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 computationally expensive "fine" model evaluations. SM procedures iteratively update and optimize surrogates based on a fast physically based "coarse" model. Proposed approaches to SM-based optimization include the original algorithm, the Broyden-based aggressive SM algorithm, various trust-region approaches, neural SM, and implicit SM. Parameter extraction is an essential SM subproblem. It is used to align the surrogate (enhanced coarse model) with the fine model. Different approaches to enhance uniqueness are suggested, including the recent gradient parameter-extraction approach. Novel physical illustrations are presented, including the cheese-cutting and wedge-cutting problems. Significant practical applications are reviewed.

https://desi.iteso.mx/erayas/documents/CirOpt_course/Lectures/Space_Mapping/Bandler_04Jan_SM_State_of_the_Art.pdf

Space mapping: the state of the art

[1] Surrogate modeling, also called metamodeling, has evolved and been extensively used over the past decades. A wide variety of methods and tools have been introduced for surrogate modeling aiming to develop and utilize computationally more efficient surrogates of high-fidelity models mostly in optimization frameworks. This paper reviews, analyzes, and categorizes research efforts on surrogate modeling and applications with an emphasis on the research accomplished in the water resources field. The review analyzes 48 references on surrogate modeling arising from water resources and also screens out more than 100 references from the broader research community. Two broad families of surrogates namely response surface surrogates, which are statistical or empirical data-driven models emulating the high-fidelity model responses, and lower-fidelity physically based surrogates, which are simplified models of the original system, are detailed in this paper. Taxonomies on surrogate modeling frameworks, practical details, advances, challenges, and limitations are outlined. Important observations and some guidance for surrogate modeling decisions are provided along with a list of important future research directions that would benefit the common sampling and search (optimization) analyses found in water resources.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2011WR011527

Review of surrogate modeling in water resources

We offer space mapping (SM), a fundamental new theory to circuit optimization utilizing a parameter space transformation. This technique is demonstrated by the optimization of a microstrip structure for which a convenient analytical/empirical model is assumed to be unavailable. For illustration, we focus upon a three-section microstrip impedance transformer and a double folded stub microstrip filter and explore various design characteristics utilizing an electromagnetic (EM) field simulator. We propose two distinct EM models: coarse for fast computations, and the corresponding fine for a few more accurate and well-targeted simulations. The coarse model, useful when circuit-theoretic models are not readily available, permits rapid exploration of different starting points, solution robustness, local minima, parameter sensitivities, yield-driven design and other design characteristics within a practical time frame. The computationally intensive fine model is used to verify the space-mapped designs obtained exploiting the coarse model, as well as in the SM process itself. >

Space mapping technique for electromagnetic optimization

The integration of optimization methodologies with computational analyses/simulations has a profound impact on the product design. Such integration, however, faces multiple challenges. The most eminent challenges arise from high-dimensionality of problems, computationally-expensive analysis/simulation, and unknown function properties (i.e., black-box functions). The merger of these three challenges severely aggravates the difficulty and becomes a major hurdle for design optimization. This paper provides a survey on related modeling and optimization strategies that may help to solve High-dimensional, Expensive (computationally), Black-box (HEB) problems. The survey screens out 207 references including multiple historical reviews on relevant subjects from more than 1,000 papers in a variety of disciplines. This survey has been performed in three areas: strategies tackling high-dimensionality of problems, model approximation techniques, and direct optimization strategies for computationally-expensive black-box functions and promising ideas behind non-gradient optimization algorithms. Major contributions in each area are discussed and presented in an organized manner. The survey exposes that direct modeling and optimization strategies to address HEB problems are scarce and sporadic, partially due to the difficulty of the problem itself. Moreover, it is revealed that current modeling research tends to focus on sampling and modeling techniques themselves and neglect studying and taking the advantages of characteristics of the underlying expensive functions. Based on the survey results, two promising approaches are identified to solve HEB problems. Directions for future research are also discussed.

