In the last decades, the advancements in both computer hardware/architectures
and algorithms enabled numerical simulations at unprecedented scales. In parallel,
Uncertainty Quantification (UQ) evolved as a crucial task to enable predictive
numerical simulations. Therefore, a great effort has been devoted in advancing the UQ algorithms
in order to enable UQ for expensive numerical simulations, however the combination of an extremely
large computational cost associated to the evaluation of a high-fidelity model and the presence of a moderate/large
set of uncertainty parameters (often correlated to the complexity of the numerical/physical assumptions)
still represents a formidable challenge for UQ.
Multilevel and multifidelity strategies have been introduced to circumvent these difficulties by
reducing the computational cost required to perform UQ with high-fidelity simulations. The
main idea is to optimally combine simulations of increasingly resolution levels or model fidelities
in order to control the overall accuracy of the surrogates/estimators. This task is accomplished by
combining large number of less accurate numerical simulations with only a limited number of high-fidelity,
numerically expensive, code realizations. In this minisymposium we present contributions related to the state-of-the-art in both forward and inverse multilevel/multifidelity UQ and related areas as optimization under uncertainty.
14:00
Recent Advances on IGA-based Multi-Index Stochastic Collocation
Lorenzo Tamellini | Istituto di Matematica Applicata e Tecnologie Informatiche (CNR-IMATI) | Italy
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Authors:
Lorenzo Tamellini | Istituto di Matematica Applicata e Tecnologie Informatiche (CNR-IMATI) | Italy
Joakim Beck | KAUST | Saudi Arabia
Yang Liu | KAUST | Saudi Arabia
Raul F. Tempone | KAUST, Saudi Arabia and RWTH Aachen, Germany | Saudi Arabia
Multi-Index Stochastic Collocation (MISC) is a method of the multi-level family, aimed at
reducing computational costs when repeatedly solving a parametric PDE for Uncertainty Quantification purposes. This cost reduction is achieved by exploiting multiple hierarchies of discretizations; in particular, anisotropic grids are considered. Moreover, the random variables are sampled in a deterministic way, by using tensor grids instead of Monte Carlo samples. In this talk, we employ Isogeometric Analysis (IGA) instead of the more traditional finite elements solvers. IGA solvers employ splines as basis functions, which enables a simpler meshing process, exact geometry representation and high-continuity basis functions. Finally, IGA solvers fit particularly well in the MISC framework due to their tensor-based construction. The effectiveness of the methodology is showcased by a few numerical examples.
14:30
Recent advancements in multilevel-multifidelity surrogate-based approaches
Michael Eldred | Sandia National Laboratories | United States
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Authors:
Michael Eldred | Sandia National Laboratories | United States
John Jakeman | Sandia National Laboratories | United States
Gianluca Geraci | Sandia National Laboratories | United States
Alex Gorodetsky | University of Michigan | United States
In the simulation of complex physics, multiple model forms of varying fidelity and resolution are commonly available. While we seek results that are consistent with the highest fidelity and finest resolution, the computational cost of directly performing UQ in high random dimensions quickly becomes prohibitive. To address these challenges, we have focused on the development and deployment of algorithms that adaptively fuse information from multiple model fidelities / resolutions in order to reduce the overall computational burden. In this presentation, we focus on forward uncertainty quantification using surrogate methods, including polynomial chaos, stochastic collocation, and functional tensor train approaches. Of particular interest is the ability to perform optimal resource allocation across a diverse set of models employing either theoretical sufficiency arguments or greedy adaptive refinement, and to retain robustness with respect to non-ideal model correlations in realistic engineering deployments.
15:00
Derivative-free multilevel optimization under uncertainty employing higher order moments
Friedrich Menhorn | Technical University of Munich | Germany
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Authors:
Friedrich Menhorn | Technical University of Munich | Germany
Gianluca Geraci | Sandia National Laboratories | United States
Michael Eldred | Sandia National Laboratories | United States
Hans-Joachim Bungartz | Technical University of Munich | Germany
Youssef Marzouk | Massachusetts Institute of Technology | United States
In optimization under uncertainty we are interested in finding robust and reliable solutions under uncertain parameters. Common optimization formulations include not only the expected value of the quantity of interest but also higher moments like the variance, standard deviation or even tail probabilities. Using a standard Monte Carlo approach to evaluate the formulations quickly runs into limitations with respect to computational resources since the problems of interest are often computationally expensive and a high number of evaluations is necessary. A remedy is the usage of multifidelity estimators instead to leverage a hierarchy of models and decrease the computational burden. We developed new strategies to estimate multilevel sampling hierarchies for higher order moments tailed for the usage in optimization under uncertainty. Coupled to the stochastic derivative-free constrained optimization method SNOWPAC and DAKOTA we present results on simple model problems to describe the different algorithm components. As a final demonstration we consider wind plant design problems for which we combine fluid dynamics tools with different numerical accuracy. For each problem we compare the performance of the multifidelity strategies with their single fidelity counterpart.
15:30
Accurate MLMC estimators for engineering design under uncertainties
Quentin Ayoul-Guilmard | EPFL | Switzerland
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Authors:
Fabio Nobile | EPFL | Switzerland
Quentin Ayoul-Guilmard | EPFL | Switzerland
Sundar Ganesh | EPFL | Switzerland
The importance of meteorological conditions in civil engineering motivates risk-averse approaches to structural design. Such robust design is typically sought as a solution of a problem of optimisation under uncertainties (OUU) whose objective function features a risk measure on a random quantity of interest depending on the solution of a boundary-value problem. This OUU problem can be solved in a number of ways, amongst which are gradient-based, methods which require computations of the sensitivities of the objective function with respect to the design parameters. Therefore, an accurate and efficient estimation of the risk measure and the statistics involved is paramount.
We propose multi-level Monte Carlo estimators for parametric expectations from which can be constructed common risk measures, such as mean, variance, conditional value at risk, etc. We
present algorithms to adaptively calibrate these estimators, based on error indicators. Their use in gradient-based optimisation techniques is also investigated and discussed. High-performance computing is leveraged through an efficient task scheduler in a parallelised implementation. Finally, we apply these methods to numerical examples inspired by civil engineering, featuring fluid-flow problems and uncertain wind profiles.