Space mapping

Concept

The space mapping methodology employs a "quasi-global" formulation that intelligently links companion "coarse" (ideal or low-fidelity) and "fine" (practical or high-fidelity) models of different complexities. In engineering design, space mapping aligns a very fast coarse model with the expensive-to-compute fine model so as to avoid direct expensive optimization of the fine model. The alignment can be done either off-line (model enhancement) or on-the-fly with surrogate updates (e.g., aggressive space mapping).

Development

Following John Bandler's concept in 1993,^[1][2] algorithms have utilized Broyden updates (aggressive space mapping),^[3] trust regions,^[4] and artificial neural networks. New developments include implicit space mapping, in which we allow preassigned parameters not used in the optimization process to change in the coarse model, and output space mapping, where a transformation is applied to the response of the model.^[5] A paper reviews the state of the art after the first ten years of development and implementation.^[6] Tuning space mapping utilizes a so-called tuning model—constructed invasively from the fine model—as well as a calibration process that translates the adjustment of the optimized tuning model parameters into relevant updates of the design variables.^[7]

Category

Space mapping optimization belongs to the class of surrogate-based optimization methods.^[8]

Terminology

There is a wide spectrum of terminology associated with space mapping: ideal model, coarse model, fine model, companion model, cheap model, expensive model, low fidelity (resolution) model, high fidelity (resolution) model, empirical model, simplified physics model, physics-based model, quasi-global model, physically expressive model, device under test, electromagnetics-based model, simulation model, computational model, tuning model, calibration model, surrogate model, surrogate update, mapped coarse model, surrogate optimization, parameter extraction, target response, optimization space, validation space, neuro-space mapping, implicit space mapping, output space mapping, predistortion (of design specifications), manifold mapping, defect correction, model management, multi-fidelity models, variable fidelity/variable complexity, multigrid methods, coarse grid, fine grid, surrogate-driven, simulation-driven, model-driven.

Methodology

At the core of the process is a pair of models: one very accurate but too expensive to use directly with a conventional optimization routine, and one significantly less expensive and, accordingly, less accurate. The latter is usually referred to as the "coarse" model. The former is usually referred to as the fine model. A validation space (“reality”) represents the fine model, for example, a high-fidelity physics model. The optimization space, where conventional optimization is carried out, incorporates the coarse (or surrogate) model, for example, the low-fidelity physics or “knowledge” model. In a space-mapping design optimization phase, there is a prediction or “execution” step, where the results of an optimized "mapped coarse model" (updated surrogate) are assigned to the fine model for validation. After the validation process, if the design specifications are not satisfied, relevant data is transferred to the optimization space (“feedback”), where the mapping-augmented coarse model or surrogate is updated (enhanced, realigned with the fine model) through an iterative optimization process termed “parameter extraction.” The mapping formulation itself incorporates “intuition,” part of the engineer's so-called “feel” for a problem.^[9]

Applications

Applications of space mapping continue to appear. Disciplines covered include microwaves, antennas, electronics, photonics, and magnetic systems; civil, mechanical, aeronautical and aerospace engineering systems, including

• Automotive crashworthiness design.^[10][11]
• EEG source analysis.^[12]
• Handset antenna design.^[13][14]
• Electric machines design and optimization .^[15]

Simulators

Various simulators can be involved in a space mapping optimization and modeling processes.

• In microwave and RF area
  • Agilent ADS
  • Agilent Momentum
  • Ansys HFSS
  • CST Microwave Studio
  • FEKO
  • Sonnet em

Nonlinear Device Modeling

The space mapping concept has been extended to neural-based space mapping for large-signal statistical modeling of nonlinear microwave devices.^[16][17]

Conferences

Three international workshops have focused significantly on the art, the science and the technology of space mapping. ^{[18] [19] [20]}

See also

• Mathematical optimization
• Engineering optimization
• Computer-aided design
• Computational electromagnetics
• Semiconductor device modeling
• Multiphysics
• Simulation
• Adaptive control
• Performance tuning
• Linear approximation
• Response surface methodology
• Kriging
• Surrogate model
• Support vector machine

References