The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design optimization of a system. The knowledge is updated with new validation information from the system when available.
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).
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 (fast model) is usually referred to as the "coarse" model (coarse space). The former (slow model) 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 model (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. In particular, the Aggressive Space Mapping (ASM) process displays key characteristics of cognition (an expert's approach to a problem), and is often illustrated in simple cognitive terms.
Development Following John Bandler's concept in 1993,J.W. Bandler, "Have you ever wondered about the engineer's mysterious 'feel' for a problem?" IEEE Canadian Review, no. 70, pp. 50-60, Summer 2013. Reprinted in IEEE Microwave Magazine , vol. 19, no. 2, pp.112-122, Mar./Apr. 2018.J.W. Bandler, R.M. Biernacki, S.H. Chen, P.A. Grobelny, and R.H. Hemmers, "Space mapping technique for electromagnetic optimization," IEEE Trans. Microwave Theory Tech., vol. 42, no. 12, pp. 2536-2544, Dec. 1994. algorithms have utilized Broyden updates (aggressive space mapping),J.W. Bandler, R.M. Biernacki, S.H. Chen, R.H. Hemmers, and K. Madsen,"Electromagnetic optimization exploiting aggressive space mapping," IEEE Trans. Microwave Theory Tech., vol. 43, no. 12, pp. 2874-2882, Dec. 1995. trust regions,M.H. Bakr, J.W. Bandler, R.M. Biernacki, S.H. Chen and K. Madsen, "A trust region aggressive space mapping algorithm for EM optimization," IEEE Trans. Microwave Theory Tech., vol. 46, no. 12, pp. 2412-2425, Dec. 1998. and artificial neural networks.M.H. Bakr, J.W. Bandler, M.A. Ismail, J.E. Rayas-Sánchez and Q.J. Zhang, "Neural space mapping EM optimization of microwave structures," IEEE MTT-S Int. Microwave Symp. Digest (Boston, MA, 2000), pp. 879-882. Developments include implicit space mapping,J.W. Bandler, Q.S. Cheng, N.K. Nikolova and M.A. Ismail, "Implicit space mapping optimization exploiting preassigned parameters," IEEE Trans. Microwave Theory Tech., vol. 52, no. 1, pp. 378-385, Jan. 2004. 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. A 2004 paper reviews the state of the art after the first ten years of development and implementation.J.W. Bandler, Q. Cheng, S.A. Dakroury, A.S. Mohamed, M.H. Bakr, K. Madsen and J. Søndergaard, "Space mapping: the state of the art," IEEE Trans. Microwave Theory Tech., vol. 52, no. 1, pp. 337-361, Jan. 2004. Tuning space mappingS. Koziel, J. Meng, J.W. Bandler, M.H. Bakr, and Q.S. Cheng, "Accelerated microwave design optimization with tuning space mapping," IEEE Trans. Microwave Theory Tech., vol. 57, no. 2, pp. 383-394, Feb. 2009. 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. The space mapping concept has been extended to neural-based space mapping for large-signal statistical modeling of nonlinear microwave devices.L. Zhang, J. Xu, M.C.E. Yagoub, R. Ding, and Q.J. Zhang, "Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling," IEEE Trans. Microwave Theory Tech., vol. 53, no. 9, pp. 2752-2767, Sep. 2005.L. Zhang, Q.J. Zhang, and J. Wood, "Statistical neuro-space mapping technique for large-signal modeling of nonlinear devices," IEEE Trans. Microwave Theory Tech., vol. 56, no. 11, pp. 2453-2467, Nov. 2008. Space mapping is supported by sound convergence theory and is related to the defect-correction approach.D. Echeverria and P.W. Hemker, "Space mapping and defect correction" Computational Methods in Applied Mathematics, vol. 5, no, 2, pp. 107-136, Jan. 2005.
A 2016 state-of-the-art review is devoted to aggressive space mapping.J.E. Rayas-Sanchez,"Power in simplicity with ASM: tracing the aggressive space mapping algorithm over two decades of development and engineering applications", IEEE Microwave Magazine, vol. 17, no. 4, pp. 64-76, April 2016. It spans two decades of development and engineering applications. A comprehensive 2021 review paper J.E. Rayas-Sánchez, S. Koziel, and J.W. Bandler, “Advanced RF and microwave design optimization: a journey and a vision of future trends,” (invited), IEEE J. Microwaves, vol. 1, no. 1, pp. 481-493, Jan. 2021. discusses space mapping in the context of radio frequency and microwave design optimization; in the context of engineering surrogate model, feature-based and cognition-driven design; and in the context of machine learning, intuition, and human intelligence.
The space mapping methodology can also be used to solve inverse problems. Proven techniques include the Linear Inverse Space Mapping (LISM) algorithm,J.E. Rayas-Sanchez, F. Lara-Rojo and E. Martanez-Guerrero,"A linear inverse space-mapping (LISM) algorithm to design linear and nonlinear RF and microwave circuits", IEEE Trans. Microwave Theory Tech., vol. 53, no. 3, pp. 960-968 2005. as well as the Space Mapping with Inverse Difference (SM-ID) method.M. Şimsek and N. Serap Şengör "Solving Inverse Problems by Space Mapping with Inverse Difference Method," Mathematics in Industry, vol. 14, 2010, pp 453-460.
