Space mapping: the state of the art
TL;DR: For the first time, a mathematical motivation is presented and SM is placed into the context of classical optimization to achieve a satisfactory solution with a minimal number of computationally expensive "fine" model evaluations.
Abstract: 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.
Citations
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TL;DR: The present state of the art of constructing surrogate models and their use in optimization strategies is reviewed and extensive use of pictorial examples are made to give guidance as to each method's strengths and weaknesses.
1,919 citations
TL;DR: 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‐f fidelity physically based surrogates which are simplified models of the original system are detailed in this paper.
Abstract: [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.
663 citations
Cites background from "Space mapping: the state of the art..."
...Many approaches have been proposed to address this problem of nonuniqueness [Bakr et al., 1999; Bandler et al., 1996; Bandler et al., 2004]....
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...The space mapping approach, explained in section 3.2.1, is a means to derive these empirical relationships [Bandler et al., 1994; Bandler et al., 2004]....
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...Once the corresponding points in the two spaces are available, different linear or nonlinear functions may be used to relate the two spaces by fitting over these points [Bandler et al., 2004]....
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TL;DR: A survey on related modeling and optimization strategies that may help to solve High-dimensional, Expensive (computationally), Black-box (HEB) problems and two promising approaches are identified to solve HEB problems.
Abstract: 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.
535 citations
Cites background from "Space mapping: the state of the art..."
...Mapping (Bakr et al. 1999a, b, 1998; Bandler et al. 1994, 1995a, b; Leary et al. 2001) aims to establish a relationship between the input space of the coarse model and that of the fine model such that the coarse model with the mapped parameter accurately mirrors the behavior of the fine model....
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...Bandler et al. (1994) proposed a space-mapping (SM) technique aiming at optimization....
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TL;DR: The extent to which the use of metamodeling techniques inmultidisciplinary design optimization have evolved in the 25 years since the seminal paper on design and analysis of computer experiments is addressed.
Abstract: The use of metamodeling techniques in the design and analysis of computer experiments has progressed remarkably in the past 25 years, but how far has the field really come? This is the question addressed in this paper, namely, the extent to which the use of metamodeling techniques in multidisciplinary design optimization have evolved in the 25 years since the seminal paper on design and analysis of computer experiments by Sacks et al. (“Design and Analysis of Computer Experiments,” Statistical Science, Vol. 4, No. 4, 1989, pp. 409–435). Rather than a technical review of the entire body of metamodeling literature, the focus is on the evolution and motivation for advancements in metamodeling with some discussion on the research itself; not surprisingly, much of the current research motivation is the same as it was in the past. Based on current research thrusts in the field, multifidelity approximations and ensembles (i.e., sets) of metamodels, as well as the availability of metamodels within commercial soft...
330 citations
Journal Article•
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TL;DR: A generic space-mapping optimization algorithm is formulated, explained step-by-step using a simple microstrip filter example, and its robustness is demonstrated through the fast design of an interdigital filter.
Abstract: In this article we review state-of-the-art concepts of space mapping and place them con- textually into the history of design optimization and modeling of microwave circuits. We formulate a generic space-mapping optimization algorithm, explain it step-by-step using a simple microstrip filter example, and then demonstrate its robustness through the fast design of an interdigital filter. Selected topics of space mapping are discussed, including implicit space mapping, gradient-based space mapping, the optimal choice of surrogate model, and tuning space mapping. We consider the application of space mapping to the modeling of microwave structures. We also discuss a software package for automated space-mapping optimization that involves both electromagnetic (EM) and circuit simulators.
327 citations
References
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9,153 citations
Book•
01 Jan 2009
TL;DR: The aim of this book is to provide a Discussion of Constrained Optimization and its Applications to Linear Programming and Other Optimization Problems.
Abstract: Preface Table of Notation Part 1: Unconstrained Optimization Introduction Structure of Methods Newton-like Methods Conjugate Direction Methods Restricted Step Methods Sums of Squares and Nonlinear Equations Part 2: Constrained Optimization Introduction Linear Programming The Theory of Constrained Optimization Quadratic Programming General Linearly Constrained Optimization Nonlinear Programming Other Optimization Problems Non-Smooth Optimization References Subject Index.
7,278 citations
Book•
01 Feb 1996
TL;DR: In this paper, Schnabel proposed a modular system of algorithms for unconstrained minimization and nonlinear equations, based on Newton's method for solving one equation in one unknown convergence of sequences of real numbers.
