain a reduced-order model with order smaller than rmax and satisfying the required tol. The smallest order of the reducedorder model is automatically adjusted to col instead of rmax . c) If not all of the errors are smaller than tol and col > rmax , and if only the 0th-order moments are included in range(V ), then it means that the initially given rmax is too small to obtain a reduced-order model satisfying the required tol and rmax . One should modify the initial rmax to a larger value, e.g., rmax = 2 ∗ col, which is double the number of the columns in the current projection matrix V, and one should add the current left midpoints (except for those deleted points s j0 with error smaller than tol) as the new expansion points, and then repeat E8. d) If not all of the errors are smaller than tol and col > rmax , and if higher order moments are also included in the current range(V ), then it tells us that we still have the opportunity to use more expansion points to reduce the order of the reduced-order model and meet the accuracy requirement tol. For this case, we do not have to modify rmax and simply need to add the current left midpoints (except for those deleted points s j0 with an error smaller than tol) as the new expansion points, and then go to E8. 2) Adaptively Finding the Order of the Reduced-Order Model: It can be seen from Step E9 that the order of the reduced-order model can also be adaptively determined if the initially given rmax is too small to obtain a sufficiently accurate reduced-order model. For example, if rmax is set to be 10 and the reduced-order model is still not accurate enough, then we can set a larger rmax and continue adding more expansion points to obtain a new reduced-order model. The previously computed moment vectors can still be made use of, and the algorithm does not have to be restarted. After a few iterations, a new reduced-order model with an adaptively determined order is obtained. C. Modification Scheme for the Tested Midpoints In E5, because s2 is the point that is farthest away from both s0 and s1 , we assume that the point that causes the largest error is usually around s2 . In fact, this is not always true. There are cases for which the error at s2 is much smaller than the errors at other points between s0 and s1 . The same may also happen to other midpoints in E8. We will deal with these cases in this section. Taking s2 as an example, if the two expansion points s0 and s1 separately cause very different errors at s2 , then the reduced-order model obtained using both expansion points may not have the biggest error at s2 . This means that if we compute two individual reduced transfer functions Hr0 and Hr1 by expanding the original transfer function H (s) at s0 and s1 separately and by matching the same number of moments, the error of Hr0 at s2 , ε0 (s2 ) = ||H (s2) − Hr0 (s2 )||/||H (s2)||, is either much smaller or much larger than the error of

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Hr1 at s2 , ε1 (s2 ) = ||H (s2)− Hr1 (s2 )||/||H (s2)||. For example, it could happen that ε0 (s2 ) = 1 and ε1 (s2 ) = 10−5 . In this case, if we derive a single reduced transfer function using multipoint expansion for H (s), i.e., expanding H (s) at both s0 and s1 , the error ε∗ (s2 ) of the single reduced transfer function at s2 could also turn out to be much smaller than ε0 (s2 ). This is because s1 contributes more to the accuracy of the reduced-order model at s2 than s0 does and therefore speeds up the decrease of ε∗ (s2 ). If we continue matching more moments at s0 and s1 and checking ε∗ (s2 ), then it [ε∗ (s2 )] probably becomes smaller than tol more quickly than would the errors at other points that are closer to s0 . This is because those points are farther away from s1 , to which s1 could not contribute as much as to s2 . As a result, the algorithm stops before the error of the reduced-order model is smaller than tol at all the frequencies. One solution to the above problem is to modify the current tested midpoint s2 to a new point s ∗ such that s ∗ satisfies min < ε0 (s ∗ )/ε1 (s ∗ ) < max. Here, max and min are taken as some reasonable values, which show that the difference between ε0 (s ∗ ) and ε1 (s ∗ ) should not be too big. One can take, e.g., max = 10 and min = 0.1. In order to find s ∗ , we take a few samples between s0 and s2 if ε0 (s2 ) > ε1 (s2 ); otherwise, we take a few samples between s2 and s1 . We compare ε0 and ε1 at the samples one after the other until s ∗ is found. During the process, the values of the original transfer function H (s) at the sample points have to be estimated. However, the samples are usually very few and less than ten. Furthermore, it is not necessary that H (s) be computed at all the sample points; usually s ∗ has been found before all the sample points are estimated. Therefore, the computational complexity of finding s ∗ is equivalent to matching a few more moments. If we look at min < ε0 (s ∗ )/ε1 (s ∗ ) < max, it tells us that the difference between Hr0 (s ∗ ) and Hr1 (s ∗ ) should not be large. Therefore, we can also use the criterion min < ||Hr0 (s ∗ )||/||Hr1 (s ∗ )|| < max as the stopping criterion of searching s ∗ . In this way, we avoid calculating the original transfer function at the sample points, which will save computation time. Our modification scheme applies to any tested midpoint between any pair of neighboring expansion points. This means that if the above case happens to other expansion points other than s0 and s1 , we can treat those tested midpoints in the same way. Note that for the case that any midpoint smid is modified to a new tested point s ∗ , s ∗ rather than smid will be selected as the new expansion point in E8. In this way, we have realized the fully adaptive scheme, which only checks the errors of the points that really produce the largest errors. The resulting expansion points may not be located equidistantly between sl and sh , a situation that is best explained in Table V. IV. S IMULATION R ESULTS A. Examples In this section, we use several examples to test the efficiency of the adaptive scheme. The first example is an RLC tree circuit, which can be instantiated for any level l.

