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We review the space-mapping (SM) technique and the SM-based surrogate (modeling) concept and their applications in engineering design optimization. For the first time, we present a mathematical motivation and place SM into the context of classical optimization. The aim of SM is to achieve a satisfactory solution with a minimal number of computation...
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... I NTRODUCTION E NGINEERS device, component, have been and using system optimization modeling and techniques computer- for aided design (CAD) for decades [1]. The target of component design is to determine a set of physical parameters to satisfy certain design specifications. Traditional optimization techniques [2], [3] directly utilize the simulated responses and possibly available derivatives to force the responses to satisfy the design specifications. Circuit-theory-based simulation and CAD tools using empirical device models are fast: analytical solutions or available exact derivatives may promote optimization convergence. Electromagnetic (EM) simulators, long used for design verification, need to be exploited in the optimization process. However, the higher the fidelity (accuracy) of the simulation, the more expensive direct optimization is expected to be. For complex problems, this cost may be prohibitive. Alternative design schemes combining the speed and matu- rity of circuit simulators with the accuracy of EM solvers are desirable. The recent exploitation of iteratively refined surrogates of fine, accurate, or high-fidelity models, and the imple- mentation of space mapping (SM) methodologies address this issue. Through the construction of an SM, a suitable surrogate is obtained. This surrogate is faster than the “fine” model and at least as accurate as the underlying “coarse” model. The SM approach updates the surrogate to better approximate the corresponding fine model. This paper reviews the state of the art of the SM approach, conceived by Bandler in 1993, for modeling and design of engineering devices and systems, e.g., RF and microwave components using EM simulators. Bandler et al. [4], [5] demonstrated how SM intelligently links companion “coarse” (ideal, fast, or low fidelity) and “fine” (accurate, practical, or high fidelity) models of different complexities. An EM simulator could serve as a fine model. A low-fidelity EM simulation or empirical circuit model could be a coarse model (see Fig. 1). More model classifications are listed in Table I. Generally, SM-based optimization algorithms comprise four steps. They are as follows. Step 1) Fine model simulation (verification). Step 2) Extraction of the parameters of a coarse or surrogate model. Step 3) Updating the surrogate. Step 4) (Re)optimization of the surrogate. The original SM-based optimization algorithm was introduced in 1994 [4], where a linear mapping is assumed between the parameter spaces of the coarse and fine models. It is evaluated by a least squares solution of the linear equations resulting from associating corresponding data points in the two spaces. Hence, the surrogate is a linearly mapped coarse model. The aggressive space mapping (ASM) approach [5] elimi- nates the simulation overhead required in [4] by exploiting each fine model iterate as soon as it is available. This iterate, determined by a quasi-Newton step, optimizes the (current) surrogate model. Parameter extraction (PE) is the key to establishing the mapping and updating the surrogate. In this step, the surrogate is locally aligned with a given fine model through various techniques. However, nonuniqueness of the PE step may cause breakdown of the algorithm [6]. Many approaches are suggested to improve the uniqueness of the PE step. Multipoint parameter extraction (MPE) [6], [7], a statistical PE [7], a penalty PE [8], and an aggressive PE [9] are such approaches. A recent gradient parameter extraction (GPE) approach [10] takes into account not only the responses of the fine model and the surrogate, but the corresponding gradients with respect to design parameters as well. In this paper, we present for the first time a mathematical motivation and place SM into the context of classical optimization based on local Taylor approximations. If the nonlinearity of the fine model is reflected by the coarse model, then the SM is expected to involve less curvature (less nonlinearity) than the two physical models. The SM model is then expected to yield a good approximation over a large region, i.e., it generates large descent iteration steps. Close to the solution, however, only small steps are needed, in which case, the classical optimization strategy based on local Taylor models is better. A combination of the two strategies gives the highest solution accuracy and fast convergence. Trust-region strategies were introduced into optimization algorithms to stabilize the iterative process [11]. The