Category
Space mapping optimization belongs to the class of surrogate-based optimization methods,A.J. Booker, J.E. Dennis, Jr., P.D. Frank, D.B. Serafini, V. Torczon, and M.W. Trosset,"A rigorous framework for optimization of expensive functions by surrogates," Structural Optimization, vol. 17, no. 1, pp. 1-13, Feb. 1999. that is to say, optimization methods that rely on a surrogate model.
Applications The space mapping technique has been applied in a variety of disciplines including microwave and electromagnetic design, civil and mechanical applications, aerospace engineering, and biomedical research. Some examples:
Optimizing aircraft wing curvatureT.D. Robinson, M.S. Eldred, K.E. Willcox, and R. Haimes, "Surrogate-Based Optimization Using Multifidelity Models with Variable Parameterization and Corrected Space Mapping," AIAA Journal, vol. 46, no. 11, November 2008. Automotive crashworthiness designM. Redhe and L. Nilsson, "Optimization of the new Saab 9-3 exposed to impact load using a space mapping technique," Structural and Multidisciplinary Optimization, vol. 27, no. 5, pp. 411-420, July 2004.T. Jansson, L. Nilsson, and M. Redhe, "Using surrogate models and response surfaces in structural optimization—with application to crashworthiness design and sheet metal forming," Structural and Multidisciplinary Optimization, vol. 25, no.2, pp 129-140, July 2003. EEG source analysisG. Crevecoeur, H. Hallez, P. Van Hese, Y. D'Asseler, L. Dupré, and R. Van de Walle,"EEG source analysis using space mapping techniques," Journal of Computational and Applied Mathematics, vol. 215, no. 2, pp. 339-347, May 2008.G. Crevecoeur, H. Hallez, P. Van Hese, Y. D'Asseler, L. Dupré, and R. Van de Walle,"A hybrid algorithm for solving the EEG inverse problem from spatio-temporal EEG data," Medical & Biological Engineering & Computing, vol. 46, no. 8, pp. 767-777, August 2008. Handset antenna optimizationS. Tu, Q.S. Cheng, Y. Zhang, J.W. Bandler, and N.K. Nikolova, "Space mapping optimization of handset antennas exploiting thin-wire models," IEEE Trans. Antennas Propag., vol. 61, no. 7, pp. 3797-3807, July 2013.]N. Friedrich, "Space mapping outpaces EM optimization in handset-antenna design," microwaves&rf, Aug. 30, 2013.Juan C. Cervantes-González, J. E. Rayas-Sánchez, C. A. López, J. R. Camacho-Pérez, Z. Brito-Brito, and J. L. Chavez-Hurtado,"Space mapping optimization of handset antennas considering EM effects of mobile phone components and human body," Int. J. RF and Microwave CAE, vol. 26, no. 2, pp. 121-128, Feb. 2016. Design centering of microwave circuitsHany L. Abdel-Malek, Abdel-karim S.O. Hassan, Ezzeldin A. Soliman, and Sameh A. Dakroury, "The Ellipsoidal Technique for Design Centering of Microwave Circuits Exploiting Space-Mapping Interpolating Surrogates," IEEE Trans. Microwave Theory Tech., vol. 54, no. 10, October 2006. Design of electric machines using multi-physical modelingR. Khlissa, S. Vivier, L.A. Ospina Vargas, and G. Friedrich, "Application of Output Space Mapping method for Fast Optimization using Multi-physical Modeling" . Control of partial differential equations.M. Hintermüller and L.N. Vicente, "Space Mapping for Optimal Control of Partial Differential Equations". Voice coil actuator designL. Encica, J. Makarovic, E.A. Lomonova, and A.J.A. Vandenput, "Space mapping optimization of a cylindrical voice coil actuator", IEEE Trans. Ind. Appl., vol. 42, no. 6, pp.1437-1444, 2006. Reconstruction of local magnetic propertiesG. Crevecoeur, L. Dupre, L. Vandenbossche, and R. Van de Walle, "Reconstruction of local magnetic properties of steel sheets by needle probe methods using space mapping techniques," Journal of Applied Physics, vol. 99, no. 08H905, 2006. Structural optimizationO. Lass, C. Posch, G. Scharrer and S. Volkwein, "Space mapping techniques for a structural optimization problem governed by the p-Laplace equation" , Optimization Methods and Software, 26:4-5, pp. 617-642, 2011. Design of microwave filters and multiplexersM.A. Ismail, D. Smith, A. Panariello, Y. Wang, and M. Yu, "EM-based design of large-scale dielectric-resonator filters and multiplexers by space mapping," IEEE Trans. Microwave Theory Tech., vol. 52, no. 1, pp. 386-392, Jan. 2004.J. Ossorio, J.C. Melgarejo, V.E. Boria, M. Guglielmi, and J.W. Bandler, "On the alignment of low-fidelity and high-fidelity simulation spaces for the design of microwave waveguide filters," IEEE Trans. Microwave Theory Tech., vol. 66, no. 12, pp. 5183-5196, Dec. 2018. Optimization of delay structuresQ. Zhang, J.W. Bandler, and C. Caloz, "Design of dispersive delay structures (DDSs) formed by coupled C-sections using predistortion with space mapping," IEEE Trans. Microwave Theory Tech., vol. 61, no. 12, pp. 4040-4051, Dec. 2013.