Fig. 1.

Clock tree example.

Fig. 2.

Bus line example.

Between two consecutive levels, the circuit branch segments  ofi the lower level each split into two children, yielding l−1 i=0 2 in the circuit of level l. Each segment is made up of four RL pairs in series, representing the wiring on a chip with four capacitors to ground and the wire–substrate interaction (see Fig. 1). We construct two models for the example with respect to different levels of l. The dimensions of the two models are n = 6134 and n = 24 566, respectively. The second example shown in Fig. 2 is a three conductor radio-frequency bus line model, split into 16 RLC segments. The two outer bus line conductors of the coplanar waveguide are terminated at both ends with resistors to ground. The middle signal line is terminated by a capacitor to ground at the one end and is driven at the other end by a current source shunted by a resistor to ground. The segment model represents each conductor segment by a line resistance, an intersegment capacitance, a line self-conductance, and an intersegment mutual inductance. The original system is of size n = 147. For both examples, the input for each example is a step signal with a 0.1-ns rise time, and the frequency range of interest is set as [ fl , fh ] = [0, 3 GHz] [7]. The third example results from a PEEC discretization of an on-chip metallic spiral inductor in a square geometry. The inductor features a ground plane shield that extends beyond the inductor’s outer windings. In addition, the inductor features a grounded guard ring, which helps to reduce parasitics even further (see Fig. 3).1 This example has been tested by many methods, including those based on balanced truncation [11]. 1 The details of this example can be found at URL: http://simulation.unifreiburg.de/downloads/benchmark/Peek inductor (38891).

FENG et al.: FULLY ADAPTIVE SCHEME FOR MOR BASED ON MOMENT MATCHING

Fig. 3.

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Spiral inductor with part of the overhanging copper plane.

Fig. 4. Package model (picture courtesy of MAGWEL NV, Leuven, Belgium).

Fig. 5. Error of the reduced transfer function of the clock tree model with n = 24 566.

The resulting descriptor system is of dimension n = 1434. The interesting parameter for this example is the resistance and inductance of the inductor. The frequency range of interest is [ fl , fh ] = [0, 10 GHz]. All of the above three examples are SISO systems. We further present two examples with MIMO. One is from the Subroutine Library in Systems and Control Theory (SLICOT) benchmark collection.2 It is derived from an electrical circuit of a CMOS-inverter-driven two-bit bus modeled by 40 RLC sections whose discretized equation system was determined using the modified modal analysis available in Simulation Program with Integrated Circuit Emphasis, yielding a numerical system with order n = 980, which has four inputs and four outputs [12]. The frequency of interest is [ fl , fh ] = [0, 3 GHz]. The last example is a package model with 11 inputs and 22 outputs, and order n = 21 760, as shown in Fig. 4. The interesting frequency interval is [ fl , fh ] = [0, 107 ] Hz.

and therefore, we have set sl = 0 and sh = 6πj 109 in the adaptive scheme. For the spiral inductor, we have sh = 2πj 1010 , and for the package model, sh = 2πj 107 . All of the simulation results are obtained with version 2010b of MATLAB. For the adaptive scheme for the SISO models, the error in the adaptive scheme ε(s 0 ) = ||H (s 0) − Hr (s 0 )||/||H (s 0)|| at a certain point s 0 is defined as ε(s 0 ) = |H (s 0 ) − Hr (s 0 )|/ |H (s 0)|, i.e., the relative error of the reduced transfer function at s 0 . When applying the adaptive scheme to the MIMO examples, the error ε(s 0 ) at a certain point s 0 is computed as ε(s 0 ) = maxik |Hik (s 0 ) − Hrik (s 0 )|/|Hik (s 0 )|, which is the maximal relative error of the entries Hrik at s 0 . It can also be explained as the maximal relative error of the reduced transfer functions Hrik relating the kth input signal to the i th output signal, k = 1, . . . , m 1 , i = 1, . . . , m 2 . Once the errors ε(s i ) at all of the tested points s i are smaller than tol, the reduced-order model is obtained.

