Robust Optimization With Parameter and Model Uncertainties Using Gaussian Processes With Limited Samples

2017 ◽  
Author(s):  
Yanjun Zhang ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Engineering Design ◽  
Gaussian Processes ◽  
Parameter Uncertainty ◽  
Model Uncertainty ◽  
Computational Cost ◽  
Random Errors ◽  
Model Uncertainties ◽  
Uncertainty Model ◽  
Gp Model

Uncertainty is inevitable in engineering design. The existence of uncertainty may change the optimality and/or the feasibility of the obtained optimal solutions. In simulation-based engineering design, uncertainty could have various types of sources, such as parameter uncertainty, model uncertainty, and other random errors. To deal with uncertainty, robust optimization (RO) algorithms are developed to find solutions which are not only optimal but also robust with respect to uncertainty. Parameter uncertainty has been taken care of by various RO approaches. While model uncertainty has been ignored in majority of existing RO algorithms with the hypothesis that the simulation model used could represent the real physical system perfectly. In the authors’ earlier work, a RO framework was proposed to consider both parameter and model uncertainties using the Bayesian approach with Gaussian processes (GP), where metamodeling uncertainty introduced by GP modeling is ignored by assuming the constructed GP model is accurate enough with sufficient training samples. However, infinite samples are impossible for real applications due to prohibitive time and/or computational cost. In this work, a new RO framework is proposed to deal with both parameter and model uncertainties using GP models but only with limited samples. The compound effect of parameter, model, and metamodeling uncertainties is derived with the form of the compound mean and variance to formulate the proposed RO approach. The proposed RO approach will reduce the risk for the obtained robust optimal designs considering parameter and model uncertainties becoming non-optimal and/or infeasible due to insufficiency of samples for GP modeling. Two test examples with different degrees of complexity are utilized to demonstrate the applicability and effectiveness of the proposed approach.

Robust Optimization With Mixed Interval and Probabilistic Parameter Uncertainties, Model Uncertainty, and Metamodeling Uncertainty

2019 ◽  
Author(s):  
Yanjun Zhang ◽  
Tingting Xia ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Real World ◽  
Parameter Uncertainty ◽  
Model Uncertainty ◽  
Optimization Problems ◽  
Probability Distributions ◽  
Parameter Uncertainties ◽  
True Value ◽  
Uncertainty Model ◽  
The Difference

Abstract Various types of uncertainties, such as parameter uncertainty, model uncertainty, metamodeling uncertainty may lead to low robustness. Parameter uncertainty can be either epistemic or aleatory in physical systems, which have been widely represented by intervals and probability distributions respectively. Model uncertainty is formally defined as the difference between the true value of the real-world process and the code output of the simulation model at the same value of inputs. Additionally, metamodeling uncertainty is introduced due to the usage of metamodels. To reduce the effects of uncertainties, robust optimization (RO) algorithms have been developed to obtain solutions being not only optimal but also less sensitive to uncertainties. Based on how parameter uncertainty is modeled, there are two categories of RO approaches: interval-based and probability-based. In real-world engineering problems, both interval and probabilistic parameter uncertainties are likely to exist simultaneously in a single problem. However, few works have considered mixed interval and probabilistic parameter uncertainties together with other types of uncertainties. In this work, a general RO framework is proposed to deal with mixed interval and probabilistic parameter uncertainties, model uncertainty, and metamodeling uncertainty simultaneously in design optimization problems using the intervals-of-statistics approaches. The consideration of multiple types of uncertainties will improve the robustness of optimal designs and reduce the risk of inappropriate decision-making, low robustness and low reliability in engineering design. Two test examples are utilized to demonstrate the applicability and effectiveness of the proposed RO approach.

scholarly journals Computer-aided SPT-based reliability model for probability of liquefaction using hybrid PSO and GA

2020 ◽  
Vol 7 (1) ◽  
pp. 107-127 ◽  
Author(s):  
Maral Goharzay ◽  
Ali Noorzad ◽  
Ahmadreza Mahboubi Ardakani ◽  
Mostafa Jalal
Keyword(s):  
Engineering Design ◽  
Parameter Uncertainty ◽  
Model Uncertainty ◽  
Probabilistic Method ◽  
Limit State ◽  
Soil Liquefaction ◽  
Probability Models ◽  
Reliability Indices ◽  
Hybrid Particle Swarm Optimization ◽  
Reliability Method

