-In this paper we show that space-mapping optimization can be understood in the framework of defect correction. Then, space-mapping algorithms can be seen as special cases of defect correction iteration. In order to analyze the properties of space mapping and the space-mapping function, we introduce the new concept of flexibility of the underlying models. The best space-mapping results are obtained for so-called equally flexible models. By introducing an affine operator as a left preconditioner, two models can be made equally flexible, at least in the neighborhood of a solution. This motivates an improved space-mapping (or manifold-mapping) algorithm. The left preconditioner complements traditional space mapping where only a right preconditioner is used. In the last section a few simple examples illustrate some of the phenomena analyzed in this paper.2000 Mathematics Subject Classification: 65K05; 65B05.

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationSpace mapping and defect correction D. Echeverría, P.W. Hemker Space mapping and defect correction ABSTRACT In this paper we show that space-mapping optimization can be understood in the framework of defect correction. Then, space-mapping algorithms can be seen as special cases of defect correction iteration. In order to analyze properties of space mapping and the space-mapping function we introduce the new concept of flexibility of the underlying models. The best spacemapping results are obtained for so-called equally flexible models. By introducing an affine operator as a left preconditioner, two models can be made equally flexible, at least in the neighborhood of a solution. This motivates an improved space-mapping (or manifold-mapping) algorithm. The left preconditioner complements traditional space mapping where only a right preconditioner is used. In the last section a few simple examples illustrate some of the phenomena analyzed in this paper. REPORT MAS-E0506 MARCH 2005 ABSTRACTIn this paper we show that space-mapping optimization can be understood in the framework of defect correction. Then, space-mapping algorithms can be seen as special cases of defect correction iteration. In order to analyze properties of space mapping and the space-mapping function we introduce the new concept of flexibility of the underlying models. The best space-mapping results are obtained for so-called equally flexible models.By introducing an affine operator as a left preconditioner, two models can be made equally flexible, at least in the neighborhood of a solution. This motivates an improved space-mapping (or manifold-mapping) algorithm. The left preconditioner complements traditional space mapping where only a right preconditioner is used. In the last section a few simple examples illustrate some of the phenomena analyzed in this paper. Mathematics Subject Classification: 65K05, 65B05Keywords and Phrases: Space mapping, defect correction, two-level optimization IntroductionThe space-mapping idea was conceived by Bandler [4] in the field of microwave filter design. It aims at reducing the cost of accurate optimization computations by iteratively correcting a sequence of rougher approximations. In technological applications this allows us to couple simple rules that represent expert knowledge accumulated over the years with the accuracy of expensive simulation techniques based on the numerical solution of partial differential equations. This combination may yield an efficient method with good accuracy of the final solution. As an example, in Section 5.2 we significantly accelerate the solution process for an optimization problem from magnetostatics by combining finite elements for the precise computations of the magnetic field with rather simple magnetic circuit calculations. The space-mapping technique has been mainly applied in electromagnetics [6,8] but, since the underlying principles...

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationManifold-mapping optimization applied to linear actuator design ABSTRACT Optimization procedures in practice are based on highly accurate models that typically have an excessive computational cost. By exploiting auxiliary models that are less accurate but much cheaper to compute, space-mapping has been reported to accelerate such procedures. However, the space-mapping solution does not always coincide with the accurate model optimum. We introduce manifold mapping, an improved version of space mapping that finds this precise solution with the same computational efficiency. By an example in linear actuator design we show that our technique delivers a significant speed-up compared to other optimization schemes.2000 Mathematics Subject Classification: 65K10, 65M60, 65N55, 65Y20, 90C31

In this paper, we analyze in some detail the manifold-mapping optimization technique introduced recently [Echeverría and Hemker in space mapping and defect correction. Comput Methods Appl Math 5(2): 107--136, 2005]. Manifold mapping aims at accelerating optimal design procedures that otherwise require many evaluations of time-expensive cost functions. We give a proof of convergence for the manifold-mapping iteration. By means of two simple optimization problems we illustrate the convergence results derived. Finally, the performances of several variants of the method are compared for some design problems from electromagnetics. Mathematics Subject Classification (2000)90C30 · 65K05 · 49M99 · 49Q99 Dedicated to Wolfgang Hackbusch on the occasion of his 60th birthday.Communicated by G. Wittum.