B. Rules of Implementation

C. Results of the Adaptive Scheme for a Fixed tol

We set rmax = 20 for the SISO models and set rmax = 50 for the MIMO model. However, the proposed adaptive scheme is flexible for various choices of rmax . For the SISO examples, the relative error e R between the reduced transfer function and the original transfer function is measured by e R = maxsi |H (si ) − Hr (si )|/|H (si )|, where we have taken 2000 frequency samples si = 2πj f i , i = 1, 2, . . . , 2000. For the MIMO example, we use eik R instead of e R to indicate the relative error of the (i, k)th entry of the reduced transfer matrix Hr (s). We use Hik (s) to represent the (i, k)th entry of H (s) and Hrik (s) to represent the (i, k)th entry of Hr (s). Hik (s) is the transfer function relating the kth input to the i th output. We define eik R = maxs j |Hik (s j ) − Hrik (s j )|/ |Hik (s j )|, j = 1, 2, . . . , 2000. Here, | · | means the magnitude of the transfer function. For all but the spiral inductor example and the package model, the frequency range of interest is [0, 3 GHz],

In this section, we show the results of the adaptive scheme when tol = 1 × 10−2 for all the examples. 1) Results for the SISO Examples: For the model of the clock tree with n = 24 566, the expansion points are adaptively chosen as s0 = sl = 2πj f 0 = 0, s1 = sh = 2πj f h = 6πj 109 , and s2 = sl + (sh − sl )/2, the midpoint of sl and sh . Here, f 0 = 0 GHz and f h = 3 GHz are the corresponding frequency points. The moments matched at each expansion point is q = 4. The order of the reduced-order model is r = 20. The relative error e R of the reduced transfer function is plotted in Fig. 5. According to our adaptive scheme, the tested point for error control is the midpoint s3 between sl and s2 and the midpoint s4 between s2 and sh . From Fig. 5, we observe that the frequency points with the largest errors are very close to f 3 = 0.75 GHz and f 4 = 2.25 GHz (si = 2πj f i , i = 3, 4), which is in agreement with our adaptive scheme. f 4 is also the final tested midpoint that produces the largest error. The other midpoint f3 corresponds to s j0 in E10, and it is removed from the list of midpoints during the adaptive scheme.

2 URL: http://www.icm.tu-bs.de/NICONET/benchmodred.html

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Fig. 6. Error of the reduced transfer function of the clock tree model with n = 6134.

Fig. 7. Comparison of the transfer functions of the clock tree model with n = 6134.

For the model of the clock tree with n = 6134, two expansion points have been adaptively chosen, s0 = sl = 0 and s1 = sh . The number of moments matched at each expansion point is q = 6. The reduced-order model is of order r = 18. Fig. 6 gives the relative error of the reduced transfer function. It shows that the error around the midpoint between sl and sh is the largest; this further justifies our adaptive scheme. From Fig. 7, we observe that the reduced transfer function matches the original transfer function very well for each peak. The error of the reduced transfer function of the clock tree model with n = 24 566 (see e R in Fig. 5) is even smaller than e R in Fig. 6, and therefore, the reduced transfer function of the model with n = 24 566 should match the original transfer function even better. To avoid repetition, we do not show it here. We obtain a reduced-order model of order r = 3 for the bus line model. For this example, only s0 = sl is used for expansion and q = 3 moments are matched. The tested point for error control is sh = 6πj 109. The error of the reduced transfer function is plotted in Fig. 8. As we can see, the error at f ∗ = 3 GHz is the largest. The relative error e R of the reduced transfer function of the spiral inductor model is plotted in Fig. 9, which remains below tol = 10−2 for all frequencies. It shows that the error

Fig. 8.

Error of the reduced transfer function of the bus line model.

Fig. 9.

Error of the reduced transfer of the spiral inductor.

Fig. 10.

Resistance of the spiral inductor.

at f ∗ = 10 GHz is the largest, which corresponds to the tested point sh . The reduced system is of order r = 9 and only s0 = sl is used as the expansion point; the tested point picked by the adaptive scheme is sh . Fig. 10 includes the resistance of the spiral inductor obtained by full simulation of the original system as well as the resistance computed from the reduced system. Fig. 11 compares the inductances of the spiral inductor computed,

FENG et al.: FULLY ADAPTIVE SCHEME FOR MOR BASED ON MOMENT MATCHING

Fig. 11.