trust-region ASM algorithm [12] integrates such a methodology with the SM technique. SM techniques require sufficiently faithful coarse models to assure good results. Sometimes the coarse model and fine models are severely misaligned, i.e., it is hard to make the PE process work. The hybrid ASM algorithm [13] overcomes this by alternating between (re)optimization of a surrogate and direct response matching. More recently, the surro- gate-model-based SM [14] optimization algorithm combines a mapped coarse model with a linearized fine model and defaults to direct optimization of the fine model. Neural space-mapping (NSM) approaches [15] – [17] utilize artificial neural networks (ANNs) in EM-based modeling and design of microwave devices. This is consistent with the knowledge-based modeling techniques of Zhang and Gupta [18]. After updating an ANN-based surrogate [15], a fine model optimal design is predicted in NSM [16] by (re)optimizing the surrogate. Neural inverse space mapping (NISM) simplifies (re)optimization by inversely connecting the ANN [17]. The next fine model iterate is then only an ANN evaluation. The latest development of SM is implicit space mapping (ISM) [19]. An auxiliary set of parameters (selected preassigned parameters such as dielectric constant or substrate height) is extracted to match the coarse and fine model responses. The resulting (calibrated) coarse model, the surrogate, is then (re)optimized to evaluate the next iterate (fine model point). The SMX [20] system is a first attempt to automate SM optimization through linking different simulators. Several SM-based model enhancement approaches have been proposed: the generalized space-mapping (GSM) tableau approach [21], space derivative mapping [22], and SM-based neuromodeling [15]. The SM technology has been recognized as a contribution to engineering design [18], [23] – [27], especially in the microwave and RF arena. Zhang and Gupta [18] have considered the integration of the SM concept into neural-network modeling for RF and microwave design. Hong and Lancaster [23] describe the ASM algorithm as an elegant approach to microstrip filter design. Conn et al. [24] have stated that trust-region methods have been effective in the SM framework, especially in circuit design. Bakr [25] introduces advances in SM algorithms, Rayas-S á nchez [26] employs ANNs, and Ismail [27] studies SM-based model enhancement. Mathematicians are addressing mathematical interpretations of the formulation and convergence issues of SM algorithms [28] – [35]. For example, Madsen ’ s group [28] – [31] considers the SM as an effective preprocessor for engineering optimizations. Madsen and S ø ndergaard investigate convergence proper- ties of SM algorithms [32]. Vicente studies convergence prop- erties of SM for design using the least squares formulation [33], [34] and introduces SM to solve optimal control problems [35]. Section II presents a formulation of the SM concept. Section III addresses the original SM optimization algorithm. The ASM algorithm is described in Section IV. PE and different approaches for ensuring uniqueness are reviewed in Section V. In Section VI, a mathematical motivation is presented: SM is placed into the context of classical optimization. Trust-region algorithms are discussed in Section VII, the hybrid- and surro- gate-model-based optimization algorithms in Section VIII, the ISM approach in Section IX, device model enhancement (quasi- global modeling) in Section X, neural approaches in Section XI, and a review of various applications and implementations in Section XII. The discussion and a glossary of terms in Section XIII are followed by conclusions in Section XIV. II. SM C ONCEPT The SM approach introduced by Bandler et al. [4] involves a calibration of a physically based “ coarse ” surrogate by a “ fine ” model to accelerate design optimization. This simple CAD methodology embodies the learning process of a designer. It makes effective use of the surrogate ’ s fast evaluation to sparingly manipulate the iterations of the fine ...
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... A paradigmatic example of how the design optimization of computationally expensive models can be dramatically accelerated by exploiting prior engineering knowledge is space mapping (SM) [3], [4]. Deeply rooted in engineering expertise and intuition, SM emerged [5], [6] as a feasible technique to optimize computationally expensive EM models (known as fine models) by exploiting physics-based inaccurate but computationally cheap models (known as coarse models). ...
... Classical PE is formulated by matching the full fine model response of interest [3], [4], [7]. In this paper, a cognitiondriven PE formulation is proposed for the first time, where instead of using the full fine model response as target, key features of fine model response are used to make a more meaningful local alignment from an engineering perspective. ...
