Multi-Fidelity Optimization with High-Fidelity Analysis and Low-Fidelity Gradients

Abstract

The paper introduces a new approach to multi-fidelit y optimization. The approach employs gradient-based optimization, where the one-dimensional search points are evaluated using high-fidelity analysis, while the gradients are evaluated using low-fidelity analysis. Correlation between the results of the high- and low-fidelity analyses is not required. The approach is demonstrated using two example problems. Computational savings in terms of time and the number of high-fidelity analyses are discussed. I. Introduction NE of the obstacles in practical implementation of optimization in industry is a potential high computational cost. An analysis of a complex system may take several hours and even days to complete and optimization requires performing many of these analyses. The number of design variables in optimization directly affects the number of analyses: the more design variables in the problem, the more analyses should be performed. This is especially true for a gradient-based optimization, where the gradients are evaluated using finite-difference calculations. A partial answer to the computational cost problem is Response Surface optimization methods 1-7 ,12 , which do not require gradient information for optimization, thus reducing the required number of analyses. One difficulty with the Response Surface optimization methods is that their range of application is typically limited by about 20 design variables. Another approach to reducing the computational cost is multi-fidelity optimization methods 8-12 . These methods combine high and low-fidelity analyses. One example of employing multi-fidelity optimization is creating a response surface from a relatively small number of high-fidelity analyses, then performing low-fidelity analyses for the same points and creating a response surface for low-fidelity analyses. Next, a correction factor is introduced that helps converting low-fidelity analysis results into the high-fidelity analysis results. The correction may be done for the response surfaces or for the analysis results themselves. Finally, when optimization is performed using the lowfidelity analysis, the results of each low-fidelity analysis is updated using the obtained correction factor. At some intermediate stage of the optimization and at the optimum a high-fidelity analysis is performed to verify the results. If the correlation is not satisfactory, the response surfaces for high and low-fidelity analyses are recreated and the correction factor is reevaluated. The process may be repeated several times. And the correction factor itself may constitute a response surface 12 . One of the disadvantages of this approach is that the results of high and low-fidelity analyses have to be correlated periodically during the course of optimization. For a relatively large number of design variables and responses the correlation may become rather involved, particularly, if each response employs its own correction factor, bringing up the limitation in the number of design variables and responses used. The current paper proposes a modified approach to multi-fidelit y optimization, where the one-dimensional search points in gradient-base d optimization are evaluated using high-fidelity analysis and the finite difference gradient calculations are performed using low-fidelity analysis. One of the advantages of the proposed approach is that with the proper selection of high and low-fidelity analysis models there is no need to correlate the results of the two during optimization. Another advantage is that such an approach removes the potential limitation on the number of design variables and responses employed in response surface based multi-fidelity optimization.