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Inductance of the spiral inductor. TABLE I

eik R FOR E ACH PAIR ik FOR THE SLICOT B ENCHMARK MIMO E XAMPLE W ITH tol = 1 × 10−2 , rmax = 50, AND [ fl , f h ] = [0, 3 GHz]

respectively, from the original system and the reduced system. Unfortunately, the reduced-order model produces an inaccurate resistance. This may be because the resistance is reciprocally related to the real part of the transfer function. If we set tol = 1 × 10−3 , we see that the resistance (see the star symbol line in Fig. 10) is already indistinguishable from the full simulation result. With tol = 1 × 10−3 , the reduced-order model is of the order r = 9, but has used two expansion points instead. 2) Results of the MIMO Examples: For the first MIMO example, we compare Hik with Hrik for each pair of i k, where i , k = 1, 2, . . . , 4, and list the relative error eik R between Hik and Hrik in Table I. The order of the reduced-order model is r = 48. Two expansion points s0 = sl = 0, s1 = sh = 2πj f h = 6πj 109 have been used, and four moments are matched for each expansion point. The tested point is the midpoint s2 = s0 + (s0 − s1 )/2. We observe from Table I that the error of the reduced transfer function Hrik , i , k = 1, 2, . . . , 4, in the whole frequency interval is very close to tol, though not below tol. The second MIMO system of the package model has many more inputs and outputs: m 1 = 11 inputs and m 2 = 22 outputs, and to avoid redundancy, we only present the maximal, minimal, and average errors defined as emax = max eik R R ik

ik emin R = min e R ik

  (m 1 m 2 ), i = 1, . . . m 2 , k = 1, . . . , m 1 . eave eik R = R ik

(7)

Fig. 12. Error of the reduced transfer function with s0 = sl and s1 = sh for the spiral inductor with the mid and modified tested point.

When tol = 10−2 , the reduced-order model for the package model has order r = 22. Only a single expansion point s0 = 0 is adaptively chosen, and two moments are adaptively matched. The errors of the reduced-order model are ave −4 min −5 emax R = 6.4446 × 10 , e R = 0, and e R = 4.5727 × 10 . It is worth mentioning that if we run PRIMA with the above-indicated expansion points and number of moments, then the computed reduced-order model produces almost the same results as that produced by our adaptive scheme. D. Efficiency of the Modification Scheme In this section, we demonstrate the efficiency of the modification scheme proposed in Section III-C to find the smallest sufficiently accurate system. We take the spiral inductor as an example for explanation. More simulation results to which the modification scheme has been applied can be found in Table V in Section IV-E. From Section IV-C, we know that if tol = 1 × 10−2 , we cannot obtain a reduced-order model that gives the correct resistance of the spiral inductor within acceptable accuracy. We need to set tol = 1 × 10−3 , where two expansion points s0 = sl and s1 = sh are adaptively chosen. If we do not use the modification scheme in Section III-C, the tested point for the error control is the midpoint s2 = 2πj f 2 with f 2 = 5 × 109 between sl and sh . The reduced-order model is of order r = 6. The error of the transfer function is plotted in Fig. 12 (solid line). We see that the error is the largest at the frequency around f = 1 × 108 rather than at the midpoint f2 . And the error of the reduced transfer function is actually not smaller than tol = 1 × 10−3 at all frequencies, though the error at s2 is already around 10−6 . As is analyzed in Section III-C, this is because the errors of the individual reduced transfer functions computed, respectively, by sl and sh at s2 show a large difference. For this example, when q = 1, we have that ε0 (s2 )/ε1 (s2 ) = 5 × 103 . However, if we use the modification scheme of Section III-C and set max = 10, min = 0.1, a new tested point s ∗ = 2πj f ∗ , f ∗ = 1.58 × 108 is found. Using this new tested

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TABLE II

TABLE III

A DAPTIVITY OF THE S CHEME FOR D IFFERENT tol

A DAPTIVITY OF THE S CHEME FOR D IFFERENT tol, SLICOT B ENCHMARK MIMO E XAMPLE W ITH n = 980, rmax = 50, AND [ f l , f h ] = [0, 3 GHz]

point, the reduced-order model we obtain is of order r = 9. q = 3 moments have been matched for each expansion point. The error plot of the reduced transfer function is the line with stars in Fig. 12. The frequency point with the largest error is around f = 1 × 108 , which agrees with the result of our modification scheme and is close to the new tested frequency point f ∗ . E. Adaptivity to Different tol In some applications, the above-assigned acceptable error tol = 10−2 for the reduced-order models may be insufficient, and one may need more accurate models. In order to show the adaptivity of the scheme to different requirements for the accuracy of the reduced-order models, we list in Tables II–V the data of the reduced-order models derived by the adaptive scheme for different tol, and for different examples. For the SISO examples, we list e R Table II. For the MIMO examples, min we only list the maximal error emax R , the minimal error e R , ave as well as the average error e R defined in (7). For each model, we set an initial rmax , and the reduced-order models are derived by requiring that r ≤ rmax . If the initially set rmax is too small (see Step E8 in the adaptive scheme), then a reduced-order model with an order larger than rmax could be obtained. From Table II, we observe that almost all of the reducedorder models meet the requirements of tol. There are exceptional cases for the spiral inductor example. This is reasonable because the tested point between each two neighboring expansion points chosen by the adaptive scheme sometimes is not exactly the point (e.g., s  ) with the largest error. In such cases,

