Knowledge-Based Variable-Fidelity Optimization of Expensive Objective Functions through Space Mapping

In this paper, we study the use of space-mapping (SM) techniques within the transmission-line matrix (TLM) method environment. Previous work on SM relies on an idealized coarse model in the design process of a computationally expensive fine model. For the first time, we examine the case when the coarse model is not capable of providing an ideal optimal response. We exploit a coarse-grid TLM solver with relaxed boundary conditions. Such a coarse model may be incapable of satisfying design specifications and traditional SM may fail. Our approach, which exploits implicit SM (ISM) and the novel output SM (OSM), overcomes this failure. Dielectric constant, an expedient preassigned parameter, is first calibrated to roughly align the coarse and fine TLM models. Our OSM scheme absorbs the remaining deviation between the implicitly mapped coarse-grid and fine-grid TLM responses. Because the TLM simulations are on a fixed grid, response interpolation is crucial. We also create a database system to avoid repeating simulations unnecessarily. Our optimization routine employs a trust region methodology. The TLM-based design of an inductive post, a single-resonator filter, and a six-section H-plane waveguide filter illustrate our approach. In a few iterations, our coarse-grid TLM surrogate, with approximate boundary conditions, achieves a good design of the fine-grid TLM model in spite of poor initial responses. Our results are verified with MEFiSTo simulations.

In this brief, a new systematic approach of matching network (MN) layout synthesis and optimization for power amplifiers (PAs) is proposed based on implicit space mapping (ISM). Commonly, the performance discrepancy between schematic and initial layout is huge, especially for millimeter wave (mmWave) PAs. To address this issue, the proposed modified ISM can shift the optimization work to the circuit schematic simulations by calibrating carefully-selected auxiliary parameters in the circuit based on EM data. To validate the proposed method, an eligible dual-band inter-stage MN layout is rapidly optimized with only 10 runs of EM-simulations, while a corresponding 16/26.5 GHz concurrent dual-band PA is implemented in a 0.15- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> GaAs pHEMT technology. It exhibits a measured small-signal gain of 21.7/14.1 dB, peak power-added efficiency of 30.2%/40.5%, and saturated output power of 24.8/28.1 dBm with a compact chip size of 2.4 mm2.

Convergence is a well-known issue for space mapping (SM) optimization algorithms. One possible convergence safeguard is the trust region (TR) approach where the surrogate model is optimized in a restricted neighborhood of the current iteration point. We demonstrate that although formal conditions for applying trust regions are not strictly satisfied for SM surrogate models, TR improves the stability and convergence properties of the SM optimization process. Further improvement can be realized when approximate fine model Jacobian information is exploited in the construction of the SM surrogate.

The implicit space mapping technique is implemented as the optimization algorithm for the reflected group delay method in designing a 6-pole microstrip hairpin filter. A technique is proposed to reduce simulation points by matching a few selected group delay points. A computer-aided EM based design approach is proposed for the integration of the implicit space mapping technique and the reflected group delay method. The design steps are summarized and the filter is designed in Sonnet. By using the proposed method, the computation time and space mapping iterations are significantly reduced.

A New Implicit Space Mapping with High Convergence for Microwave Filter Design

In this work, we introduce a robust and efficient technique for rapid design of compact RF circuits. Our approach exploits two-level space mapping (SM) correction of an equivalent circuit model of the structure under design. The first SM layer (local correction) is utilized to ensure good matching between the equivalent circuit and the electromagnetic model at the component level. On the other hand, the global correction allows for taking into account the electromagnetic couplings between the components. The important advantage of decomposing the low-fidelity model correction into local and global level is that small number of parameter is utilized at each stage, which substantially simplifies and speeds up the parameter extraction procedure. Another advantage is that the local-global SM results in considerably better generalization capability of the surrogate model (as compared to conventional SM). This leads to improved reliability of the optimization process and its reduced computational cost. The proposed technique is validated using an example of a compact microstrip rat-race coupler (RRC) and compared to previous surrogate-assisted attempts to compact RF structure optimization, including implicit SM and sequential SM.

A novel design methodology for multiplexer design is presented. For the first time, finite element EM based simulators and space-mapping optimization are combined to produce an accurate design for manifold coupled output multiplexers with dielectric resonator (DR) loaded filters. Finite element EM based simulators are used as a fine model of each multiplexer channel and a coupling matrix representation is used as a coarse model. Fine details such as tuning screws are included in the fine model. Therefore channel dispersion and spurious modes are taken into account. The DR filter channel design parameters are kept bounded during optimization. Our approach has been used to design large-scale manifold coupled output multiplexers and it has significantly reduced the overall tuning time compared to traditional techniques. The technique is illustrated through design of a 10-channel output multiplexer with 5-pole DR filter based channels.

We introduce a new framework for the global optimization of computationally expensive multimodal functions when derivatives are unavailable. The proposed Stochastic Response Surface (SRS) Method iteratively utilizes a response surface model to approximate the expensive function and identifies a promising point for function evaluation from a set of randomly generated points, called candidate points. Assuming some mild technical conditions, SRS converges to the global minimum in a probabilistic sense. We also propose Metric SRS (MSRS), which is a special case of SRS where the function evaluation point in each iteration is chosen to be the best candidate point according to two criteria: the estimated function value obtained from the response surface model, and the minimum distance from previously evaluated points. We develop a global optimization version and a multistart local optimization version of MSRS. In the numerical experiments, we used a radial basis function (RBF) model for MSRS and the resulting algorithms, Global MSRBF and Multistart Local MSRBF, were compared to 6 alternative global optimization methods, including a multistart derivative-based local optimization method. Multiple trials of all algorithms were compared on 17 multimodal test problems and on a 12-dimensional groundwater bioremediation application involving partial differential equations. The results indicate that Multistart Local MSRBF is the best on most of the higher dimensional problems, including the groundwater problem. It is also at least as good as the other algorithms on most of the lower dimensional problems. Global MSRBF is competitive with the other alternatives on most of the lower dimensional test problems and also on the groundwater problem. These results suggest that MSRBF is a promising approach for the global optimization of expensive functions.