Parameter extraction (PE) is a key subproblem of space mapping (SM) design optimization. It consists of a local alignment between the coarse and fine models at each SM iteration. In this work, cognition-driven PE is proposed for SM. In contrast to classical PE, where the full fine model responses are used as targets, the proposed cognitive PE focuses on key features of the fine model response selected from an engineering perspective. It is demonstrated that the proposed cognitive PE approach: 1) yields more accurate extracted parameters regardless of the type of PE objective function employed; and 2) achieves a more meaningful matching to the fine model target response and with less variability. To proof this with independence of the optimization method employed for PE, plots of the PE objective functions are presented over large regions of the coarse model design space. Two synthetic examples are used to support these findings.
... Our contribution to this line of research is a calibration algorithm for the Heston model that is independent of a specific characteristic function and easily extendable to time-dependent parameters. The core of the algorithm is based on the space mapping [2], an iterative procedure that minimizes the residuum of a fine and a coarse model. In fact, to calibrate the parameters of the fine model, the coarse model is optimized and the fine model is only evaluated. ...
... Below we summarize the main ideas of the space mapping approach. For more details we recommend [2,8,26,23]. The purpose of space mapping is to optimize (or calibrate) an accurate and computationally expensive model (fine) that cannot be optimized using a surrogate model (coarse) for which efficient optimization algorithms are available. ...
We present a novel approach for parameter calibration of the Heston model for pricing an Asian put option, namely space mapping. Since few parameters of the Heston model can be directly extracted from real market data, calibration to real market data is implicit and therefore a challenging task. In addition, some of the parameters in the model are non-linear, which makes it difficult to find the global minimum of the optimization problem within the calibration. Our approach is based on the idea of space mapping, exploiting the residuum of a coarse surrogate model that allows optimization and a fine model that needs to be calibrated. In our case, the pricing of an Asian option using the Heston model SDE is the fine model, and the surrogate is chosen to be the Heston model PDE pricing a European option. We formally derive a gradient descent algorithm for the PDE constrained calibration model using well-known techniques from optimization with PDEs. Our main goal is to provide evidence that the space mapping approach can be useful in financial calibration tasks. Numerical results underline the feasibility of our approach.
... It should be noted that there are several techniques for reducing the search space when solving complex optimization problems. One proposal involves using the concept of space mapping, where the strategy consists of solving a simpler problem and, through an iterative process of parameter adjustment, finding the solution to a more complex problem [25], [26]. e idea of space mapping can also be used as part of a metaheuristic to solve nonlinear programming problems [27]. ...
Transmission network expansion planning is a critical and complex problem related to the operation and development of electrical power systems. It is typically formulated as a mixed-integer nonlinear programming (MINLP) problem with combinatorial characteristics. Various mathematical models have been proposed to better approximate real-world system behavior, but even the most relaxed formulations remain computationally challenging. This paper introduces a search space reduction strategy to reduce the gap between the optimal solution of the MINLP model and its relaxed counterpart by strategically considering surrogate constraints. This approach enhances computational efficiency, significantly reducing processing time when using an optimization solver. By applying this method, we successfully determined the previously unknown optimal solution for the Brazilian north-northeast system.
... However, it is time-consuming. The space-mapping technique employs the coarse model to align with a fine model for reducing the computation burden [10], [11], [12] However, the accuracy of the coarse model is greatly affected by the quality of evaluation in the fine model, which still requires significant computation. ...
... The parameter k q represents the influence of the slot number on the air-gap field of the SPM motor. Therefore, the positions of scalar magnetic potential (r, θ) and the equivalent current (r kϕ , θ kϕ ) are transformed to that of (r ϕ , θ ϕ ) and (r kϕ , θ kϕ ) in the annulus of ϕ g plane for the new SPM motor using (8)- (10). The analytical solution of A k (r ϕ , θ ϕ , r kϕ , θ kϕ ) in (6) can be calculated accordingly. ...
