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 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.

Towards Multi-level Optimization: Space-Mapping and Manifold-Mapping

“…A space-mapping function is perfect iff p(x * f ) = x * c . Then, if p is perfect and injective, we have x * sm = x * f [6]. Though injectivity is not really a strong requirement, only in very special problems we can expect the spacemapping function to be perfect (it is easy to check that this is the case for any reachable design).…”

“…In [6] we analyze under which circumstances space-mapping yields the precise optimum. Essentially it depends on the behavior of the manifolds f (X) and c(Z).…”

“…According to the definition of p, we can expect that c(p(x)) ≈ f (x) and thus that the space-mapping solution problem, we obtain linear preconditioning [6]. The actual space-mapping algorithm used in this report is shown in Figure 3.…”

The state of the art of microwave CAD: EM-based optimization and modeling

“…A vector of space-mapping parameters, , is obtained using the parameter extraction procedure (4) An example of the generic surrogate model is an input space mapping of the form , where . A variety of other space-mapping surrogates can be found in [1]- [4], [27].…”

Implicit space mapping with adaptive selection of preassigned parameters

“…Since the space mapping concept [1][2][3][4][5] was first developed, it has been successfully applied to microwave engineering design problems as well as in other engineering fields (e.g. [6 -18]).…”