Improved microwave circuit design using multipoint-response-correction space mapping and trust

Improved Microwave Circuit Design using Multipoint-Response-Correction Space Mapping and Trust RegionsSlawomir Koziel, Senior Member, IEEEEngineering Optimization & Modeling Center, School of Science and Engineering, Reykjavik University, Kringlunni 1, IS-103 Reykjavik, IcelandAbstract—Space mapping (SM) technology includes so-called output SM that ensures exact matching between the surrogate and the fine model at the current design. Output SM exploits the fine model data at a single design and is not able to align the models’ sensitivity. Here, a multipoint response correction is introduced that generalizes the concept of output SM. By using a design-variable-dependent correction term and exploiting all available fine model information, the proposed technique provides exact match between the surrogate and the fine model at several designs. This not only retains the benefits of output SM but also enhances sensitivity matching between the two models, which results in improved performance of the SM optimization process. Verification using two microwave design problems is provided.Index Terms—Computer-aided design(CAD), EM optimization, surrogate modeling, response correction, space mapping.I. I NTRODUCTIONEfficient EM-simulation-based optimization of CPU-intensive structures can be realized using surrogate models [1]. One of the most successful approaches of this kind in microwave engineering is space mapping (SM) [2]-[6]. SM replaces the optimization of expensive or “fine” model by means of the iterative optimization and updating of so-called “coarse” model which is less accurate but cheaper to evaluate. Early versions of space mapping, here referred to as input-like-SM, were focused on transformation of the coarse model domain using shifts and/or variable scaling [2]. Flexibility of the coarse model can be enhanced by implicit SM [4] that exploits so-called preassigned parameters: non-optimizable variables that are fixed in the fine model (e.g., dielectric constants) but adjustable in the coarse model. In many cases, however, neither input nor implicit SM is able to provide sufficient alignment between the coarse and the fine model. Performance of SM algorithms can be improved using output SM [2], in which the coarse model is enhanced using a correction term that allows us to obtain a perfect matching between the coarse and the fine model at the current iteration point [3]. This correction term is a constant vector calculated using fine model data at a single (most recent) design [3]. Output SM usually improves the performance of the SM algorithm, particularly when used with the trust region (TR) This work was supported in part by the Reykjavik University Development Fund under Grant T09009.S. Koziel is with the Engineering Optimization & Modeling Center, School of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, Iceland. approach [7], [8]. However, output SM ensures, at most, zero-order consistency between the coarse and the fine model [8], which may result in premature termination of the optimization process at the design that is far from the optimum [8]. This problem arises, in particular, when the model responses are very nonlinear, a common situation in microwave engineering.In this paper, a multi-point response correction is introduced that generalizes the concept of output SM. A design-variable-dependent correction term is constructed that ensures perfect matching between the coarse and the fine model at a number of previously considered designs. The proposed approach ensures approximate satisfaction of the first-order consistency condition [9] and improves the performance of the SM optimization process when compared to the output SM. Two microwave design problems are presented for verification purposes.II. R ESPONSE C ORRECTION:O UTPUT S PACE M APPINGLet R f : X f →R m,X f⊆R n, denote the response vector of a fine model of the device of interest, e.g., S-parameters evaluated at m different frequencies. Our goal is to solve*arg min(())ff fXU∈=xx R x(1) where U is a scalar objective function, e.g., minimax [2]. Space mapping avoids solving (1) directly. Instead, it produces a sequence of approximate solutions to (1), x(i), i = 0, 1, 2, …, as:()(1)()arg min()i isU+=xx R x(2) where R s(i) is the surrogate model at iteration i.Let R c : X c→R m,X c⊆R n,denote the response vectors of the coarse model: computationally cheap but relaxed-accuracy representation of the fine model. The surrogate model R s(i) is constructed from the coarse model as follows()()().