if the tested point s T is not close enough to s  , the error of Hr at s  could have a relatively large difference from Hr (s) at s T . As a result, the error of Hr (s  ) maybe still a bit larger than tol, although the error of Hr (s T ) is already below tol. However, the accuracy of the reduced-order model can be further improved at least for one possible cause of the exceptional cases. If the inaccuracy of such cases is caused by the large range [min, max] assigned for ε0 (s ∗ )/ε1 (s ∗ ) in Section III-C, then using a smaller range, we have a more accurate estimation of the tested point s T , which could be even closer to s  . All of the results in Table II are derived using max = 10 and min = 0.1. However, when we use tighter values, e.g., max = 2 and min = 0.5, we can obtain more accurate reduced-order models. Taking the spiral inductor as an example, the reduced-order model actually does not satisfy the accuracy tol = 1 × 10−3 in Table II, and the reduced transfer function has an error that is slightly larger than tol. If we use max = 2 and min = 0.5 to recompute the reducedorder model, the newly obtained reduced-order model is of the order r = 12 and has an error = 4.2634 × 10−5 , which is smaller than tol = 1 × 10−3 . We have used the same two expansion points, but the tested point has been modified from s ∗ = 2πj f ∗ = 2πj 1.58 × 108 to s ∗ = 2πj f ∗ = 2πj 9.98 × 107. The new tested point is that one which causes almost the largest error, because the error of Hr (s) at the new s ∗ is now H (s ∗) = 4.0015×10−5 and the largest error of H (s) at 2000 frequency samples si = 2πj fi , i = 1, 2, . . . , 2000, is maxsi e R = 4.2634 × 10−5 . The two errors are of course very close. Therefore, with the modified max and min, we can test for the point that produces the largest error even more accurately. The accuracy of the reduced-order model corresponding to tol = 1 × 10−6 can also be improved using the same technique. Unfortunately, the accuracy of the reduced-order models for the SLICOT MIMO example in Table III cannot be improved any more, even if the new max = 2 and min = 0.5 are used. This means that the inaccuracy of the reduced-order model may not be due to the large range [min, max]. Through observing the transfer functions corresponding to different pairs of input–output ports, we find that the transfer functions have many resonances (please also see Figs. 15 and 16). Furthermore, the locations of the resonances are different from one transfer function to another; for example, the locations of the resonances of H14 in Fig. 15 are different from the locations of those of H44 in Fig. 16. If only considering the SISO system corresponding to H44, we can get a reduced transfer function for H44 that is below

FENG et al.: FULLY ADAPTIVE SCHEME FOR MOR BASED ON MOMENT MATCHING

Fig. 13. Comparison of the transfer function of the clock tree example with n = 6134 and [ fl , f h ] = [0, 10 GHz].

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Fig. 16. Comparison of the transfer function of the SLICOT MIMO example with n = 980 and [ fl , f h ] = [0, 5 GHz]: I4O4. TABLE IV A DAPTIVITY OF THE S CHEME FOR D IFFERENT tol, THE MIMO S YSTEM OF THE PACKAGE M ODEL W ITH n = 21760, rmax = 50, AND [ f l , f h ] = [0, 107 Hz]

TABLE V

Fig. 14.

Relative error of the transfer function in Fig. 13, tol = 1 × 10−2 .

Fig. 15. Comparison of the transfer function of the SLICOT MIMO example with n = 980 and [ f l , f h ] = [0, 5 GHz]: I1O4.

tol = 0.01 [though it is not the case for the higher frequency f h = 5 GHz (see Section IV-F)]. However, taking all the SISO systems together, i.e., MOR for the whole MIMO system, we have the results in Table I, which are not below tol = 0.01. This indicates that the proposed modification scheme may not catch all the resonances of all the transfer functions at one time, i.e., it may not be able to identify the test points corresponding to all the resonances of different transfer functions.