/pdf/survey-of-modeling-and-optimization-strategies-to-solve-high-1tg25zsec9.pdf

Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions

We propose a significantly improved space mapping (SM) strategy for electromagnetic (EM) optimization. Instead of waiting for upfront EM analyses at several base points, our new approach aggressively exploits every available EM analysis, producing dramatic results right from the first step. We establish a relationship between the novel SM optimization and the quasi-Newton iteration for solving a system of nonlinear equations. Approximations to the matrix of first-order derivatives are updated by the classic Broyden formula. A high-temperature superconducting microstrip filter design solution emerges after only six EM simulations with sparse frequency sweeps. Furthermore, less CPU effort is required to optimize the filter than is required by one single detailed frequency sweep. We also extend the SM concept to the parameter extraction phase, overcoming severely misaligned responses induced by inadequate empirical models. This novel concept should have a significant impact on parameter extraction of devices.

Electromagnetic optimization exploiting aggressive space mapping

The authors present the foundation of a sophisticated hierarchical multidimensional response surface modeling system for efficient yield-driven design. The scheme dynamically integrates models and database updating in real optimization time. The method facilitates a seamless, smart, optimization-ready interface. It has been specially designed to handle circuits containing complex subcircuits or components whose simulation requires significant computational effort. This approach makes it possible, for the first time, to perform direct gradient-based yield optimization of circuits with components or subcircuits simulated by an electromagnetic simulator. The efficiency and accuracy of the technique are demonstrated by yield optimization of a three-stage microstrip transformer and a small-signal microwave amplifier. The authors also perform yield sensitivity analysis for the three-stage microstrip transformer. >

Yield-driven electromagnetic optimization via multilevel multidimensional models

We present novel approaches to electromagnetic design of high-temperature superconducting (HTS) quarter-wave parallel coupled-line microstrip filters. The desired narrow bandwidths (less than 2%) coupled with the large dielectric constant of substrate materials used in the HTS technology (er ≈ 24) make the design problems difficult to treat accurately for many traditional microwave circuit design software packages with analytical/empirical models. We have successfully performed an HTS filter design with accurate electromagnetic field simulations. We discuss problems related to electromagnetic design of such filters. We describe a look-up table method and a powerful space mapping optimization technique, which dramatically reduce the CPU time needed in the design process. Finally, we address the issue of improved modeling of the HTS filter using analytical/empirical models. © 1995 John Wiley & Sons, Inc.

Electromagnetic design of high‐temperature superconducting microwave filters

We present novel approaches to electromagnetic design of high-temperature superconducting quarter-wave parallel coupled-line microstrip filters. The dielectric constant of substrate materials used in high-temperature superconductor technology is too large to be accurately treated by traditional microwave circuit design software packages with analytical/empirical models. We employ electromagnetic field simulation and develop a look-up table method and a powerful space mapping optimization technique, which dramatically reduce the CPU time for the design process. >

Electromagnetic design of high-temperature superconducting microwave filters

We offer space mapping (SM), a fundamental new theory to circuit optimization utilizing a parameter space transformation. This technique is demonstrated by the optimization of a microstrip structure for which a convenient analytical/empirical model is assumed to be unavailable. For illustration, we focus upon a three-section microstrip impedance transformer and a double folded stub microstrip filter and explore various design characteristics utilizing an electromagnetic (EM) field simulator. We propose two distinct EM models: coarse for fast computations, and the corresponding fine for a few more accurate and well-targeted simulations. The coarse model, useful when circuit-theoretic models are not readily available, permits rapid exploration of different starting points, solution robustness, local minima, parameter sensitivities, yield-driven design and other design characteristics within a practical time frame. The computationally intensive fine model is used to verify the space-mapped designs obtained exploiting the coarse model, as well as in the SM process itself

P.A. Grobelny

Papers

Space mapping technique for electromagnetic optimization

Yield-driven electromagnetic optimization via multilevel multidimensional models

Electromagnetic design of high‐temperature superconducting microwave filters

Electromagnetic design of high-temperature superconducting microwave filters

Space mapping technique for electromagnetic optimization