This article develops a scalable analytical model that builds a relationship between any surface-mounted permanent magnet (SPM) motor and a general motor with variable slot-opening and air-gap length, considering both the saturation effect and slotting effect. It can not only give the performance of the single SPM motor but also show internal connection among different SPM motors regardless of the motor power and dimension. To account for the iron saturation, the equivalent saturation current combined with simplified BH curves of iron is introduced to directly present the iron magnetic potential distribution without an iterative process. Thus, the proposed model can be used as the surrogate model in the motor design with little computation and provide great insight into the relationship among motors with different power and dimension. Both finite-element analysis and experiment are carried out to validate the proposed model.
... Computational time may also motivate the creation of a metamodel. This is often the case with complex simulation models, where techniques such as space mapping can be used to reduce computation time for optimization [30]. For this, the analytical model is superior to finite element analysis, as ascertained in section III, since it can be solved considerably faster. ...
A method for optimizing multiple responses in solenoids utilized in a magnetic nozzle assembly for an electric plasma thruster is presented. The primary goal is to minimize power usage while maintaining a sufficiently high magnetic flux density at the coil axis. This is especially critical during testing in a vacuum chamber, where heat dissipation is a challenge, especially for small components and intricate assemblies. This paper presents a method for quickly calculating the magnetic field along the solenoid axis that provides accurate estimates, independent of coil size. Additionally, we present general and concrete results for the employed magnetic nozzle design to illustrate the applied optimization method. The paper details a methodology for optimizing multiple parameters of a solenoid, including its radius, length, current, number of turns, and wire diameter. The aim is to minimize, maximize, or meet specific targets for magnetic flux density, power, or mass. Additional studies could utilize a suitable metric to optimize the homogeneity of the magnetic field of multiple coils. The coils designed using the algorithm have a radius of 17.5mm, a width of 21mm, consist of 300 turns with a wire diameter of 1mm, and operate at a current of about 5A. The potential maximum magnetic field is 40mT with a power consumption of around 30 W.
... If not, a "feedback" procedure (④, typically by parameter extraction) is implemented to enhance the mapping P, from which the next fine model design is predicted. SM has been proven to efficiently solve complex microwave and other engineering problems within a few iterations [1], [63], [64], [65]. ...
In this article, we honor Prof. John W. Bandler and his legacy in RF and microwave modeling and automated design optimization. We showcase his pioneering breakthroughs in minimax optimization,
p-th norm formulations, yield optimization, and nonlinear circuit design optimization. We highlight advances in direct electromagnetic (EM) microwave optimization, circuit response sensitivities, and efficient S-parameters sensitivity calculations. We explore the port-tuning version of space mapping (SM) for EM-based analysis, techniques for industrial microwave design of satellite systems, and post-manufacture hardware tuning. The integration of artificial neural networks (ANNs) with SM for enhanced EM-based design optimization and yield prediction, cognition-driven microwave filter design, and parallels between SM and artificial intelligence (AI) is examined. Finally, we speculate on the future integration of cognitive science with engineering design, leveraging the synergy of AI, machine learning (ML), and SM.
... To further enhance the design process, SAO and TR methodologies are incorporated within the LFM framework, expediting the optimization procedure. Acknowledging the inherent misalignment between LFM and HFM, the framework employs Space Mapping (SM) [21] technology to diminish this discrepancy, thereby refining the fidelity of the optimization process. Initially, the framework facilitates the optimization of litz wire through a numerical evaluation algorithm. ...
Litz wire is essential in minimizing eddy losses in high-frequency power devices. However, accurately evaluating litz wire impedance is challenging due to the low precision of analytical methods and the high computational demands of numerical simulations. To address this, we propose a multi-fidelity surrogate-assisted optimization framework embedding with an adaptive trust region (MF-TR-SAO) that efficiently designs litz wires through numerical simulation. This approach merges multi-fidelity modeling with space mapping to rapidly achieve global optimization. We validate the framework’s performance using Zakharov and Ackley functions and demonstrate its superiority over other optimization techniques. The MF-TR-SAO is then applied to the design of litz wires comprising hundreds of strands. The optimized litz wire design exhibits an 18%-36% performance enhancement in terms of power loss reduction compared to a conventional single-bundle wire while also considering cost-effectiveness and computational efficiency. This algorithm significantly advances litz wire manufacturing by optimizing for power loss, cost, and design speed.