E XPANSION P OINTS AND M OMENTS M ATCHED FOR THE S PIRAL I NDUCTOR E XAMPLE W ITH n = 1434 AND [ fl , f h ] = [0, 10 GHz]

However, using the recently proposed error estimation for LTI systems [18], we may solve this issue. There, an a posteriori error bound for the reduced transfer function is proposed. Using the error bound within a greedy scheme, to iteratively choose the expansion points corresponding to the biggest error, one should get more reliable reduced-order models for such MIMO systems. This process nevertheless undergoes a bit higher computational complexity. The results for the MIMO system of the package model are listed in Table IV. The proposed scheme derives expected reduced-order models for all the different tol. In Table V, we show the expansion points used, and the moments matched for each expansion point according to different tol for the spiral inductor model. To obtain a clearer

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view, we show the frequency points f i corresponding to each expansion point si = 2πj f i , i = 1, 2, . . . instead of si . As we can see, in some cases, the expansion points are not equidistantly distributed. The moments matched are not equal from one point to another either. The expansion points are also not the direct result of using the bisection principle. This is because the modification scheme in Section III-C has been used during model reduction, where the mid tested points may have been modified to other tested points, which are again the candidates for the new expansion points. F. Robustness of the Scheme for Systems With Many Resonances In this section, we especially show the robustness of the adaptive scheme for systems with many resonances. We observe that if we extend the frequency range of the clock tree example and the SLICOT MIMO example, their corresponding transfer functions have many resonances. Since we do not have any other examples available, we show the efficiency of our adaptive scheme using these two examples with extended frequency range, although high frequency might be unnecessary for practical applications. 1) First Example: We extend the highest frequency of the clock tree example to f h = 10 GHz, so that there are many more resonances than what is shown in Fig. 7 (see Fig. 13). We see no difference of the transfer function Hr (s) produced by the reduced-order model from the original transfer function H (s). Fig. 14 shows the error of the transfer function computed by the reduced-order model, which is below the tolerance tol = 1 × 10−2 . 2) Second Example: We extend the highest frequency of the SLICOT MIMO example to fh = 5 GHz. We show the transfer functions corresponding to two different pairs of ports: the transfer function H41 for the input port 1 and the output port 4 (I1O4), and the transfer function H44 for the input port 4 and the output port 4 (I4O4). Figs. 15 and 16 plot the transfer functions H41 and H44, and their approximations generated by the reduced-order models.3 Over the whole frequency range [0, 5 GHz], the maximal error for H41 is around 0.05, and the maximal error for H44 is around 0.02. Both of the average errors are below 3 × 10−4 (see Fig. 17 for an illustration). We further compare the phases of the transfer functions with their corresponding approximations in Fig. 18, and both approximations match the original ones very well.

Fig. 17. Relative error of the transfer function in Figs. 15 and 16, respectively, with tol = 1e − 2.

Fig. 18.

Phase comparison for the SLICOT MIMO example.

If one knows the magnitude of the input, one may infer the error requirement in the frequency domain using the error requirement in the time domain. Conversely, one can also estimate the error in the time domain from the error in the frequency domain. In order to check if the reduced-order models are acceptable in the time domain, we take the clock tree example with n = 6134 and the bus line example. In Figs. 19 and 20, the outputs yr (t) of the reduced-order models and y(t) of the original system, corresponding to the step input with rise time 0.1 ns, are presented. The reduced-order models of both examples are constructed by setting tol = 1 × 10−2 . The absolute errors4 are all below 6 × 10−5 . H. Comparison With PRIMA

G. Time-Domain Comparison

where ||H (s) − Hr (s)||∞ = sups σmax (H (s) − Hr (s)). For an SISO system, ||H (s) − Hr (s)||∞ = |H (s) − Hr (s)|, the magnitude of the difference between H (s) and Hr (s).

We use the clock tree model with n = 24 566 and the spiral inductor model to compare our adaptive scheme with PRIMA. We set tol = 1 × 10−5 and rmax = 20 for both methods. In Fig. 21, the errors of the reduced transfer function of the clock tree model are compared. Our adaptive scheme uses 17 expansion points and the order of the reduced-order model is r = 22. The error is below tol = 1 × 10−5 . If single expansion s0 = sl is used in PRIMA and if q = 20 moments are matched, the reduced-order model is of the order

3 Note that we compute two different reduced-order models of the two SISO examples with I1O4 and I4O4 as input and output, respectively.

4 For these two examples, it make little sense to use relative error, since the outputs are zero at the beginning of the time interval.

From system theory, it is known that the output error in the time domain can be bounded by the error of the transfer function and the magnitude of the input ||y − yr (t)||2 ≤ ||H (s) − Hr (s)||∞ ||u||2

(8)

FENG et al.: FULLY ADAPTIVE SCHEME FOR MOR BASED ON MOMENT MATCHING

Fig. 19. Time-domain output response: clock tree example with n = 6134, r = 18, u(t) = 0, t ≤ 0.1 ns, and u(t) = 1, where t ≥ 0.1 ns.

Fig. 22.

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Comparison with PRIMA for the spiral inductor model.

