Algorithms and analyses for stochastic optimization for turbofan noise reduction using parallel reduced-order modeling

Yang, Huanhuan; Gunzburger, Max

doi:10.1016/j.cma.2017.02.030

Cited by 22 publications

(10 citation statements)

References 26 publications

(57 reference statements)

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“…This can be handled by adding safety factors to the threshold; however, it has been shown before that probabilistic approaches lead to safer designs with optimized performance compared to the safety factor approach [7,8,9]. Superquantile/CVaR has been recently used in specific formulations in civil [10,11], naval [12,13] and aerospace [14,15] engineering, as well as general PDE-constrained optimization [16,17,18,19]. The bPoF risk measure has been shown to possess beneficial properties when used in optimization [6,20,21,22], yet has been seldom used in engineering to-date [23,24,25,26].…”

Section: Introductionmentioning

confidence: 99%

Certifiable Risk-Based Engineering Design Optimization

Chaudhuri¹,

Kramer²,

Norton³

et al. 2021

Preprint

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Reliable, risk-averse design of complex engineering systems with optimized performance requires dealing with uncertainties. A conventional approach is to add safety margins to a design that was obtained from deterministic optimization. Safer engineering designs require appropriate cost and constraint function definitions that capture the risk associated with unwanted system behavior in the presence of uncertainties. The paper proposes two notions of certifiability. The first is based on accounting for the magnitude of failure to ensure data-informed conservativeness. The second is the ability to provide optimization convergence guarantees by preserving convexity. Satisfying these notions leads to certifiable risk-based design optimization (CRiBDO). In the context of CRiBDO, risk measures based on superquantile (a.k.a. conditional value-at-risk) and buffered probability of failure are analyzed. CRiBDO is contrasted with reliability-based design optimization (RBDO), where uncertainties are accounted for via the probability of failure, through a structural and a thermal design problem. A reformulation of the short column structural design problem leading to a convex CRiBDO problem is presented. The CRiBDO formulations capture more information about the problem to assign the appropriate conservativeness, exhibit superior optimization convergence by preserving properties of underlying functions, and alleviate the adverse effects of choosing hard failure thresholds required in RBDO.

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Section: Introductionmentioning

confidence: 99%

Certifiable Risk-Based Engineering Design Optimization

Chaudhuri¹,

Kramer²,

Norton³

et al. 2021

Preprint

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“…It is critical to incorporate this uncertainty in the optimization problem to make the optimal solution more reliable and robust. Optimization under uncertainty has become an important research area and received increasing attentions in recent years [74,10,45,41,70,76,49,26,78,19,52,63,20,51,48,8,3,5,84,85,42,69,53,50,55,33,59,47,79,80,81,27,18,86,82,35,54,61,6,38,39,31,37,36]. To account for the uncertainty in the optimization problem, different statistical measures of the objective function have been studied, e.g., mean, variance, conditional value-at-risk, worst case scenario, etc., [70,48,85,3,…”

Section: Introductionmentioning

confidence: 99%

“…This challenge prevents direct application of most of the conventional numerical methods for computing statistics of the objective function, since they require a large number of evaluations of the objective function and thus the PDE solution. To tackle this challenge, multigrid, multilevel, and model reduction methods have been successfully applied to solve stochastic PDE-constrained optimization problems [9,63,19,20,5,53,85,86,6]. The multigrid discretization and multilevel statistical evaluation rely on a hierarchical discretization of the PDE model and an efficient algorithm to balance the discretization error and the number of samples required for statistical evaluation at each level.…”

Section: Introductionmentioning

confidence: 99%

Taylor approximation for chance constrained optimization problems governed by partial differential equations with high-dimensional random parameters

Chen¹,

Ghattas²

2020

Preprint

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We propose a fast and scalable optimization method to solve chance or probabilistic constrained optimization problems governed by partial differential equations (PDEs) with high-dimensional random parameters. To address the critical computational challenges of expensive PDE solution and high-dimensional uncertainty, we construct surrogates of the constraint function by Taylor approximation, which relies on efficient computation of the derivatives, low rank approximation of the Hessian, and a randomized algorithm for eigenvalue decomposition. To tackle the difficulty of the non-differentiability of the inequality chance constraint, we use a smooth approximation of the discontinuous indicator function involved in the chance constraint, and apply a penalty method to transform the inequality constrained optimization problem to an unconstrained one. Moreover, we design a gradientbased optimization scheme that gradually increases smoothing and penalty parameters to achieve convergence, for which we present an efficient computation of the gradient of the approximate cost functional by the Taylor approximation. Based on numerical experiments for a problem in optimal groundwater management, we demonstrate the accuracy of the Taylor approximation, its ability to greatly accelerate constraint evaluations, the convergence of the continuation optimization scheme, and the scalability of the proposed method in terms of the number of PDE solves with increasing random parameter dimension from one thousand to hundreds of thousands.

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“…There are a number of other works that have combined reduced-order, or more generally, surrogate models and/or sparse grids, to accelerate optimization under uncertainty [22,35,28,39] and uncertainty quantification [32,33] and others that have used reduced-order models to estimate risk measures, including the conditional value-at-risk [16,38]. Ref.…”

mentioning

confidence: 99%

An Efficient, Globally Convergent Method for Optimization Under Uncertainty Using Adaptive Model Reduction and Sparse Grids

Zahr¹,

Carlberg²,

Kouri³

2019

SIAM/ASA J. Uncertainty Quantification

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This work introduces a new method to efficiently solve optimization problems constrained by partial differential equations (PDEs) with uncertain coefficients. The method leverages two sources of inexactness that trade accuracy for speed: (1) stochastic collocation based on dimension-adaptive sparse grids (SGs), which approximates the stochastic objective function with a limited number of quadrature nodes, and (2) projection-based reduced-order models (ROMs), which generate efficient approximations to PDE solutions. These two sources of inexactness lead to inexact objective function and gradient evaluations, which are managed by a trust-region method that guarantees global convergence by adaptively refining the sparse grid and reduced-order model until a proposed error indicator drops below a tolerance specified by trust-region convergence theory. A key feature of the proposed method is that the error indicator-which accounts for errors incurred by both the sparse grid and reduced-order model-must be only an asymptotic error bound, i.e., a bound that holds up to an arbitrary constant that need not be computed. This enables the method to be applicable to a wide range of problems, including those where sharp, computable error bounds are not available; this distinguishes the proposed method from previous works. Numerical experiments performed on a model problem from optimal flow control under uncertainty verify global convergence of the method and demonstrate the method's ability to outperform previously proposed alternatives.

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Algorithms and analyses for stochastic optimization for turbofan noise reduction using parallel reduced-order modeling

Cited by 22 publications

References 26 publications

Certifiable Risk-Based Engineering Design Optimization

Certifiable Risk-Based Engineering Design Optimization

Taylor approximation for chance constrained optimization problems governed by partial differential equations with high-dimensional random parameters

An Efficient, Globally Convergent Method for Optimization Under Uncertainty Using Adaptive Model Reduction and Sparse Grids

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