MULTILEVEL MONTE CARLO ESTIMATORS FOR DERIVATIVE-FREE OPTIMIZATION UNDER UNCERTAINTY
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Published:2024
Issue:3
Volume:14
Page:21-65
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ISSN:2152-5080
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Container-title:International Journal for Uncertainty Quantification
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language:en
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Short-container-title:Int. J. UncertaintyQuantification
Author:
Menhorn Friedrich,Geraci Gianluca,Seidl D. Thomas,Marzouk Youssef M.,Eldred Michael S.,Bungartz Hans-Joachim
Abstract
Optimization is a key tool for scientific and engineering applications; however, in the presence of models affected
by uncertainty, the optimization formulation needs to be extended to consider statistics of the quantity of interest.
Optimization under uncertainty (OUU) deals with this endeavor and requires uncertainty quantification analyses
at several design locations; i.e., its overall computational cost is proportional to the cost of performing a forward
uncertainty analysis at each design location. An OUU workflow has two main components: an inner loop strategy for
the computation of statistics of the quantity of interest, and an outer loop optimization strategy tasked with finding
the optimal design, given a merit function based on the inner loop statistics. In this work, we propose to alleviate the cost of the inner loop uncertainty analysis by leveraging the so-called multilevel Monte Carlo (MLMC) method, which is able to allocate resources over multiple models with varying accuracy and cost. The resource allocation problem in MLMC is formulated by minimizing the computational cost given a target variance for the estimator. We consider
MLMC estimators for statistics usually employed in OUU workflows and solve the corresponding allocation problem.
For the outer loop, we consider a derivative-free optimization strategy implemented in the SNOWPAC library; our novel strategy is implemented and released in the Dakota software toolkit. We discuss several numerical test cases to
showcase the features and performance of our approach with respect to its Monte Carlo single fidelity counterpart.
Subject
Control and Optimization,Discrete Mathematics and Combinatorics,Modeling and Simulation,Statistics and Probability
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