# 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)

### Introduction

- 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.

### III. OSM

- OSM addresses residual misalignment between the optimal coarse-model response and the true fine-model optimum response .
- The results of Bakr et al. [4] 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.

### A. Surrogate

- 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 [1] and OSM [5] 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 [4].
- 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) [7].
- 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.

### VII. CONCLUSION

- 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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