Multi-fidelity non-intrusive polynomial chaos based on regression

© 2016 Elsevier B.V. In this paper we present a multi-fidelity (MF) extension of non-intrusive polynomial chaos based on regression (point collocation) for uncertainty quantification purposes. The proposed method uses the principle of a global correction function from a previous similar method that uses spectral projection to estimate the coefficients. Due to its usage of regression to estimate the coefficients, the present method offers high flexibility in the sampling and generation of the polynomial basis. The method takes advantage of a nested sampling plan to create the samples for the low-fidelity (LF) and correction expansions where the high-fidelity (HF) samples are a subset of the LF ones. To build the polynomial basis, a total order or hyperbolic truncation strategy is used with a highly flexible combination of the LF and correction polynomial expansions. The method is demonstrated on some artificial test problems and aerodynamic problems of the Euler flow around an airfoil and common three-dimensional research models. In order to derive the strategies for successful MF approximation, the effect of the correlation and the errors between the LF and HF functions is also studied. The results show that high correlation and moderately low errors are important to improve the MF approximation's accuracy. On a common research model problem, the MF approach with partially-converged simulations as the LF samples can successfully reduce the computational cost to about 40% for similar accuracy compared to an approach using a single HF expansion

Comparison of scalarization functions within a local surrogate assisted multi-objective memetic algorithm framework for expensive problems

Combining a surrogate model and a heuristic-based optimizer for multi-objective optimization is now a common approach to make best use of the available computational budget. One possible combination is to use a local surrogate that acts as a guide for local search as a module of the heuristic algorithm. The local search works by optimizing the scalarizing function and uses the local surrogate as a cheap replacement of the original function. Various scalarizing functions exist and an understanding of the advantages and disadvantages of these functions is needed for further improvement of the optimization algorithms. In this paper, various scalarizing functions implemented inside a single surrogate assisted local search memetic algorithm (SS-MOMA) framework are compared. The scalarizing functions studied here are the Tchebycheff type (SS-MOMA-TC) and weighted sum (SS-MOMA-WS) with 15-dimensional ZDT1, ZDT2, and ZDT3 test problems as the benchmark problems using the generational distance and diversity metrics as performance indicators. On the ZDT1, ZDT2, and ZDT3 problems, SS-MOMA-TC clearly outperforms SS-MOMA-WS. The results show that the Tchebycheff scalarizing function can enhance the diversity of the non-dominated solutions independent of the convexity of the problem, but it encounters a slight difficulty with the discontinuous Pareto front of ZDT3

Decomposition-based evolutionary aerodynamic robust optimization with multi-fidelity point collocation non-intrusive polynomial chaos

Evolutionary algorithms are powerful optimizers often used to explore the trade-off between performance and robustness in robust optimization. A popular methodology for un- certainty quantiffication (UQ) in evaluating robustness is through use of a polynomial chaos (PC) expansion. To make best use of the available computational budget, improvements in the performance of both optimizer and UQ method are desired. In this paper we present an approach for aerodynamic robust optimization which consists of a decomposition-based optimizer (MOEA/D) and multi-ffidelity point collocation non-intrusive PC. The inherited diversity that a decomposition-based optimizer produces is a benefficial trait for multi-objective robust optimization applications. In our UQ approach, the availability of the multi-ffidelity simulations is incorporated within the point collocation PC to allow exible numbers of samples and calculate the effiect of uncertainty efficiently. A comparison between MOEA/D and NSGA-II is performed for a subsonic application. This shows that the decomposition-based optimizer is able to find a more diverse Pareto front. The multi-ffidelity robust optimization framework is then demonstrated on a transonic airfoil robust optimization application with the goals of maximizing lift-to-drag ratio (L/D) while minimizing the sensitivity to aleatory uncertainties. The multi-ffidelity point collocation non-intrusive PC approach is able reduce the computational time needed for UQ. In the transonic case, the solution with the maximum mean of L/D is preferred over the other airfoils even though it is accompanied by a high standard deviation in L/D. The stochastic response surface shows that its minimum value of L/D over the response surface is still higher than that of the airfoil with the minimum standard deviation. This shows that it is important to examine the trend of the stochastic response surface before conclusions are drawn and decisions made

Erratum: Corrigendum to “Multi-Fidelity Non-Intrusive Polynomial Chaos Based on Regression” (Computer Methods in Applied Mechanics and Engineering (2016) 305 (579–606) (S0045782516301049) (10.1016/j.cma.2016.03.022))

This corrigendum corrects equations (18), (21) and (24) from Palar, Tsuchiya, and Parks [Comput. Methods Appl. Mech. Engrg. 305 (2016) 579–606]. These errors do not change the results, figures, discussions and main conclusions of the paper

A comparative study of local search within a surrogate-assisted multi-objective memetic algorithm framework for expensive problems

© 2016 Elsevier B.V. All rights reserved. A comparative study of the impacts of various local search methodologies for the surrogate-assisted multi-objective memetic algorithm (MOMA) is presented in this paper. The base algorithm for the comparative study is the single surrogate-assisted MOMA (SS-MOMA) with the main aim being to solve expensive problems with a limited computational budget. In addition to the standard weighted sum (WS) method used in the original SS-MOMA, we studied the capabilities of other local search methods based on the achievement scalarizing function (ASF), Chebyshev function, and random mutation hill climber (RMHC) in various test problems. Several practical aspects, such as normalization and constraint handling, were also studied and implemented to deal with real-world problems. Results from the test problems showed that, in general, the SS-MOMA with ASF and Chebyshev functions was able to find higher-quality solutions that were more robust than those found with WS or RMHC; although on problems with more complicated Pareto sets SS-MOMA-WS appeared as the best. SS-MOMA-ASF in conjunction with the Chebyshev function was then tested on an airfoil-optimization problem and compared with SS-MOMA-WS and the non-dominated sorting based genetic algorithm-II (NSGA-II). The results from the airfoil problem clearly showed that SS-MOMA with an achievement-type function could find more diverse solutions than SS-MOMA-WS and NSGA-II. This suggested that for real-world applications, higher-quality solutions are more likely to be found when the surrogate-based memetic optimizer is equipped with ASF or a Chebyshev function than with other local search methods