We compare the results for the spiral inductor in Fig. 22. Two expansion points s0 = sl and s1 = sh have been adaptively selected by our adaptive scheme. The resistance computed by the reduced-order model is indistinguishable from the results by full simulation. However, if we use q = 15 and s0 = sl in PRIMA, the reduced-order model is of order r = 15, but with remarkably high error at high frequencies. Even if we increase q to q = 20, the reduced-order model with r = 20 is still obviously inaccurate. V. C ONCLUSION Fig. 20. Time-domain output response: bus line example with n = 147, r = 3. u(t) = 0, where t ≤ 0.1 ns, and u(t) = 1, where t ≥ 0.1 ns.

A fully adaptive scheme for reduced-order modeling of large-scale LTI systems is proposed. The adaptive scheme is based on a bisection principle for the interesting frequency range. Given the desired accuracy and the acceptable order of the reduced-order model, the adaptive scheme aims to automatically generate the required reduced-order model by adaptively choosing the expansion points for the transfer function as well as the number of moments for each expansion point. If the given rmax is too small, the adaptive scheme can also automatically determine a proper order of the reducedorder model. Various examples have been tested, which show the robustness of the adaptive scheme. Further efforts should be made toward developing more efficient algorithms for MIMO examples with many resonances. ACKNOWLEDGMENT

Fig. 21.

Comparison with PRIMA for the clock tree model.

r = 20 but with large error [maxsi e R (si ) = 0.8767, where si = 2πj f i , i = 1, 2, . . . , 2000, are the frequency samples] at high frequencies. If we increase the matched moments to q = 30, the reduced-order model of the order r = 30 has smaller errors, but is still not acceptable [maxsi e R (si ) = 0.015]. If using q = 40 moments, the reduced-order model is of the order r = 40 and the accuracy is acceptable. However, the reduced-order model is more than double the size of the one derived by the adaptive scheme.

The authors would like to thank Prof. M. S. Nakhla for his discussions and excellent suggestions. R EFERENCES [1] B. C. Moore, “Principal component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Trans. Autom. Control, vol. 26, no. 1, pp. 17–32, Feb. 1981. [2] A. Bodendiek and M. Bollhöfer, “Adaptive-order rational Arnoldi-type methods in computational electromagnetism,” BIT Numer. Math., vol. 54, no. 2, pp. 357–380, 2014. [3] P. Benner, “System-theoretic methods for model reduction of large-scale systems: Simulation, control, and inverse problems,” in Proceedings of MathMod, I. Troch and F. Breitenecker, Eds. Vienna, Austria: ARGESIM, Feb. 2009, pp. 126–145.

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[4] T. Bonin, H. Faßbender, A. Soppa, and M. Zaeh, “A fully adaptive rational global Arnoldi method for the model-order reduction of second-order MIMO systems with proportional damping,” Math. Comput. Simul., DOI: 10.1016/j.matcom.2015.08.017. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0378475415001822 [5] E. Chiprout and M. S. Nakhla, “Analysis of interconnect networks using complex frequency hopping (CFH),” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 14, no. 2, pp. 186–200, Feb. 1995. [6] L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation for timing analysis,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 9, no. 4, pp. 352–366, Apr. 1990. [7] R. Achar and M. S. Nakhla, “Simulation of high-speed interconnects,” Proc. IEEE, vol. 89, no. 5, pp. 693–728, May 2001. [8] S. Gugercin, A. C. Antoulas, and C. A. Beattie, “H2 model reduction for large-scale linear dynamical systems,” SIAM J. Matrix Anal. Appl., vol. 30, no. 2, pp. 609–638, 2008. [9] U. Hetmaniuk, R. Tezaur, and C. Farhat, “An adaptive scheme for a class of interpolatory model reduction methods for frequency response problems,” Int. J. Numer. Methods Eng., vol. 93, no. 10, pp. 1109–1124, 2013. [10] H.-J. Lee, C.-C. Chu, and W.-S. Feng, “An adaptive-order rational Arnoldi method for model-order reductions of linear time-invariant systems,” Linear Algebra Appl., vol. 415, nos. 2–3, pp. 235–261, 2006. [11] J.-R. Li, F. Wang, and J. White, “An efficient Lyapunov equation-based approach for generating reduced-order models of interconnect,” in Proc. 36th Design Autom. Conf., 1999, pp. 1–6. [12] A. Odabasioglu, M. Celik, and L. T. Pileggi, “PRIMA: Passive reducedorder interconnect macromodeling algorithm,” IEEE Trans. Comput.Aided Design Integr. Circuits Syst., vol. 17, no. 8, pp. 645–654, Aug. 1998. [13] P. Feldmann and R. W. Freund, “Efficient linear circuit analysis by Padé approximation via the Lanczos process,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 14, no. 5, pp. 639–649, May 1995. [14] T. Bechtold, E. B. Rudnyi, and J. G. Korvink, “Error indicators for fully automatic extraction of heat-transfer macromodels for MEMS,” J. Micromech. Microeng., vol. 15, no. 3, pp. 430–440, 2005. [15] T. Bui-Thanh, K. Willcox, and O. Ghattas, “Model reduction for largescale systems with high-dimensional parametric input space,” SIAM J. Sci. Comput., vol. 30, no. 6, pp. 3270–3288, 2008. [16] G. Rozza, D. B. P. Huynh, and A. T. Patera, “Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations,” Arch. Comput. Methods Eng., vol. 15, no. 3, pp. 229–275, 2008. [17] L. M. Silveira and J. R. Phillips, “Resampling plans for sample point selection in multipoint model-order reduction,” IEEE Trans. Comput.Aided Design Integr. Circuits Syst., vol. 25, no. 12, pp. 2775–2783, Dec. 2005. [18] L. Feng, A. C. Antoulas, and P. Benner, “Some a posteriori error bounds for reduced order modelling of (non-)parametrized linear systems,” MaxPlanck Institute Magdeburg Preprints, Rep. Preprint MPIMD/15-17, 2014. [Online]. Available: http://www.mpi-magdeburg.mpg.de/preprints/ Lihong Feng received the Ph.D. degree in computational mathematics from Fudan University, Shanghai, China, in 2002. She held post-doctoral positions with Fudan University, the University of Freiburg, Freiburg im Breisgau, Germany, and the Chemnitz University of Technology (TU Chemnitz), Chemnitz, Germany. Since 2010, she has been a Senior Scientist with the Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany, and a Team Leader with the CSC Group headed by P. Benner. Her current research interests include model order reduction, fast simulation of complex differential equations arising from engineering applications, such as chemical engineering, microelectromechanical systems simulation, and integrated circuit simulation, numerical analysis, and scientific computing. Dr. Feng was the recipient of an Alexander-von-Humboldt Fellowship at the host institute TU Chemnitz from 2007 to 2009.

