Uncertainty quantification methods for evolutionary optimization under uncertainty

In this paper, we discuss the role of uncertainty quantification (UQ) in assisting optimization under uncertainty. UQ plays a significant role in quantifying the robustness of solutions so as to help the optimizer in achieving robust optimum solutions. In this respect, the scientific discipline of UQ addresses various theoretical and practical aspects of uncertainty, which include representations of uncertainty and also efficient computation of the output uncertainty, to name a few. However, the UQ community and the evolutionary computation community rarely interact with each other despite the potential of utilizing the advancement in UQ for research in evolutionary computation. To that end, this paper serves as a short introduction to the science of UQ for the evolutionary computation community. We discuss several aspects of UQ for robust optimization such as aleatory and epistemic uncertainty and objective functions when uncertainties are considered. A tutorial on an aerodynamic design problem is also given to illustrate the use of UQ in a real-world problem.

[1]  Dan-Xia Xu,et al.  Genetic Algorithm and Polynomial Chaos Modelling for Performance Optimization of Photonic Circuits Under Manufacturing Variability , 2018, 2018 Optical Fiber Communications Conference and Exposition (OFC).

[2]  Bret Stanford,et al.  Uncertainty Quantification in Aeroelasticity , 2017, Uncertainty Quantification in Computational Fluid Dynamics.

[3]  Pramudita Satria Palar,et al.  Gaussian Process Surrogate Model with Composite Kernel Learning for Engineering Design , 2020 .

[4]  Bernhard Sendhoff,et al.  Trade-Off between Performance and Robustness: An Evolutionary Multiobjective Approach , 2003, EMO.

[5]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[6]  Ming Zhao,et al.  Uncertainty quantification for chaotic computational fluid dynamics , 2006, J. Comput. Phys..

[7]  Ilias Bilionis,et al.  Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification , 2018, J. Comput. Phys..

[8]  Sergey Oladyshkin,et al.  Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion , 2012, Reliab. Eng. Syst. Saf..

[9]  Omar M. Knio,et al.  Sparse Pseudo Spectral Projection Methods with Directional Adaptation for Uncertainty Quantification , 2016, J. Sci. Comput..

[10]  Joe Wiart,et al.  A new surrogate modeling technique combining Kriging and polynomial chaos expansions - Application to uncertainty analysis in computational dosimetry , 2015, J. Comput. Phys..

[11]  G. Karniadakis,et al.  Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures , 2006, SIAM J. Sci. Comput..

[12]  Thierry A. Mara,et al.  Bayesian sparse polynomial chaos expansion for global sensitivity analysis , 2017 .

[13]  Serhat Hosder,et al.  Quantification of margins and mixed uncertainties using evidence theory and stochastic expansions , 2014, Reliab. Eng. Syst. Saf..

[14]  Pénélope Leyland,et al.  A Continuation-Multilevel Monte Carlo Evolutionary Algorithm for Robust Aerodynamic Shape Design , 2018 .

[15]  Nicholas Zabaras,et al.  Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification , 2018, J. Comput. Phys..

[16]  Mohammad Javad Azizi,et al.  A robust simulation optimization algorithm using kriging and particle swarm optimization: Application to surgery room optimization , 2021, Commun. Stat. Simul. Comput..

[17]  Kalyanmoy Deb,et al.  Introducing Robustness in Multi-Objective Optimization , 2006, Evolutionary Computation.

[18]  Pietro Marco Congedo,et al.  Kriging-sparse Polynomial Dimensional Decomposition surrogate model with adaptive refinement , 2019, J. Comput. Phys..

[19]  Jeroen A. S. Witteveen,et al.  Simplex stochastic collocation with ENO-type stencil selection for robust uncertainty quantification , 2013, J. Comput. Phys..

[20]  Geoffrey T. Parks,et al.  Robust Aerodynamic Design Optimization Using Polynomial Chaos , 2009 .

