Electromagnetic optimisation using sensitivity analysis in the frequency domain D. Li, J. Zhu, N.K. Nikolova, M.H. Bakr and J.W. Bandler Abstract: Gradient-based optimisation relies on the response Jacobian whose evaluation constitu- tes a major computational overhead in full-wave numerical analysis. Adjoint-based techniques may offer numerically efficient solutions, but their implementation is too involved in the case of full- wave computations. A simple approach that uses the self-adjoint sensitivity analysis and Broyden’s update is proposed. The overhead of the Jacobian computation is greatly reduced because an adjoint system analysis is not needed and because Broyden’s update is used to compute the system matrix derivatives. To improve the robustness of the Broyden update in the sensitivity analysis, we propose a switching criterion between the Broyden and the finite-difference estimation of the system matrix derivatives. We illustrate and validate the proposed method using full-wave commercial electromagnetic solvers based on the finite-element method as well as on the method of moments. Different gradient-based optimisation algorithms are exploited in the examples where efficiency is compared in terms of CPU time savings. 1 Introduction Gradient-based optimisation is widely used to solve non- linear design and inverse-imaging problems [1–5]. It employs algorithms such as quasi-Newton, sequential quad- ratic programming (SQP) and trust-region methods. These algorithms exploit the objective function Jacobian and/or Hessian in addition to the objective function itself in order to search for a local optimal point. Typically, they converge much faster, that is, with significantly fewer system ana- lyses, than algorithms utilising the objective function only (e.g. pattern search and the family of global-search algor- ithms). Naturally, the solution provided by a gradient-based optimisation algorithm depends on the quality of the initial design or model. And yet, because of the relatively small number of required forward solutions, gradient-based optimisation is preferred when design or inverse problems are solved with the aid of time-intensive 3-D electromag- netic (EM) simulations. The EM structure representing the starting point of the optimisation is typically the result of approximate, linearised inverse-problem solutions, equivalent-circuit designs, and so on. The efficiency of a successful gradient-based optimis- ation process depends mainly on two factors: (i) the number of iterations required to achieve convergence and (ii) the number of simulation calls per iteration. The first factor depends largely on the nature of the algorithm, on the proper formulation of the objective or cost function and on the accuracy of the response Jacobian and/or Hessian. The second factor depends on the nature of the algorithm and on the method used to compute the Jacobian and/or Hessian, which are necessary to determine the search direction and the step in the parameter space. The sensitivity analysis, which provides the Jacobian, is very time consuming when finite differences or higher order approximations are used at the response level. At least N þ 1 full-wave simulations are needed to obtain a Jacobian for N design parameters. This is unacceptable when N is large. It is well known that adjoint variable methods offer superior efficiency since they yield the Jacobian with only one additional (adjoint) system analysis. They have been exploited widely for design, for yield and tolerance analysis [1–4], for system stability and uncertainty analysis [5], for imaging and inverse scattering problems based on acoustic, microwave and/or near-infrared technology, and so on. In addition to [1–5], some representative examples and valu- able reviews can be found in [6–10]. The adjoint variable methods have their shortcomings. A common feature in their applications is the reliance on analytical system matrix derivatives. These are not only specific to the numerical technique, but also often difficult to derive and even more difficult to implement for tasks such as shape and topology optimisation. Almost exclusively, applications are based on the finite-element method (FEM), which is relatively amenable to obtain analytical system matrix derivatives with respect to shape parameters. The major computational overhead of this adjoint-based Jacobian cal- culation comes from the adjoint system analysis whose computational requirements are usually comparable with those of the original system analysis. Recently, we proposed finite-difference self-adjoint sen- sitivity analysis (FD-SASA) methods for the efficient com- putation of network parameter sensitivities, for example, the S parameters, in the frequency and time domains [11–12]. The S parameters, or functions thereof, are widely used in RF/microwave design and inverse imaging to evaluate the design or model performance and to define the objective # The Institution of Engineering and Technology 2007 doi:10.1049/iet-map:20060303 Paper first received 2nd November 2006 and in revised form 9th April 2007 The authors are with the Department of Electrical and Computer Engineering, McMaster University, 1280 Main St. West, Hamilton, Ontario, Canada L8S 4K1 J.W. Bandler is also with Bandler Corporation, P.O. Box 8083, Dundas, Ontario, Canada L9H 5E7 D. Li and J. Zhu are currently with the Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, Ontario, Canada M5S 3G4 E-mail: talia@mail.ece.mcmaster.ca IET Microw. Antennas Propag., 2007, 1, (4), pp. 852–859 852