... This algorithm can construct surrogate models that mimic the behavior of the simulation model as closely as possible to generate sample points from a random sequence. This is an effective way to reduce computational costs [11,12]. The objective function is defined as ...
The information regarding the layered structure is pivotal to the stability analysis of soft-hard interlayered rock slopes. The variation in the shear wave velocity offers insights into the characteristics of the strata structure within a rock slope. In this study, a joint analysis method that combines the multichannel analysis of surface waves (MASW) and the microtremor surface method (MSM) was used to identify the layered structure of a soft-hard interlayered rock slope located in the Shandong Province, China. The survey lines were positioned on slope crests. For the MASW, the receivers were placed in line with an active seismic source. The MSM method uses a concentric circular array comprising seven receivers. Fundamental-mode dispersion data were extracted from the active and passive microtremor sources. The 1D velocity models were then inverted from the combined dispersion curves using a surrogate optimization algorithm. A 2D velocity profile of the investigated slope was constructed through linear interpolation, and it exhibited distinct characteristics of high- and low-velocity interlayers, consistent with the borehole data. Our results demonstrate that the joint analysis method provides a complementary approach to image velocity profiling of slopes, thus contributing to enhanced investigation depth and resolution in shallow layers owing to the wider frequency range of the dispersion curves.
... In spatiotemporal sequence prediction, temporal mapping and spatial mapping techniques transform time and space dimensions into continuous deep feature representations [36,37], assisting the model in capturing patterns, trends, and correlations in both domains. By enhancing feature representations and promoting generalization, these mappings contribute to improved prediction accuracy and overall model performance [38]. ...
Railway signaling electromechanical devices (RSEDs) play a pivotal role in the railway industry. Normal wear and tear of these devices occur during day-and-night operation and even progressively develop into failures. Hence, it is imperative to predict the remaining useful life (RUL) for reliable services. However, there are three existing challenges in addressing the issue. To overcome these challenges, we introduce M2BIST-SPNet for RSEDs RUL prediction. The model incorporates advanced spatiotemporal feature extraction mechanisms, including a spatial-temporal attention-based convolution networks (STACN), multiple branches with multi-scale multifrequency module (MBM), and communication module (CM). Extensive experiments conducted on three typical object datasets (i.e., railway safety relay, EMU contactor, switch circuit controller) shows that the proposed approach has demonstrated state-of-the-art performance in long-term multiparameter prediction and RUL prediction.
... So far, some researchers have studied the equivalent circuit of the stub waveguide [12][13][14], and a series of tuning algorithms have been proposed [15][16][17][18][19]. In [20], a four-stub waveguide was studied and analyzed as a radio frequency circuit, and impedance matching was accomplished by adjusting the stubs to change the reflection coefficient of the port. ...
In high-power microwave applications, the reflection of energy can be effectively reduced by adjusting the three-stub waveguide. However, most of the existing tuning algorithms do not make an arrangement for the adjustment sequence of stubs, and accurately calculating the depth of the stubs requires a great deal of time via electromagnetic (EM) simulations, which may cause large reflection in the matching process. To solve these problems, we first propose an improved calculation method that can accurately calculate the input impedance of a three-stub waveguide. Then, an impedance matching algorithm is designed based on the equivalent circuit model that can quickly and accurately calculate the optimal depth of the stubs. Finally, we present a low-reflection tuning strategy to suppress the large reflection during the adjustment of the stub process. An experimental system is built to verify the calculation method and the tuning strategy. The results show that the strategy can avoid large reflection in the case of load mutation and can maintain low reflection when the load changes continuously. The algorithm meets the needs of the industry and can be used for automatic and real-time adjustment of three-stub waveguides of different specifications.