Abstract In this paper, an approach for soil liquefaction evaluation using probabilistic method based on the world-wide SPT databases has been presented. In this respect, the parameters’ uncertainties for liquefaction probability have been taken into account. A calibrated mapping function is developed using Bayes’ theorem in order to capture the failure probabilities in the absence of the knowledge of parameter uncertainty. The probability models provide a simple, but also efficient decision-making tool in engineering design to quantitatively assess the liquefaction triggering thresholds. Within an extended framework of the first-order reliability method considering uncertainties, the reliability indices are determined through a well-performed meta-heuristic optimization algorithm called hybrid particle swarm optimization and genetic algorithm to find the most accurate liquefaction probabilities. Finally, the effects of the level of parameter uncertainty on liquefaction probability, as well as the quantification of the limit state model uncertainty in order to incorporate the correct model uncertainty, are investigated in the context of probabilistic reliability analysis. The results gained from the presented probabilistic model and the available models in the literature show the fact that the developed approach can be a robust tool for engineering design and analysis of liquefaction as a natural disaster.

Robust Optimization With Parameter and Model Uncertainties Using Gaussian Processes

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Yanjun Zhang ◽  
Mian Li ◽  
Jun Zhang ◽  
Guoshu Li
Keyword(s):  
Robust Optimization ◽  
Gaussian Processes ◽  
Parameter Uncertainty ◽  
Simulation Models ◽  
Optimal Designs ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Design Problems ◽  
Physical Systems ◽  
Design Variables

Uncertainty is unavoidable in engineering design, which may result in variations in the objective functions and/or constraints. The former may degrade the designed performance while the latter can even change the feasibility of the obtained optimal solutions. Taking uncertainty into consideration, robust optimization (RO) algorithms aim to find optimal solutions that are also insensitive to uncertainty. Uncertainty may include variation in parameters and/or design variables, inaccuracy in simulation models used in design problems, and other possible errors. Most existing RO algorithms only consider uncertainty in parameters, but overlook that in simulation models by assuming that the simulation model used can always provide identical outputs to those of the real physical systems. In this paper, we propose a new RO framework using Gaussian processes, considering not only parameter uncertainty but also uncertainty in simulation models. The consideration of model uncertainty in RO could reduce the risk for the obtained robust optimal designs becoming infeasible even if the parameter uncertainty has been considered. Two test examples with different degrees of complexity are utilized to demonstrate the applicability and effectiveness of our proposed algorithm.

Robust Optimization With Parameter and Model Uncertainties Using Gaussian Processes

2016 ◽  
Author(s):  
Yanjun Zhang ◽  
Mian Li ◽  
Jun Zhang ◽  
Guoshu Li
Keyword(s):  
Robust Optimization ◽  
Gaussian Processes ◽  
Parameter Uncertainty ◽  
Simulation Models ◽  
Optimal Designs ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Design Problems ◽  
Physical Systems ◽  
Design Variables

Uncertainty is unavoidable in engineering design, which may result in variations in the objective functions and/or constraints. The former may degrade the designed performance while the latter can even change the feasibility of the obtained optimal solutions. Taking uncertainty into consideration, robust optimization (RO) algorithms aim to find optimal solutions that are also insensitive to uncertainty. Uncertainty may include variation in parameters and/or design variables, inaccuracy in simulation models used in design problems, and other possible errors. Most existing RO algorithms only consider uncertainty in parameters, but overlook that in simulation models by assuming that the simulation model used can always provide identical outputs to those of the real physical systems. In this paper, we propose a new RO framework using Gaussian processes, considering not only parameter uncertainty but also uncertainty in simulation models. The consideration of model uncertainty in RO will significantly reduce the risk for the obtained robust optimal designs becoming infeasible even the parameter uncertainty has been considered. Two test examples with different degrees of complexity are utilized to demonstrate the applicability and effectiveness of our proposed algorithm.