Parallel radial basis function methods for the global optimization of expensive functions

In this paper, space mapping (SM) technique is combined with response surface methodology (RSM). SM is an optimization method well suited for very costly problems to find an improved design with fulfilled constraints. The SM technique use less costly models, which complements the correct models. The theory is established and compared to the corrected RSM. A multipoint version of SM is presented, where a separate evaluation is done in each iteration to improve the mapping function. Using this additional evaluation to update the mapping function, generally, the number of iterations to find the optimum solution can be reduced. Thus, the elapsed time to solve the optimization problem can be reduced if a parallel computer is utilized. Finally, one engineering optimization problem is solved to illustrate the application of SM in vehicle crashworthiness structural optimization.

Global optimization of expensive functions has important applications in physical and computer experiments. It is a challenging problem to develop efficient optimization scheme, because each function evaluation can be costly and the derivative information of the function is often not available. We propose a novel global optimization framework using adaptive radial basis functions (RBF) based surrogate model via uncertainty quantification. The framework consists of two iteration steps. It first employs an RBF-based Bayesian surrogate model to approximate the true function, where the parameters of the RBFs can be adaptively estimated and updated each time a new point is explored. Then it utilizes a model-guided selection criterion to identify a new point from a candidate set for function evaluation. The selection criterion adopted here is a sample version of the expected improvement criterion. We conduct simulation studies with standard test functions, which show that the proposed method has some advantages, especially when the true function has many local optima. In addition, we also propose modified approaches to improve the search performance for identifying optimal points.

An enhancement of the space mapping (SM) surrogate model through support vector regression is presented. This technique uses a standard SM model (trend function) and support vector regression to model the residuals between the fine model and the standard model. The latter is implemented as a additive output SM term. The proposed methodology offers efficient utilization of the available fine model data (not possible in the standard SM modeling) and accuracy comparable or better than the recently published modeling techniques combining SM with radial basis functions and fuzzy systems. Examples demonstrate the robustness of our approach.

We present a novel, hybrid aggressive space mapping (HASM) optimization algorithm. HASM is a hybrid approach exploiting both the trust region aggressive space mapping (TRASM) algorithm and direct optimization. It does not assume that the final space-mapped design is the true optimal design and is robust against severe misalignment between the coarse and the fine models. The algorithm is based on a novel lemma that enables smooth switching from the TRASM optimization to direct optimization and vice versa. The new algorithm has been tested on several microwave filters and transformers.

Antenna design optimization using full EM simulation is computationally expensive and does not include conventional design techniques or the working mechanism of antenna design. In order to design antennas in a smarter way, design techniques need to be incorporated into parameter optimization. One of the important ways to achieve this is to pixelate the metal on the designed surface. For the first research contribution, we incorporate N-port Characteristic Mode Analysis (CMA) into the formulation to obtain an objective function that characterizes the fundamental resonances of the pixel antenna. These fundamental resonant properties of the pixel antenna are found without requiring the selection of the feeding ports and therefore our method can provide optimization with better antenna performance without needing to optimize the feed positions simultaneously. For the second contribution feeding port optimization is proposed by using sequential feeding port selection. This can greatly reduce the search space and computational load. In addition, this approach does not require additional complex feeding networks or impedance matching circuits. This makes the design simple, compact and practical. For the third contribution, perturbation sensitivity is used to assist the optimization of pixel antennas, making it possible to use gradient based optimization algorithms. Specific algebraic techniques are used to calculate the perturbation sensitivity of each port, which avoids the inversion of impedance matrices at each evaluation of the objective function. This significantly reduces the computational load and results demonstrate better optimization performance compared to GA and SEBO. For the fourth contribution, space mapping (SM) is utilized to optimize multiple-input and multiple-output (MIMO) antennas. SM is formulated for the optimization of MIMO antennas and the scope of the coarse models is also expanded to include radiation efficiency. We extend parameter extraction (PE), to MIMO antennas, and also extend the IMPM approach to continuous tunable loads to optimize the isolation of external ports using SM. Throughout the thesis presentation, experimental results are provided to demonstrate the improvements that the new algorithms and techniques, developed during the thesis, provide.

Rather than optimizing expensive objective functions such as complex engineering simulations directly, Bayesian optimization methodologies fit a surrogate model (typically Kriging or a Gaussian Process) on evaluations of the objective function(s). To determine the next evaluation, an acquisition function is optimized (also referred to as infill criterion or sampling policy) which incorporates the model prediction and uncertainty and balances exploration and exploitation. Therefore, Bayesian optimization methodologies replace a single optimization of the objective function by a sequence of optimization problems: this makes sense as the acquisition function is cheap-to-evaluate whereas the objective is not. Depending on the goal different acquisition functions are available: multi-objective acquisition functions are relatively new and this chapter gives a state-of-the-art overview and illustrates some approaches based on hypervolume improvement. It is shown that the quality of the model is crucial for the performance of Bayesian optimization and illustrate this by using the more flexible Student-t processes as surrogate models.