A space-mapping interpolating surrogate algorithm for highly optimized EM-based design of microwave devices
Abstract: We justify and elaborate in detail on a powerful new optimization algorithm that combines space mapping (SM) with a novel output SM. In a handful of fine-model evaluations, it delivers for the first time the accuracy expected from classical direct optimization using sequential linear programming. Our new method employs a space-mapping-based interpolating surrogate (SMIS) framework that aims at locally matching the surrogate with the fine model. Accuracy and convergence properties are demonstrated using a seven-section capacitively loaded impedance transformer. In comparing our algorithm with major minimax optimization algorithms, the SMIS algorithm yields the same minimax solution within an error of 10/sup -15/ as the Hald-Madsen algorithm. A highly optimized six-section H-plane waveguide filter design emerges after only four HFSS electromagnetic simulations, excluding necessary Jacobian estimations, using our algorithm with sparse frequency sweeps.
Summary (2 min read)
- The SMIS is required to match both the responses and derivatives of the fine model within a local region of interest.
- An algorithm based on it is outlined in Section V. Convergence is compared with two classical minimax algorithms, and a hybrid aggressive space-mapping (HASM) surrogate-based optimization algorithm using a seven-section capacitively loaded impedance transformer.
- OSM addresses residual misalignment between the optimal coarse-model response and the true fine-model optimum response .
- The results of Bakr et al.  indicate that “input” SM-based surrogates are good approximations to the fine model over a large region, which makes them useful in the early stages of an optimization process.
- Fig. 2 depicts model effectiveness plots for the two-section capacitively loaded impedance transformer corresponding to Fig.
- The dark grid shows the deviation of the fine model from its classical Taylor approximation as in Fig.
- The SM-based interpolating surrogate is defined as a transformation of a coarse model through input and output mappings at each sampled response.
- Using the function with individually adjusted coarse responses, defined as , where , the surrogate can be expressed as a composed mapping .
- The constants are determined in such a way that the alignment (5) holds and the requirements in (6) are satisfied as well as possible (in some sense to be specified).
- The alignment (5) is satisfied by choosing and appropriately.
- In summary, the surrogate used in the th iteration is given by (9) In each iteration, the surrogate is optimized to find the next iterate by solving (10).
B. Surface Fitting Approach for Parameter Extraction (PE)
- The authors employ a surface fitting approach for PE, which involves the minimization of residuals between the surrogate and fine models, and extracting the parameters , and .
- Since (5) is automatically satisfied by using (7), the aim is to ensure the matching (6).
- The residual (11) is used during the PE optimization process (12) which extracts the mapping parameters for the th response, and for iteration .
- Hence, the authors have the complete set of mapping parameters after PE optimizations.
V. PROPOSED SMIS ALGORITHM
- The proposed algorithm begins with the coarse model as the initial surrogate.
- The algorithm incorporates explicit SM  and OSM  to speed up the convergence to the optimal solution.
- TABLE I FINE MODEL CAPACITANCES, AND THE CHARACTERISTIC IMPEDANCES FOR THE SEVEN-SECTION CAPACITIVELY LOADED IMPEDANCE TRANSFORMER.
- Step 5) Terminate if the stopping criteria are satisfied.
- Step 6) Update the input and output mapping parameters through PE by solving (12).
A. Seven-Section Capacitively Loaded Impedance Transformer
- The authors consider the benchmark synthetic example of a seven-section capacitively loaded impedance transformer .
- Design parameters are normalized lengths with respect to the quarter-wave length at the center frequency of 4.35 GHz.
- The fine-model response at the optimal coarse-model solution is shown in Fig.
- In these figures, the authors show the HASM surrogate exploiting exact gradients.
- Using the adjoint technique, the SMIS algorithm was able to obtain the same optimum solution as the Hald–Madsen algorithm within an error of 10 after only five iterations.
B. Six-Section -Plane Waveguide Filter
- The physical structure of the six-section -plane waveguide filter is shown in Fig. 10(a) .
- The authors simulate the fine model using Agilent High Frequency Structure Simulator (HFSS).
- The design parameters are the lengths and widths, namely, 2Agilent HFSS, ver.
- The authors have presented a powerful algorithm based on a novel SMIS framework that delivers the solution accuracy expected from direct gradient-based optimization using SLP, yet converges in a handful of iterations.
- It aims at matching a surrogate (mapped coarse model) with the fine model within a local region of interest by introducing more degrees of freedom into the SM.
- Convergence is demonstrated through a seven-section capacitively loaded impedance transformer.
- The authors compare the SMIS algorithm with major direct minimax optimization algorithms.
- A highly optimized -plane filter design emerges after only four EM simulations (three iterations), excluding necessary Jacobian estimations, using the new algorithm with sparse frequency sweeps.
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