Anirban Chaudhuri

Efficient Global Optimization with Adaptive Target Setting

S URROGATE-BASED optimization is becoming increasingly popular due to savings in computational time [1–12]. Surrogatebased optimization proceeds through cycles, selecting new sampling points that contribute toward global optimization in each cycle. Algorithms like the popular efficient global optimization (EGO) of Jones et al. [11,12] use both the surrogate prediction and its error estimates. The most common EGO variant uses prediction and prediction variance to seek the point of maximum expected improvement (EI) [12]. Jones [12] also discusses a version of EGO that uses the probability of improvement (PI) beyond a given target as the selection criterion. The use of PI was first introduced by Kushner in 1964 [13] for a one-dimensional algorithm. It was extended heuristically to higher dimensions by Stuckman [14], Elder [15], and Mockus [16]. Maximizing PI can balance local and global searches, but its performance can be sensitive to the target value [12]. If the target is too ambitious, the search is excessively global and slow to focus on promising areas. If the target is too modest, there is exhaustive search around the present best solution (PBS) before moving to global search. This issuemay account for the lack of popularity of EGOwith PI. Jones [12] proposed to address the issue of target setting by considering several target values to add multiple points per cycle, calling the method “a highly promising approach”. Queipo et al. [17] proposed away to estimate the optimum that could be used as a target. In this work, we propose an adaptive target method that adapts the target for each EGO cycle according to the success of meeting the target in the previous cycle. We dub this variant EGO-AT (for “adaptive target”). EGO-AT learns from the history of progress to predict what to expect in the next cycle. Traditionally, EGO-like algorithms add one point per cycle. When simulations take long to complete, it is attractive to run in parallel multiple simulations per cycle. Consequently, there has beenwork on including multiple points [18–21]. Selecting multiple points to maximize EI is computationally expensive [20]. However, there are methods such as kriging believer [20] and constant liar [20] to moderate the expense of multipoint EI. Using PI for finding multiple points using a single surrogate has been shown to be quite cheap [12,22]. The joint probability of all the points being added can also be easily calculated for a single target [22]. The major objective of this work is to bring PI on equal footing with EI and remove issues with using EGO-PI. In addition, EGO-AT provides two ingredients that may be useful for decisions on stopping: amount of improvement to target and the probability of targeted improvement [23,24].

NASA Uncertainty Quantification Challenge: An Optimization-Based Methodology and Validation

The focus of this work is on uncertainty characterization, sensitivity analysis, uncertainty propagation, and extreme-case analysis. To deal with the computationally expensive and complex NASA problem, a simpler toy problem is devised to mimic the NASA problem for which the true results were known. The toy problem helped in thoroughly testing the current methods and their repeatability. For uncertainty characterization, a novel cumulative density function matching method is proposed, which gave similar results as a standard Markov chain–Monte Carlo-based Bayesian approach. An efficient reliability reanalysis-based probability-box sensitivity analysis method is employed to identify the most sensitive parameters to the risk analysis metrics. Uncertainty propagation to find extreme values for the risk analysis metrics is done using a single-loop efficient reliability reanalysis-based method. A modified version of the efficient reliability reanalysis is proposed that uses self-normalizing weights and caps on ...

Efficient Global Optimization with Adaptive Target for Probability of Targeted Improvement

The use of Surrogate-based optimization has become increasingly prevalent in the design of engineering systems. Each optimization cycle consists of fitting a surrogate through an initial number of designs and performing optimization based on the surrogate to get a new design, where an exact simulation is performed. Algorithms like Efficient Global Optimization use uncertainty estimates available with the Kriging surrogate to guide the selection of new point(s). With access to parallel computation, adding multiple points per optimization cycle has become increasingly attractive. With EGO, Expected Improvement is commonly used as the criterion for selection of new point, but it is difficult and computationally expensive to add multiple points using this criterion. It is much easier to select multiple points when Probability of targeted Improvement is used as the selection criterion, but it suffers from the issue of target setting due to lack of knowledge of the true function. An adaptive target setting method is proposed in this paper, which changes the target to reflect the improvement achieved in each optimization cycle. This method is demonstrated to be highly efficient for three analytic examples. For these examples the proposed method is compared to a constant target setting and is shown to give better results. It is also seen that when multiple points are added per cycle using this method, the convergence rate becomes faster and also, the confidence in the final result becomes much higher.

Multifidelity Uncertainty Propagation in Coupled Multidisciplinary Systems

Fixed point iteration is a common strategy to handle interdisciplinary coupling within a coupled multidisciplinary analysis. For each coupled analysis, this requires a large number of disciplinary high-fidelity simulations to resolve the interactions between different disciplines. When embedded within an uncertainty analysis loop (e.g., with Monte Carlo sampling over uncertain parameters) the number of high-fidelity disciplinary simulations quickly becomes prohibitive, since each sample requires a fixed point iteration and the uncertainty analysis typically involves thousands or even millions of samples. This paper develops a method for uncertainty analysis in feedback-coupled black-box systems that leverages adaptive surrogates to reduce the number of cases for which fixed point iteration is needed. The multifidelity coupled uncertainty propagation method is an iterative process that uses surrogates for approximating the coupling variables and adaptive sampling strategies to refine the surrogates. The adaptive sampling strategies explored in this work are residual error, information gain, and weighted information gain. The surrogate models are adapted in a way that does not compromise accuracy of the uncertainty analysis relative to the original coupled high-fidelity problem.