()()i i is c sm=+R x R x d(3) Here, R c.sm(i) denotes R c composed with suitable SM transformations. For example, in case of input SM we have R c.sm(i)(x) = R c(B(i)·x + c(i)), where the n×n matrix B(i) and n×1 vector c(i) are obtained using parameter extraction [2].The output SM correction term d(i) is defined as [3]()()()().()()i i i if c sm=−d R x R x(4) This definition ensures zero-order consistency between the fine model and the SM surrogate at x(i), i.e., R f(x(i)) = R s(i)(x(i)).III. M ULTIPOINT R ESPONSE C ORRECTION T ECHNIQUE The output SM correction term d (i )is defined using the R f response at the current iteration point x (i ). Here, a multipoint response correction is proposed that exploits most of the available R f data. In particular, we want to define the model()()()().()()()i i i i s c sm g =+R x R x ΔR x (5)so that the correction term ΔR g (i ) has the following features: (i) ΔR g (i )(x (i )) = R f (x (i )) – R c.sm (i )(x (i )), i.e., R s (i )(x (i )) = R f (x (i )), (ii) R s (i )(x (k )) = R f (x (k )), i.e., ΔR g (i )(x (k )) = R f (x (k )) – R c.sm (i )(x (k )) for as many k < i as possible,(iii) ΔR g (i ) is a linear function of x (so that it does not degrade the ability of R c to model higher order nonlinearities of R f ), Let S (i ) be a set of all affinely independent subsets of X (i ) = {x (1), x (2), …, x (i )} containing x (i ). Let s (i )∈S (i ) be such that: (i) s (i ) is maximal, i.e., |s (i )| = n i = max{|s | : s ∈ S (i )}, and (ii) s (i ) is the smallest in the sense of its diameter, i.e., d (s (i )) = min{d (s ) : s ∈ S (i ) ∧ |s | = n i }, where d (s ) ≡ max{||x – y || : x , y ∈ s }. We have |s (i )| ≤ min(i , n + 1). Let x k , k = 1, 2, …, n i , denote the elements of s (i ) (here, x 1 = x (i )). Define R k = R f (x k ) – R c.sm (i )(x k ), k = 1, 2, …, n i . Let us also define the m ×n i matrix R (i ) = [R 1 R 2 … R n i ].Let M (i ) ⊆ R n be a linear manifold generated by s (i ), i.e.,{}()11:,1i i n ni n k k k k k M R αα===∈==∑∑x x x (6)M (i ) is well defined because s (i ) is affinely independent. Let B (i ) = {b 1, …, b n i –1}, where b k = x k +1 – x 1, k = 1, 2, …, n i – 1. Now, let Y (i ) ⊆ R n be a linear subspace generated by B (i ), i.e.,{}1()1:,,1,2,...,1i n i n k k k i k Y R R k n ββ−==∈=∈=−∑x x b (7)Clearly, Y (i ) = M (i ) – x (i ). For any x ∈ R ndefine P i (x ) as the orthogonal projection of x – x (i ) onto Y (i ). For any y ∈ M (i ) define λi (y ) = [λ1 λ2 … λn i ]T so that y = λ1x 1 + λ2x 2 + … + λn i x n i . λi is uniquely determined and λ1 + λ2 + … + λn i = 1. We can now define ΔR g (i )(x ), x ∈ R n , as()()()().()()()()()for 1()(())for 1i i i i f c sm gi i i i i i P i ⎧−==⎨⋅−+>⎩R x R x ΔR x R λx x x(8)In (8), we distinguished two cases (i = 1 and i > 1) becausesome of the above objects (e.g., Y (i )) are not defined for i = 1. Fig. 1 illustrates the concepts used to define ΔR g (i )( ⋅ ).Vector λi (y ) can be determined by solving a linear system()()i i i =T λy (9)where T (i )is (n +1) × n i matrix defined as()()1(1)0[11...1]1i i i n ×−⎡⎤=⎢⎥⎢⎥⎣⎦B T(10)with B (i ) being the n ×(n i -1) matrix such that its k th column is avector b k , and y (i ) is (n +1)×1 vector()()()()1i i i i P ⎡⎤−+=⎢⎥⎣⎦x x x y(11)Because vectors b k , k = 1, 2,…, n i – 1, are linearly independent,solution to (9) can be found as()1()()()()i T i i T i i −=λT T T y (12)Fig. 1. Illustration of the objects used to define the multipoint response correction model ΔR g (x (i )) for n = 3.It follows, that ΔR g (i ) satisfies properties (i)-(iii) listed at the beginning of this section. In particular, if x = x (i ), then P i (x – x (i )) = 0 and λi (P i (x – x (i )) + x (i )) = λi (x (i )) = [1 0 … 0]T ; consequently, ΔR g (i )(x (i )) = R (i )⋅[1 0 … 0]T = R 1 = R f (x (i )) – R c.sm (i )(x (i )). In general, ΔR g (i )(x k ) = R k = R f (x k ) – R c.sm (i )(x k ) for all k = 1, 2,…, n i ; hence, as s (i ) is maximal, we have ΔR g (i )(x k ) = R f (x k ) – R c.sm (i )(x k ) for as many k as possible taking into account other constraints for our mapping. Finally, ΔR g (i ) is linear, because P i , λi , and multiplication R (i )⋅λi are linear. It should be noted, that ΔR g (i ) contains some information about the Jacobian of R f (x (i )) – R c.sm (i )(x (i )). In particular, the Jacobian of ΔR g (i ) is the approximation of the Jacobian reduced to the subspace Y (i ). Note also that ΔR g (i ) is constant along all directions orthogonal to Y (i ). Moreover, it can be shown, that if d (s (i )) → 0, then ()()().()()()()|i i i gf c smi i Y J J −→ΔR R R x x ,i.e., given the diameter of s (i ) going to zero, the Jacobian ofΔR g (i ) becomes more and more accurate approximation of Jacobian of R f (x (i )) – R c.sm (i )(x (i )), reduced to Y (i ). In particular, if |s (i )| = n + 1, then the Jacobian of ΔR g (i ) becomes good approximation of Jacobian of R f (x (i )) – R c.sm (i )(x (i )) on R n , provided that d (s (i )) is small enough. Moreover, ΔR g (i ) assure perfect matching of responses at all points from s (i ).In practice, problem (9) can be ill-conditioned because points x (k ) tend to be affinely dependent (e.g., almost collinear) and the number of points that can be chosen as s (i ) so that the problem (9) can be efficiently solved is limited (usually smaller than n ). In practical implementation of ΔR g (i ), the choice of s (i ) is based on testing the condition number of matrix B (i )T B (i ). Also, for better accuracy it is beneficial to construct ΔR g (i ) using only designs satisfying || x (k ) – x (i ) || ≤ r (i ), where r (i ) is a user-defined threshold.To improve the algorithm convergence a trust-region approach [8] will be used in the numerical experiments. In particular, the surrogate optimization process (2) will be constrained to a neighborhood of x (i ), defined as ||x – x (i )|| ≤ δ(i ), as follows()()()(1)(),||||arg min ()i i i i s U δ+−<=x x xx R x (13) where δ(i )is a trust-region radius at iteration i . We use thestandard trust-region radius updating rules [7].III. V ERIFICATION E XAMPLESThe performance of the multipoint response correction SM described in Section II (referred to as SM MRC) is compared to the performance of the space mapping algorithm using the standard output SM (SM STD) (2)-(4), and the trust-region-enhanced space mapping (SM TR) algorithm (3), (4), (13).A. Miniature Dual-Mode Bandpass Microstrip Filter [10] Consider the miniature dual-mode bandpass filter [8] shown in Fig. 2. The design parameters are x = [L s p g]T; W = 1 mm, W c = 0.5 mm. The fine model is simulated in FEKO [11]. The coarse model is the circuit model implemented in Agilent ADS [12] (Fig. 3). The design specifications are |S21| ≥ –1 dB for 2.35 GHz ≤ω≤ 2.45 GHz, |S21| ≤ –20 dB for 1.6 GHz ≤ω≤ 2.2 GHz and for 2.6 GHz ≤ω≤ 3.2 GHz. The initial design is x(0) = [12.0 0.905 1.273 0.185]T mm (R c optimum).Table I shows the optimization results. Five types of SM surrogate models R c.sm were tested: (i) input SM model R c.sm(x) = R c(x + c) [2], (ii) implicit SM model R c.sm(x) =R c.I(x) in which R c.I is the coarse model with the substrate height and dielectric constants used as preassigned parameters [4], (iii) the combination of input and implicit SM, (iv) the combination of input and frequency SM, R c.sm(x) = R c.F(x + c) where R c.F is the coarse model evaluated at frequencies different from the original sweep according to the linear mapping ω→f1 + f2ω (f1 and f2 are obtained using the usual parameter extraction process(4)) [3], and (v) the combination of implicit and frequency SM. As expected [13], the algorithm performance depends on the proper choice of the surrogate model (here, only the combination of input and frequency SM seems to work well with the standard algorithm). Also, the standard SM algorithm has convergence problems in most cases. On the other hand, both SM TR and SM MRC algorithms converge in all cases, however, the algorithm using multipoint response correction usually yields a better final design. In first two cases, the multipoint response correction turns out to be critical for finding the solution that satisfies the design specifications. Figure 4 shows the fine model responses at the initial design and the designs obtained with the SM MRC algorithm using the R c.F(x+c) model. Convergence plots for SM STD and SM MRC working with the same model, R c.F(x+c), are shown in Fig. 5.Fig. 2. Miniature dual-mode bandpass filter: geometry [10]. Fig. 3. Miniature dual-mode bandpass filter: coarse model (Agilent ADS).2.1 2.3 2.4 2.5 2.7-10Frequency [GHz]|S21|Fig. 4. Miniature dual-mode bandpass filter: Initial (dashed line) and optimized (solid line) |S21| versus frequency; optimization using SM TR-B2 algorithm with the R c.F(x+c) model.81012141010Iteration Index i||x(i)-x(i-1)||Fig. 5. Miniature dual-mode bandpass filter: Convergence plots for SM STD (o) and SM MRC (∗), both using surrogate model R c.F(x+c), versus iteration index.T ABLE IM INIA TURE B ANDPASS F ILTER:O PTIMIZA TION R ESULTS SurrogateModelSMAlgorithmSpecification Error Number ofFine ModelEvaluationsFinal[dB]Best Found[dB]R c(x+c)SM STD +11.7 –0.2 21#SM TR +1.7 +1.7 6SM MRC –0.4 –0.4 17 R c.I(x)STDSM TR +0.2 +0.2 12SM MRC –0.4 –0.4 19 R c.I(x+c)SM STD +0.1 –0.3 21#SM TR –0.2 –0.2 11SM MRC –0.3 –0.3 8 R c.F(x+c)SM STD –0.3 –0.3 21#SM TR –0.3 –0.3 11SM MRC –0.4 –0.4 10 R c.FI(x)SM STD +0.9 +0.7 7SM TR –0.4 –0.4 11SM MRC –0.4 –0.4 8 # Convergence not obtained; algorithm terminated after 20 iterations.B. Fourth-Order Ring Resonator Bandpass Filter [14]As the second example, consider the fourth-order ring resonator bandpass filter [14] shown in Fig. 6. The designparameters are x = [L 1 L 2 L 3 S 1 S 2 W 1 W 2]Tmm. The fine model R f is simulated in FEKO [11]. The coarse model, Fig. 7, is implemented in Agilent ADS [12]. The design specifications are |S 21| ≥ –1 dB for 1.75 GHz ≤ ω ≤ 2.25 GHz, and |S 21| ≤ –20 dB for 1.0 GHz ≤ ω ≤ 1.5GHz and 2.5 GHz ≤ ω ≤ 3.0 GHz. The initial design is x (0) = [24.74 19.51 24.10 0.293 0.173 1.232 0.802]T mm (specification error +9.0 dB). The optimization results are shown in Table II. Here, we consider the following surrogate models: (i) input SM model R c.sm (x ) = R c (x + c ) [2], (ii) implicit SM model R c.sm (x ) = R c.I (x ) with the substrate height and dielectric constants used as preassigned parameters [4], and (iii) the combination of input and implicit SM. Figure 8 shows the initial fine model response and the optimized fine model response obtained using SM MRC algorithm with the R c (x +c ) model. The standard SM algorithm does not converge for any surrogate model type. SM MRC works as well as SM TR for the input and input/implicit SM surrogate but better than SM TR for the implicit SM surrogate, which indicates that the multipoint correction may improve the SM algorithm performance.Fig. 7. Fourth-order ring resonator bandpass filter: coarse model (Agilent ADS).1.61.822.2 2.4-10Frequency [GHz]|S 21|Fig. 8. Fourth-order ring resonator bandpass filter: Initial (dashed line) and optimized (solid line) |S 21| versus frequency; optimization using SM TR-B2 algorithm with the R c.F (x +c ) model.T ABLE II4TH-O RDER R ING R ESONATOR F ILTER : O PTIMIZATION R ESULTSSurrogate Model SMAlgorithmSpecification Error Number of Fine ModelEvaluations Final [dB] Best Found [dB]R c (x +c ) STD # SM TR –0.45 –0.45 18 SM MRC –0.4 –0.4 18 R c.I (x ) SM STD +0.2 –0.3 21# SM TR –0.1 –0.1 16 SM MRC –0.4 –0.4 14 R c.I (x +c )STD # SM TR –0.45 –0.45 18SM MRC –0.45 –0.45 16 #Convergence not obtained; algorithm terminated after 20 iterations.IV. 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