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationA trust-region strategy for manifold mapping optimization P.W. Hemker, D. Echeverría A trust-region strategy for manifold mapping optimization ABSTRACT As a starting point we take the space-mapping iteration technique by Bandler et al. for the efficient solution of optimization problems. This technique achieves acceleration of accurate design processes with the help of simpler, easier to optimize models. We observe the difference between the solution of the optimization problem and the computed space-mapping solutions. We repair this discrepancy by exploiting the correspondence with defect-correction iteration and we construct the manifold-mapping algorithm, which is as efficient as the space-mapping algorithm but converges to the exact solution. To increase the robustness of the algorithm we also introduce a trust-region strategy that is based on the generalized singular value decomposition of the linearized fine and coarse manifold representations. The effect of the strategy is shown by the solution of a variety of small non-linear least squares problems. simpler, easier to optimize models. We observe the difference between the solution of the optimization problem and the computed space-mapping solutions. We repair this discrepancy by exploiting the correspondence with defect-correction iteration and we construct the manifold-mapping algorithm, which is as efficient as the space-mapping algorithm but converges to the exact solution. REPORT MAS-E0618 AUGUST 2006To increase the robustness of the algorithm we also introduce a trust-region strategy that is based on the generalized singular value decomposition of the linearized fine and coarse manifold representations. The effect of the strategy is shown by the solution of a variety of small non-linear least squares problems.2000 Mathematics Subject Classification: 90C30, 65K05, 49M99, 49Q99

Engineering optimization procedures employ highly accurate numerical models that typically have an excessive computational cost, e.g. finite elements (FE). The space mapping (SM) technique speeds up the minimization procedure by exploiting also simplified (less accurate) models. We will use the SM terminology of fine and coarse in order to refer to the accurate and inaccurate models, respectively. SM implementation in the field of electromagnetic actuators design, in the context of constrained optimization, is a rather unexplored topic. A linear voice coil actuator is chosen as a benchmark test example. The key element in SM is the so-called SM function, which efficiently corrects the imprecise results that can be obtained with just coarse information. SM is used to solve a shape optimization problem. The design problem is stated as a minimization in which the computational cost lies completely in the constraint evaluation, thus SM is applied only to the constraints. A mathematical description of the approach is presented in this paper and two implementations are compared. The solution of the SM optimization is validated (locally) by means of a standard minimization routine. The numerical results obtained show a high efficiency of the SM-based optimization algorithm, reflected by a significantly low number of fine model simulations and an overall low computational effort. 2.

In this report we study space-mapping and manifold-mapping, two multi-level optimization techniques that aim at accelerating expensive optimization procedures with the aid of simple auxiliary models. Manifoldmapping improves in accuracy the solution given by space-mapping. In this report, the two mentioned techniques are basically described and then applied in the solving of two minimization problems. Several coarse models are tried, both from a two and a three level perspective. The results with these simple tests confirm the speed-up expected for the multi-level approach.

Purpose -Optimisation in electromagnetics, based on finite element models, is often very time-consuming. In this paper, we present the space-mapping (SM) technique which aims at speeding up such procedures by exploiting auxiliary models that are less accurate but much cheaper to compute. Design/methodology/approach -The key element in this technique is the SM function. Its purpose is to relate the two models. The SM function, combined with the low accuracy model, makes a surrogate model that can be optimised more efficiently. Findings -By two examples we show that the SM technique is effective. Further we show how the choice of the low accuracy model can influence the acceleration process. On one hand, taking into account more essential features of the problem helps speeding up the whole procedure. On the other hand, extremely simple auxiliary models can already yield a significant acceleration. Research limitations/implications -Obtaining the low accuracy model is not always straightforward. Some research could be done in this direction. The SM technique can also be applied iteratively, i.e. the auxiliary model is optimised aided by a coarser one. Thus, the generation of hierarchies of models seems to be a promising venue for the SM technique. Originality/value -Optimisation in electromagnetics, based on finite element models, is often very time-consuming. The results given show that the SM technique is effective for speeding up such procedures.