Article2014

Inexact Objective Function Evaluations in a Trust-Region Algorithm for PDE-Constrained Optimization under Uncertainty

Drew Kouri · Matthias Heinkenschloss B · Denis Ridzal · Bart van Bloemen Waanders

67 citations0/0 files storedSIAM Journal on Scientific Computing

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Probabilistic and Robust Engineering Design Advanced Numerical Methods in Computational Mathematics Advanced Multi-Objective Optimization Algorithms

Abstract

This paper improves the trust-region algorithm with adaptive sparse grids introduced in [SIAM J. Sci. Comput., 35 (2013), pp. A1847--A1879] for the solution of optimization problems governed by partial differential equations (PDEs) with uncertain coefficients. The previous algorithm used adaptive sparse-grid discretizations to generate models that are applied in a trust-region framework to generate a trial step. The decision whether to accept this trial step as the new iterate, however, required relatively high-fidelity adaptive discretizations of the objective function. In this paper, we extend the algorithm and convergence theory to allow the use of low-fidelity adaptive sparse-grid models in objective function evaluations. This is accomplished by extending conditions on inexact function evaluations used in previous trust-region frameworks. Our algorithm adaptively builds two separate sparse grids: one to generate optimization models for the step computation and one to approximate the objective function. These adapted sparse grids often contain significantly fewer points than the high-fidelity grids, which leads to a dramatic reduction in the computational cost. This is demonstrated numerically using two examples. Moreover, the numerical results indicate that the new algorithm rapidly identifies the stochastic variables that are relevant to obtaining an accurate optimal solution. When the number of such variables is independent of the dimension of the stochastic space, the algorithm exhibits near dimension-independent behavior.

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References

2000Interpolation of Spatial Data: Some Theory for Kriging 2005High-Order Collocation Methods for Differential Equations with Random Inputs 2007A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2010A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2013A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Cited by

2018Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization 2017Multilevel Monte Carlo Analysis for Optimal Control of Elliptic PDEs with Random Coefficients 2019An Efficient, Globally Convergent Method for Optimization Under Uncertainty Using Adaptive Model Reduction and Sparse Grids 2019Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty 2018Optimization of PDEs with Uncertain Inputs 2018Extreme value oriented random field discretization based on an hybrid polynomial chaos expansion — Kriging approach

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