Jan G. Korvink (F’13) received the M.Sc. degree in mechanical engineering with a specialization in computational mechanics from the University of Cape Town, Cape Town, South Africa, in 1987, and the Ph.D. degree in applied computer science from ETH Zurich, Zürich, Switzerland, in 1993. He joined the Physical Electronics Laboratory, ETH Zurich. In 1997, he moved to the University of Freiburg, Freiburg im Breisgau, Germany, where he is currently a Professor of Microsystems Engineering and runs the Laboratory for Microsystem Simulation. He was a Guest Professor with ETH Zurich and Ritsumeikan University, Kusatsu, Japan, and a Guest Scientist with Kyoto University, Kyoto, Japan. From 2007 to 2013, he was the Director of the Freiburg Institute for Advanced Studies. He is also a Professor and the Director of the Institute of Microstructure Technology with the Karlsruhe Institute of Technology, Karlsruhe, Germany. His current research interests include ultralow cost micromanufacturing methods, microsystem applications for Nuclear Magnetic Resonance (NMR) and NMR spectroscopy, and the design and simulation of micro and nanosystems. Prof. Korvink is a Curatory Board Member of the Fraunhofer IPM Laboratory, a Research Advisory Panel Member of the Council for Scientific Research’s Materials Science and Manufacturing activities in South Africa, and a Steering Committee Member of the International Research Group Nano Micro Systems.

Peter Benner received the Diploma degree in mathematics from RWTH Aachen University, Aachen, Germany, in 1993, the Ph.D. degree from the University of Kansas, Lawrence, KS, USA, and the Chemnitz University of Technology, Chemnitz, Germany, in 1997, and the Habilitation degree in mathematics from the University of Bremen, Bremen, Germany, in 2001. He was an Assistant Professor with the University of Bremen from 1997 to 2001. He was a Visiting Associate Professor with the Technical University of Hamburg, Hamburg, Germany. He was a Lecturer in Mathematics with the Technical University of Berlin, Berlin, Germany, from 2001 to 2003. Since 2003, he has been a Professor of Mathematics in Industry and Technology with the Chemnitz University of Technology. Since 2010, he has been one of the four directors of the Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany. Since 2011, he has been an Honorary Professor with the Otto-von-Guericke University of Magdeburg, Magdeburg. He is a Distinguished Professor with Shanghai University, Shanghai, China. His current research interests include scientific computing, numerical mathematics, systems theory, optimal control, in a particular emphasis on applying methods from numerical linear algebra and matrix theory in systems and control theory, numerical methods for optimal control of systems modeled by evolution equations (Partial Differential Equations, Differential Algebraic Equations, and Stochastic Partial Differential Equations), model order reduction, preconditioning in optimal control and Uncertainty Quantification problems, and Krylov subspace methods for structured or quadratic eigenproblems. Research in all these areas is accompanied by the development of algorithms and mathematical software suitable for modern and high-performance computer architectures.