Quality assessment of coarse models and surrogates for space mapping optimization

Abstract

One of the central issues in space mapping optimization is the quality of the underlying coarse models and surrogates. Whether a coarse model is sufficiently similar to the fine model may be critical to the performance of the space mapping optimization algorithm and a poor coarse model may result in lack of convergence. Although similarity requirements can be expressed with proper analytical conditions, it is difficult to verify such conditions beforehand for real-world engineering optimization problems. In this paper, we provide methods of assessing the quality of coarse/surrogate models. These methods can be used to predict whether a given model might be successfully used in space mapping optimization, to compare the quality of different coarse models, or to choose the proper type of space mapping which would be suitable to a given engineering design problem. Our quality estimation methods are derived from convergence results for space mapping algorithms. We provide illustrations and several practical application examples.

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

“…Moreover, as a perfect match between and at is not ensured (with respect to value and/or first-order derivatives), there is no guarantee for the space-mapping algorithm to locate the (local) fine model optimal solution [21].…”

“…Existing theoretical results for algorithm (2)-(4) or some of its sub-classes provide convergence conditions, which are, however, difficult to verify beforehand [3], [21]. Moreover, conditions for convergence are typically different from conditions for convergence to the fine model optimal solution (i.e., its firstorder stationary point) [21].…”

“…Moreover, conditions for convergence are typically different from conditions for convergence to the fine model optimal solution (i.e., its firstorder stationary point) [21]. This is because fine model sensitivity information is not utilized by current space-mapping algorithms.…”

“…The idea of surrogate-based optimization is also exploited by other researchers [17]- [20], although many of them construct a surrogate model by direct approximation of the fine model data with no underlying physically based coarse model. Probably the most serious issue for standard space mapping is convergence, depending on the similarity between the fine model and the space-mapping surrogate [3], [21]. Also, because the zero-and first-order consistency conditions [17] between the fine and surrogate models are not necessarily satisfied (i.e., the surrogate model may not match the fine model with respect to value and first-order derivative at any given iteration points), and subsequent iterations are accepted regardless of objective function improvement, there is no guarantee that the specification error will be reduced from iteration to iteration.…”

“…Given an optimization problem, the convergence properties and overall performance of the space-mapping algorithm can be improved to some extent by a proper choice of the coarse model and the space-mapping surrogate [21], [23]. On the other hand, given the models and the mapping type, good algorithm performance (measured by convergence and final design quality) is not guaranteed and has to be verified experimentally by executing the optimization process.…”