Space mapping and defect correction

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 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 pape

A trust-region strategy for manifold-mapping optimization

Studying the space-mapping technique by Bandler et al. [J. Bandler, R. Biernacki, S. Chen, P. Grobelny, R.H. Hemmers, Space mapping technique for electromagnetic optimization, IEEE Trans. Microwave Theory Tech. 42 (1994) 2536–2544] for the solution of optimization problems, we observe the possible difference between the solution of the optimization problem and the computed space-mapping solution. 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 introduce a trust-region strategy (a regularization technique) based on the generalized singular value decomposition of the linearized fine and coarse manifold representations. The effect of this strategy is shown by the solution of a variety of small non-linear least squares problems. Finally we show the use of the technique for a more challenging engineering problem

Manifold mapping: a two-level optimization technique

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 problemswe illustrate the convergence results derived. Finally, the performances of several variants of the method are compared for some design problems from electromagnetics

A trust-region strategy for manifold mapping optimization.

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 problem

Space Mapping and Defect Correction

In this chapter we present the principles of the space-mapping iteration techniques for the efficient solution of optimization problems. We also show how space-mapping optimization can be understood in the framework of defect correction. 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 true solution. In the last section we show a simple example from practice, comparing space-mapping and manifold mapping and illustrating the efficiency of the technique

Optimization in Electromagnetics with the Space-Mapping Technique

Abstract: 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

On the manifold-mapping optimization technique

In this paper, we study in some detail the manifold-mapping optimization technique introduced in an earlier paper. 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 practical design problems in electromagnetic

Space-mapping techniques applied to the optimization of a safety isolating transformer

Space-mapping optimization techniques allow to allign low-fidelity and high-fidelity models in order to reduce the computational time and increase the accuracy of the solution. The main idea is to build an approximate model from the difference of response between both models. Therefore the optimization process is computed on the surrogate model. In this paper, some recent approaches of space-mapping techniques such as agressive-space-mapping, output-mapping and manifold-mapping algorithms are applied to optimize a safety insulating transformer. The electric, magnetic and thermal phenomena of the device are modeled by an analytical model and a 3D finite element model. It is considered as a benchmark for multi-level optimization to test different algorithms

Path Integrals, Density Matrices, and Information Flow with Closed Timelike Curves

Two formulations of quantum mechanics, inequivalent in the presence of closed timelike curves, are studied in the context of a soluable system. It illustrates how quantum field nonlinearities lead to a breakdown of unitarity, causality, and superposition using a path integral. Deutsch's density matrix approach is causal but typically destroys coherence. For each of these formulations I demonstrate that there are yet further alternatives in prescribing the handling of information flow (inequivalent to previous analyses) that have implications for any system in which unitarity or coherence are not preserved.Comment: 25 pages, phyzzx, CALT-68-188