Gaussian Processes for Classification: Mean-Field Algorithms

2000 ◽  
Vol 12 (11) ◽  
pp. 2655-2684 ◽  
Author(s):  
Manfred Opper ◽  
Ole Winther
Keyword(s):  
Support Vector Machines ◽  
Gaussian Processes ◽  
Disordered Systems ◽  
Binary Classification ◽  
Computational Cost ◽  
Mean Field ◽  
Strong Support ◽  
Support Vector ◽  
Vector Machines ◽  
Leave One Out

We derive a mean-field algorithm for binary classification with gaussian processes that is based on the TAP approach originally proposed in statistical physics of disordered systems. The theory also yields an approximate leave-one-out estimator for the generalization error, which is computed with no extra computational cost. We show that from the TAP approach, it is possible to derive both a simpler “naive” mean-field theory and support vector machines (SVMs) as limiting cases. For both mean-field algorithms and support vector machines, simulation results for three small benchmark data sets are presented. They show that one may get state-of-the-art performance by using the leave-one-out estimator for model selection and the built-in leave-one-out estimators are extremely precise when compared to the exact leave-one-out estimate. The second result is taken as strong support for the internal consistency of the mean-field approach.

Comparative Analysis of Methodologies for Uncertainty Propagation and Quantification

2017 ◽  
Author(s):  
Alessandra Cuneo ◽  
Alberto Traverso ◽  
Shahrokh Shahpar
Keyword(s):  
Engineering Design ◽  
Polynomial Chaos ◽  
Epistemic Uncertainty ◽  
Uncertainty Propagation ◽  
Computational Cost ◽  
Computational Effort ◽  
Optimization Under Uncertainty ◽  
Design Parameters ◽  
Test Case ◽  
Aleatory And Epistemic Uncertainty

In engineering design, uncertainty is inevitable and can cause a significant deviation in the performance of a system. Uncertainty in input parameters can be categorized into two groups: aleatory and epistemic uncertainty. The work presented here is focused on aleatory uncertainty, which can cause natural, unpredictable and uncontrollable variations in performance of the system under study. Such uncertainty can be quantified using statistical methods, but the main obstacle is often the computational cost, because the representative model is typically highly non-linear and complex. Therefore, it is necessary to have a robust tool that can perform the uncertainty propagation with as few evaluations as possible. In the last few years, different methodologies for uncertainty propagation and quantification have been proposed. The focus of this study is to evaluate four different methods to demonstrate strengths and weaknesses of each approach. The first method considered is Monte Carlo simulation, a sampling method that can give high accuracy but needs a relatively large computational effort. The second method is Polynomial Chaos, an approximated method where the probabilistic parameters of the response function are modelled with orthogonal polynomials. The third method considered is Mid-range Approximation Method. This approach is based on the assembly of multiple meta-models into one model to perform optimization under uncertainty. The fourth method is the application of the first two methods not directly to the model but to a response surface representing the model of the simulation, to decrease computational cost. All these methods have been applied to a set of analytical test functions and engineering test cases. Relevant aspects of the engineering design and analysis such as high number of stochastic variables and optimised design problem with and without stochastic design parameters were assessed. Polynomial Chaos emerges as the most promising methodology, and was then applied to a turbomachinery test case based on a thermal analysis of a high-pressure turbine disk.

Robust Shape Optimization of Uncertain Dense Gas Flows Through a Plane Turbine Cascade

2011 ◽  
Author(s):  
Samuel J. Hercus ◽  
Paola Cinnella
Keyword(s):  
Robust Optimization ◽  
Shape Optimization ◽  
Flow Behavior ◽  
Computational Cost ◽  
Optimization Procedure ◽  
System Stability ◽  
Dense Gas ◽  
Turbine Cascade ◽  
The Mean ◽  
Inlet Conditions

A robust shape optimization procedure based on a multi-objective genetic algorithm coupled to a non-intrusive uncertainty quantification analysis was applied to a transonic inviscid flow of a dense gas over a plane turbine cascade. The goal was to simultaneously improve the mean turbine performance and the system stability under fluctuating thermodynamic inlet conditions. Despite an elevated computational cost, the optimization procedure was capable of generating a Pareto front of turbine geometries which improved the mean isentropic turbine efficiency μ(ηs) over the baseline profile, while limiting the solution variability in terms of the coefficient of variation of the power output CV(P2D). In addition to demonstrating an excellent parallel scalability over 1600 processors, the robust optimization revealed that variability of CV(P2D) depends more on the variation of inlet conditions than turbine geometry. A posteriori stochastic analyses on selected optimized turbine geometries allowed an investigation of flow behavior variability, as well as propositions for the improved selection of robust optimization cost criteria in future simulations.