[21]  Paola Cinnella,et al.  Robust optimization of dense gas flows under uncertain operating conditions , 2010 .

[22]  Claudia Schillings,et al.  Efficient shape optimization for certain and uncertain aerodynamic design , 2011 .

[23]  Bruno Sudret,et al.  Adaptive sparse polynomial chaos expansion based on least angle regression , 2011, J. Comput. Phys..

[24]  Clint Dawson,et al.  Data-driven uncertainty quantification for predictive flow and transport modeling using support vector machines , 2018, Computational Geosciences.

[25]  Michael S. Eldred,et al.  Epistemic Uncertainty in the Calculation of Margins , 2009 .

[26]  KersaudyPierric,et al.  A new surrogate modeling technique combining Kriging and polynomial chaos expansions - Application to uncertainty analysis in computational dosimetry , 2015 .

[27]  Shigeyoshi Tsutsui,et al.  Genetic algorithms with a robust solution searching scheme , 1997, IEEE Trans. Evol. Comput..

[28]  Nicholas Geneva,et al.  Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks , 2018, J. Comput. Phys..

[29]  Prasanth B. Nair,et al.  Some greedy algorithms for sparse polynomial chaos expansions , 2019, J. Comput. Phys..

[30]  Michael S. Eldred,et al.  Sparse Pseudospectral Approximation Method , 2011, 1109.2936.

[31]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[32]  Pramudita Satria Palar,et al.  Multiple Metamodels for Robustness Estimation in Multi-objective Robust Optimization , 2017, EMO.

[33]  Liu Yang,et al.  Neural-net-induced Gaussian process regression for function approximation and PDE solution , 2018, J. Comput. Phys..

[34]  Stefano Marelli,et al.  Data-driven polynomial chaos expansion for machine learning regression , 2018, J. Comput. Phys..

[35]  Gianluca Iaccarino,et al.  Padé-Legendre approximants for uncertainty analysis with discontinuous response surfaces , 2009, J. Comput. Phys..

[36]  Kaichao Zhang,et al.  Multivariate output global sensitivity analysis using multi-output support vector regression , 2019, Structural and Multidisciplinary Optimization.

[37]  Qing Li,et al.  Robust optimization of foam-filled thin-walled structure based on sequential Kriging metamodel , 2014 .

[38]  Dong-Heon Kang,et al.  A robust optimization using the statistics based on kriging metamodel , 2006 .

[39]  Houman Owhadi,et al.  Handbook of Uncertainty Quantification , 2017 .

[40]  Kai-Yew Lum,et al.  Max-min surrogate-assisted evolutionary algorithm for robust design , 2006, IEEE Transactions on Evolutionary Computation.

[41]  Ilias Bilionis,et al.  Multi-output local Gaussian process regression: Applications to uncertainty quantification , 2012, J. Comput. Phys..

[42]  Jürgen Branke,et al.  Evolutionary optimization in uncertain environments-a survey , 2005, IEEE Transactions on Evolutionary Computation.

[43]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[44]  Gao Zhenghong,et al.  Review of Robust Aerodynamic Design Optimization for Air Vehicles , 2019 .

[45]  Gareth A. Vio,et al.  Multi-particle swarm optimization used to study material degradation in aeroelastic composites including probabalistic uncertainties , 2017, 2017 IEEE Congress on Evolutionary Computation (CEC).

[46]  Tapabrata Ray,et al.  Six-Sigma Robust Design Optimization Using a Many-Objective Decomposition-Based Evolutionary Algorithm , 2015, IEEE Transactions on Evolutionary Computation.

[47]  Slawomir Koziel,et al.  Multi-Fidelity Robust Aerodynamic Design Optimization under Mixed Uncertainty , 2015 .

[48]  Pierre Sagaut,et al.  A hybrid anchored-ANOVA - POD/Kriging method for uncertainty quantification in unsteady high-fidelity CFD simulations , 2016, J. Comput. Phys..