Multi-Objective Experimental Optimization with Multiple Simultaneous Sampling for Flapping Wings

Flapping wing micro air vehicles (MAV) have recently attracted ample interest due to their capability of hovering and forward flight with high maneuverability. In our previous work, we dealt with the maximization of thrust through experimental optimization with adaptive sampling. During that study, many wing failures were attributed to overloading or high power consumption. This along with the general restriction of a limited power source on an MAV led us to look into a multi-objective set-up with power as an objective. We initially looked at a one-shot optimization using surrogates to predict Pareto fronts in order to understand the trade-offs between thrust and power. The main objective of this work is to use the knowledge from our previous work and implement a Multi-objective experimental optimization framework using adaptive sampling, based on surrogates fitted to actual experimental data. For this purpose we use a multiobjective implementation of the surrogate-based Efficient Global Optimization (EGO) algorithm (MO-EGO) with multiple surrogates and multiple sampling criteria.

Framework for Quantification and Risk Analysis for Layered Uncertainty using Optimization: NASA UQ Challenge

The focus of this work is on uncertainty characterization, sensitivity analysis, uncertainty propagation, extreme-case analysis, and robust design. This paper is being submitted to the special session, NASA Langley Multidisciplinary Uncertainty Quantification Challenge. For uncertainty characterization, an MCMC based Bayesian approach and a CDF Matching method are compared and found to give similar results; thus increasing our confidence in the methods. A surrogate based p-box sensitivity analysis method is employed to identify the most sensitive parameters to the risk analysis metrics. Uncertainty propagation to find extreme values for the risk analysis metrics are done using a single loop importance sampling (Efficient Reliability Reanalysis) and a double loop surrogates based method. The use of surrogates and importance sampling was dictated by the cost of the black-box functions provided by NASA. A simpler toy problem is also devised to mimic the NASA problem, which helped us in thoroughly testing our methods and their repeatability.

Thrust-Power Pareto Fronts based on Experiments for Small Flapping Wings

Flapping wing micro air vehicles (MAV) have recently attracted ample interest due to their capability of hovering and forward flight with high maneuverability. In our previous work, we dealt with the maximization of thrust through experimental optimization. During that study, many wing failures were attributed to overloading or high power consumption. This along with the general restriction of a limited power source on an MAV led us to look into a multi-objective set-up with power as an objective. The main aim of this paper is to understand the trade-offs between thrust and current or power consumed or efficiency of the wing, which can lead to the choice of appropriate objective functions. This is achieved by the analyzing predictions of Pareto fronts generated by multiple surrogates. In this work, surrogates are fitted to actual experimental data. The surrogate-based Pareto fronts appear to indicate that not much improvement may be achieved in the current design space, so that new design concepts need to be explored. It also indicates that maximizing thrust and efficiency are the best choice of objectives.

How to Decide Whether to Run One More Cycle in Efficient Global Optimization

The use of Surrogate-based optimization has become increasingly prevalent in the design of engineering systems. When using these optimization algorithms a major problem has been the choice of an adequate stopping criterion. The traditional goal of stopping criteria has been convergence to the optimum. But this is not very practical when each cycle (iteration) is very expensive and convergence is slow. In this paper we propose a stopping criterion to be used when continuing with one more cycle is justified only if it yields at least a specified improvement in the objective function. We develop this criterion for a variant of the Efficient Global Optimization (EGO) that maximizes the probability of improving the objective beyond a target, and where this target is adaptively set. The EGO with adaptive target (EGO-AT) is particularly suited for such a criterion, because it automatically estimates two important ingredients for the decision: What is a reasonable target for improvement in the next cycle, and what is the probability of achieving that target. The effectiveness of this stopping criterion is demonstrated for three analytic examples. For these examples it is shown that it is possible to combine the two ingredients in such a way that it leads to the correct decision about 70% of the time.

Safety of Spur Gear Design Under Non-Ideal Conditions With Uncertainty

The current practice of gear design is based on the Lewis bending and Hertzian contact models. The former provides the maximum stress on the gear base, while the latter calculates the contact pressure at the contact point between the gear and pinion. Both calculations are obtained at the reference configuration with ideal conditions; i.e., no tolerances and clearances. The first purpose of this paper is to compare these two analytical models with the numerical results, in particular, using finite element analysis. It turns out that the estimations from the two analytical equations are closely matched with those of the numerical analysis. The numerical analysis also yields the variation of contact pressures and bending stresses according to the change in the relative position between gear and pinion. It has been shown that both the maximum bending stress and contact pressure occur at non-reference configurations, which should be considered in the calculation of a safety factor. In reality, the pinion-gear assembly is under the tolerance of each part and clearance between the parts. The second purpose of this report is to estimate the effect of these uncertain parameters on the maximum bending stress and contact pressure. For the case of the selected gear-pinion assembly, it turns out that due to a 0.57% increase of clearance, the maximum bending stress is increased by 4.4%. Due to a 0.57% increase of clearance, the maximum contact pressure is increased by 17.9%.Copyright

Stiffness Investigation of Synthetic Flapping Wings for Hovering Flight

Tests on a single active degree of freedom flapping platform are used to investigate the relationship between span-wise/chord-wise stiffness and hovering performance. The intended application is to establish constraints in a multi-objective optimization (thrust-power) that avoid selection of wings that perform poorly. It can also have utility as an alternative engine for identifying favorable performance. The procedure used to make the stiffness measurements is detailed along with the post-processing approach. Twelve wing designs, adapted from a previous study, were tested in both directions to extract a figure of merit that combines both stiffness values into a non-dimensional parameter (SCratio). The wings were also tested for thrust performance and current consumption across three different flapping frequencies (20, 25, and 30 Hz). A comparison is provided that identifies the added benefit of considering power consumption when selecting a wing for favorable performance. The data for 20 and 25 Hz flapping frequencies suggest a decrease in efficiency with increased SCratio, while the 30 Hz flapping frequency data was unimodal. This suggests the presence of a point or region on the spectrum of SCratio that provides optimum efficiency.