Optimization Using Space Mapping | Pernille Brock
| | Abstract | The subject of this master thesis is non-linear optimization using the Space Mapping method with an interpolating surrogate model. The Space Mapping method is useful in optimization problems, where the fine model we wish to optimize is very computationally expensive. The interpolating surrogate is based on a cheap coarse model and serves as a replacement for the expensive model in order to minimize the number of function evaluations. An important part of the Space Mapping algorithm is the Parameter Extraction, which involves minimization of the residual between the surrogate and the fine model, which we aim to align. The Parameter Extraction problem does not always have a unique solution, and different formulations are presented in order to ensure this uniqueness.
The thesis provides a presentation of the mathematical theory followed by the Space Mapping algorithm. We then make a number of theoretical and practical investigations concerning different formulations of the residual defining the Parameter Extraction problem. The step length in forward difference approximations is analyzed, and the optimal step length suited for the considered problems is found to be approximately 10(-5). We make an analysis of the solutions to underdetermined and overdetermined problems, hereby an analysis of the Marquardt equations and of least squares problems with and without weighting factors. We look at the effect of adding a regularization term to the residual vector and find, that this residual formulation corresponds to a special case of the Marquardt equations with the damping parameter 1(+u). The presented Space Mapping algorithm is tested in the various versions on three test problems, and the results are compared. The convergence is faster than with classical optimization algorithms. It is not possible to make general conclusions on the performance of the different algorithm versions based on the included test problems. | | Keywords | Space mapping, non-linear optimization, interpolating surrogates, least squares problems, weighting factors, underdetermined and overdetermined problems | | Type | Master's thesis [Academic thesis] | | Year | 2004 | | Publisher | Informatics and Mathematical Modelling, Technical University of Denmark, DTU | | Address | Richard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby | | Series | IMM-Thesis-2004-54 | | Note | Supervised by Prof. Kaj Madsen, Assoc. Prof. Hans Bruun Nielsen, and Ph.D.stud. Frank Pedersen | | Electronic version(s) | [pdf] [ps] | | BibTeX data | [bibtex] | | IMM Group(s) | Scientific Computing |
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