Space Mapping and Defect Correction

Abstract

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...

Manifold mapping: a two-level optimization technique

“…There are many variants of the space-mapping algorithm: aggressive space mapping (ASM) [5], trust-region aggressive space mapping (TRASM) [2], neural space mapping (NSM) [3] and implicit space mapping (ISM) [7] being the most significant examples. Although all these schemes do not always converge to the right solution [20], the solution obtained is generally acceptable for practical purposes. Recently, in [8] the original space-mapping approach is modified according to the framework proposed in [1].…”

“…The second type of approximations is found in situations where, because of, e.g., experience or simple rules of thumb, the derivation of the surrogate is simplified. An example of this is the use of lumped parameter models (e.g., magnetic [20], electric [6] or thermal [31] equivalent circuits).…”

“…Though defect-correction theory was originally developed for linear or nonlinear systems of equations, it can also be applied in optimization problems. Space-mapping procedures can be seen as special cases of defect-correction iteration [20] and this can be used to explain in which case the space-mapping approach does or does not fail to find the correct solution.…”

“…Manifold mapping [20] is an alternative surrogate-based optimization technique. Manifold mapping can be used without computing exact gradient information and it has provable convergence to the right solution.…”

Neural input space mapping optimization based on nonlinear two-layer perceptrons with optimized nonlinearity

“…For cases when the actual fine model optimal solution is of practical interest, or when an extremely accurate interpolating surrogate is needed (for instance, to perform highly accurate statistical analysis), SM can be formulated to address the output space of the coarse model, by building a mapping function at the level of the coarse model responses. This approach, termed output space mapping, can be applied by itself [20, 21], or can be combined with the input space mapping techniques, as in the following approaches: linear input–output SM models exploiting fine model Jacobians [22]; implicit SM enhanced by linear input–output mappings [23]; linear input–output SM models enhanced by quadratic approximations [24]; linear input–output SM models with mapping parameters obtained using variable weighting factors [25, 26]; linear input neural output SM models [27]; input–output SM models enhanced by radial basis functions [28]; SM models with fuzzy interpolants [29]; nonlinear support vector regression output SM models [30]; and neural input–output SM models [31].…”