Abstract: Preface 1. Introduction. Problems to be considered Characteristics of 'real-world' problems Finite-precision arithmetic and measurement of error Exercises 2. Nonlinear Problems in One Variable. What is not possible Newton's method for solving one equation in one unknown Convergence of sequences of real numbers Convergence of Newton's method Globally convergent methods for solving one equation in one uknown Methods when derivatives are unavailable Minimization of a function of one variable Exercises 3. Numerical Linear Algebra Background. Vector and matrix norms and orthogonality Solving systems of linear equations-matrix factorizations Errors in solving linear systems Updating matrix factorizations Eigenvalues and positive definiteness Linear least squares Exercises 4. Multivariable Calculus Background Derivatives and multivariable models Multivariable finite-difference derivatives Necessary and sufficient conditions for unconstrained minimization Exercises 5. Newton's Method for Nonlinear Equations and Unconstrained Minimization. Newton's method for systems of nonlinear equations Local convergence of Newton's method The Kantorovich and contractive mapping theorems Finite-difference derivative methods for systems of nonlinear equations Newton's method for unconstrained minimization Finite difference derivative methods for unconstrained minimization Exercises 6. Globally Convergent Modifications of Newton's Method. The quasi-Newton framework Descent directions Line searches The model-trust region approach Global methods for systems of nonlinear equations Exercises 7. Stopping, Scaling, and Testing. Scaling Stopping criteria Testing Exercises 8. Secant Methods for Systems of Nonlinear Equations. Broyden's method Local convergence analysis of Broyden's method Implementation of quasi-Newton algorithms using Broyden's update Other secant updates for nonlinear equations Exercises 9. Secant Methods for Unconstrained Minimization. The symmetric secant update of Powell Symmetric positive definite secant updates Local convergence of positive definite secant methods Implementation of quasi-Newton algorithms using the positive definite secant update Another convergence result for the positive definite secant method Other secant updates for unconstrained minimization Exercises 10. Nonlinear Least Squares. The nonlinear least-squares problem Gauss-Newton-type methods Full Newton-type methods Other considerations in solving nonlinear least-squares problems Exercises 11. Methods for Problems with Special Structure. The sparse finite-difference Newton method Sparse secant methods Deriving least-change secant updates Analyzing least-change secant methods Exercises Appendix A. A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations (by Robert Schnabel) Appendix B. Test Problems (by Robert Schnabel) References Author Index Subject Index.
6,831 citations
Book•
01 Mar 1983
TL;DR: Newton's Method for Nonlinear Equations and Unconstrained Minimization and methods for solving nonlinear least-squares problems with Special Structure.
Abstract: Preface 1. Introduction. Problems to be considered Characteristics of 'real-world' problems Finite-precision arithmetic and measurement of error Exercises 2. Nonlinear Problems in One Variable. What is not possible Newton's method for solving one equation in one unknown Convergence of sequences of real numbers Convergence of Newton's method Globally convergent methods for solving one equation in one uknown Methods when derivatives are unavailable Minimization of a function of one variable Exercises 3. Numerical Linear Algebra Background. Vector and matrix norms and orthogonality Solving systems of linear equations-matrix factorizations Errors in solving linear systems Updating matrix factorizations Eigenvalues and positive definiteness Linear least squares Exercises 4. Multivariable Calculus Background Derivatives and multivariable models Multivariable finite-difference derivatives Necessary and sufficient conditions for unconstrained minimization Exercises 5. Newton's Method for Nonlinear Equations and Unconstrained Minimization. Newton's method for systems of nonlinear equations Local convergence of Newton's method The Kantorovich and contractive mapping theorems Finite-difference derivative methods for systems of nonlinear equations Newton's method for unconstrained minimization Finite difference derivative methods for unconstrained minimization Exercises 6. Globally Convergent Modifications of Newton's Method. The quasi-Newton framework Descent directions Line searches The model-trust region approach Global methods for systems of nonlinear equations Exercises 7. Stopping, Scaling, and Testing. Scaling Stopping criteria Testing Exercises 8. Secant Methods for Systems of Nonlinear Equations. Broyden's method Local convergence analysis of Broyden's method Implementation of quasi-Newton algorithms using Broyden's update Other secant updates for nonlinear equations Exercises 9. Secant Methods for Unconstrained Minimization. The symmetric secant update of Powell Symmetric positive definite secant updates Local convergence of positive definite secant methods Implementation of quasi-Newton algorithms using the positive definite secant update Another convergence result for the positive definite secant method Other secant updates for unconstrained minimization Exercises 10. Nonlinear Least Squares. The nonlinear least-squares problem Gauss-Newton-type methods Full Newton-type methods Other considerations in solving nonlinear least-squares problems Exercises 11. Methods for Problems with Special Structure. The sparse finite-difference Newton method Sparse secant methods Deriving least-change secant updates Analyzing least-change secant methods Exercises Appendix A. A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations (by Robert Schnabel) Appendix B. Test Problems (by Robert Schnabel) References Author Index Subject Index.
