TY - JOUR

T1 - MULTILEVEL MONTE CARLO ESTIMATORS FOR DERIVATIVE-FREE OPTIMIZATION UNDER UNCERTAINTY

AU - Menhorn, Friedrich

AU - Geraci, Gianluca

AU - Seidl, D. Thomas

AU - Marzouk, Youssef M.

AU - Eldred, Michael S.

AU - Bungartz, Hans Joachim

N1 - Publisher Copyright:
© 2024 by Begell House, Inc. www.begellhouse.com.

PY - 2024

Y1 - 2024

N2 - 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.

AB - 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.

KW - multilevel Monte Carlo

KW - optimization under uncertainty

KW - uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=85184190452&partnerID=8YFLogxK

U2 - 10.1615/Int.J.UncertaintyQuantification.2023048049

DO - 10.1615/Int.J.UncertaintyQuantification.2023048049

M3 - Article

AN - SCOPUS:85184190452

SN - 2152-5080

VL - 14

SP - 21

EP - 65

JO - International Journal for Uncertainty Quantification

JF - International Journal for Uncertainty Quantification

IS - 3

ER -