Article2011

DESIGN UNDER UNCERTAINTY EMPLOYING STOCHASTIC EXPANSION METHODS

87 citations0/0 files storedInternational Journal for Uncertainty Quantification

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Probabilistic and Robust Engineering Design Advanced Multi-Objective Optimization Algorithms Gaussian Processes and Bayesian Inference

Abstract

Nonintrusive polynomial chaos expansion (PCE) and stochastic collocation (SC) methods are attractive techniques for uncertainty quantification due to their fast convergence properties and ability to produce functional representations of stochastic variability. PCE estimates coefficients for known orthogonal polynomial basis functions based on a set of response function evaluations, using sampling, linear regression, tensor-product quadrature, cubature, or Smolyak sparse grid approaches. SC, on the other hand, forms interpolation functions for known coefficients and requires the use of structured collocation point sets derived from tensor product or sparse grids. Once PCE or SC representations have been obtained for a response metric of interest, analytic expressions can be derived for the moments of the expansion and for the design derivatives of these moments, allowing for efficient design under uncertainty formulations involving moment control (e.g., robust design). This paper presents two approaches for moment design sensitivities, one involving a single response function expansion over the full range of both the design and uncertain variables and one involving response function and derivative expansions over only the uncertain variables for each instance of the design variables. These two approaches present trade-offs involving expansion dimensionality, global versus local validity, collocation point data requirements, and L2 (mean, variance, probability) versus L∞ (minima, maxima) interrogation requirements. Given this capability for analytic moments and moment sensitivities, bilevel, sequential, and multifidelity formulations for design under uncertainty are explored. Computational results are presented for a set of algebraic benchmark test problems, with attention to design formulation, stochastic expansion type, stochastic sensitivity approach, and numerical integration method.

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References

2002The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations 1978Structural reliability under combined random load sequences 2005High-Order Collocation Methods for Differential Equations with Random Inputs 2004Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design 2008A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data 2010DAKOTA : a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis. Version 5.0, user's manual.2008An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data 2004Finite elements for elliptic problems with stochastic coefficients

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