6,217 citations
Book•
01 Jan 2001
TL;DR: In this paper, the authors present a general framework for coupling matrix for Coupled Resonator Filters with short-circuited Stubs (UWB) and Cascaded Quadruplet (CQ) filters.
Abstract: Preface to the Second Edition. Preface to the First Edition. 1 Introduction. 2 Network Analysis. 2.1 Network Variables. 2.2 Scattering Parameters. 2.3 Short-Circuit Admittance Parameters. 2.4 Open-Circuit Impedance Parameters. 2.5 ABCD Parameters. 2.6 Transmission-Line Networks. 2.7 Network Connections. 2.8 Network Parameter Conversions. 2.9 Symmetrical Network Analysis. 2.10 Multiport Networks. 2.11 Equivalent and Dual Network. 2.12 Multimode Networks. 3 Basic Concepts and Theories of Filters. 3.1 Transfer Functions. 3.2 Lowpass Prototype Filters and Elements. 3.3 Frequency and Element Transformations. 3.4 Immittance Inverters. 3.5 Richards' Transformation and Kuroda Identities. 3.6 Dissipation and Unloaded Quality Factor. 4 Transmission Lines and Components. 4.1 Microstrip Lines. 4.2 Coupled Lines. 4.3 Discontinuities and Components. 4.4 Other Types of Microstrip Lines. 4.5 Coplanar Waveguide (CPW). 4.6 Slotlines. 5 Lowpass and Bandpass Filters. 5.1 Lowpass Filters. 5.2 Bandpass Filters. 6 Highpass and Bandstop Filters. 6.1 Highpass Filters. 6.2 Bandstop Filters. 7 Coupled-Resonator Circuits. 7.1 General Coupling Matrix for Coupled-Resonator Filters. 7.2 General Theory of Couplings. 7.3 General Formulation for Extracting Coupling Coefficient k. 7.4 Formulation for Extracting External Quality Factor Qe. 7.5 Numerical Examples. 7.6 General Coupling Matrix Including Source and Load. 8 CAD for Low-Cost and High-Volume Production. 8.1 Computer-Aided Design (CAD) Tools. 8.2 Computer-Aided Analysis (CAA). 8.3 Filter Synthesis by Optimization. 8.4 CAD Examples. 9 Advanced RF/Microwave Filters. 9.1 Selective Filters with a Single Pair of Transmission Zeros. 9.2 Cascaded Quadruplet (CQ) Filters. 9.3 Trisection and Cascaded Trisection (CT) Filters. 9.4 Advanced Filters with Transmission-Line Inserted Inverters. 9.5 Linear-Phase Filters. 9.6 Extracted Pole Filters. 9.7 Canonical Filters. 9.8 Multiband Filters. 10 Compact Filters and Filter Miniaturization. 10.1 Miniature Open-Loop and Hairpin Resonator Filters. 10.2 Slow-Wave Resonator Filters. 10.3 Miniature Dual-Mode Resonator Filters. 10.4 Lumped-Element Filters. 10.5 Miniature Filters Using High Dielectric-Constant Substrates. 10.6 Multilayer Filters. 11 Superconducting Filters. 11.1 High-Temperature Superconducting (HTS) Materials. 11.2 HTS Filters for Mobile Communications. 11.3 HTS Filters for Satellite Communications. 11.4 HTS Filters for Radio Astronomy and Radar. 11.5 High-Power HTS Filters. 11.6 Cryogenic Package. 12 Ultra-Wideband (UWB) Filters. 12.1 UWB Filters with Short-Circuited Stubs. 12.2 UWB-Coupled Resonator Filters. 12.3 Quasilumped Element UWB Filters. 12.4 UWB Filters Using Cascaded Miniature High- And Lowpass Filters. 12.5 UWB Filters with Notch Band(s). 13 Tunable and Reconfigurable Filters. 13.1 Tunable Combline Filters. 13.2 Tunable Open-Loop Filters without Via-Hole Grounding. 13.3 Reconfigurable Dual-Mode Bandpass Filters. 13.4 Wideband Filters with Reconfigurable Bandwidth. 13.5 Reconfigurable UWB Filters. 13.6 RF MEMS Reconfigurable Filters. 13.7 Piezoelectric Transducer Tunable Filters. 13.8 Ferroelectric Tunable Filters. Appendix: Useful Constants and Data. A.1 Physical Constants. A.2 Conductivity of Metals at 25 C (298K). A.3 Electical Resistivity rho in 10-8 m of Metals. A.4 Properties of Dielectric Substrates. Index.
4,774 citations
"Space mapping: the state of the art..." refers background in this paper
...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]....
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