scholarly journals Unscented Kalman filter for airship model uncertainties and wind disturbance estimation

PLoS ONE ◽  
2021 ◽  
Vol 16 (11) ◽  
pp. e0257849
Author(s):  
Muhammad Wasim ◽  
Ahsan Ali ◽  
Mohammad Ahmad Choudhry ◽  
Faisal Saleem ◽  
Inam Ul Hasan Shaikh ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Precision Agriculture ◽  
Model Uncertainty ◽  
Unscented Kalman Filter ◽  
Estimation Method ◽  
Wind Disturbance ◽  
Model Uncertainties ◽  
Nonlinear Simulation ◽  
Lumped Model ◽  
Disturbance Estimation

An airship is lighter than an air vehicle with enormous potential in applications such as communication, aerial inspection, border surveillance, and precision agriculture. An airship model is made up of dynamic, aerodynamic, aerostatic, and propulsive forces. However, the computation of aerodynamic forces remained a challenge. In addition to aerodynamic model deficiencies, airship mass matrix suffers from parameter variations. Moreover, due to the lighter-than-air nature, it is also susceptible to wind disturbances. These modeling issues are the key challenges in developing an efficient autonomous flight controller for an airship. This article proposes a unified estimation method for airship states, model uncertainties, and wind disturbance estimation using Unscented Kalman Filter (UKF). The proposed method is based on a lumped model uncertainty vector that unifies model uncertainties and wind disturbances in a single vector. The airship model is extended by incorporating six auxiliary state variables into the lumped model uncertainty vector. The performance of the proposed methodology is evaluated using a nonlinear simulation model of a custom-developed UETT airship and is validated by conducting a kind of error analysis. For comparative studies, EKF estimator is also developed. The results show the performance superiority of the proposed estimator over EKF; however, the proposed estimator is a bit expensive on computational grounds. However, as per the requirements of the current application, the proposed estimator can be a preferred choice.

Reliable Calculation of Thermoacoustic Instability Risk Using an Imperfect Surrogate Model

2020 ◽  
Author(s):  
Shuai Guo ◽  
Camilo F. Silva ◽  
Wolfgang Polifke
Keyword(s):  
Surrogate Model ◽  
Learning Strategy ◽  
Epistemic Uncertainty ◽  
Computational Cost ◽  
Risk Calculation ◽  
Thermoacoustic Instability ◽  
Training Samples ◽  
Computational Budget ◽  
Model Training ◽  
Gp Model

Abstract One of the fundamental tasks in performing robust thermoacoustic design of gas turbine combustors is calculating the modal instability risk, i.e., the probability that a thermoacoustic mode is unstable, given various sources of uncertainty (e.g., operation or boundary conditions). To alleviate the high computational cost associated with conventional Monte Carlo simulation, surrogate modeling techniques are usually employed. Unfortunately, in practice it is not uncommon that only a small number of training samples can be afforded for surrogate model training. As a result, epistemic uncertainty may be introduced by such an “inaccurate” model, provoking a variation of modal instability risk calculation. In the current study, using Gaussian Process (GP) as the surrogate model, we address the following two questions: Firstly, how to quantify the variation of modal instability risk induced by the epistemic surrogate model uncertainty? Secondly, how to reduce the variation of risk calculation given a limited computational budget for the surrogate model training? For the first question, we leverage on the Bayesian characteristic of the GP model and perform correlated sampling of the GP predictions at different inputs to quantify the uncertainty of risk calculation. We show how this uncertainty shrinks when more training samples are available. For the second question, we adopt an active learning strategy to intelligently allocate training samples, such that the trained GP model is highly accurate particularly in the vicinity of the zero growth rate contour. As a result, a more accurate and robust modal instability risk calculation is obtained without increasing the computational cost of surrogate model training.

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