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
Beyond Multilevel Monte Carlo Methods for Expected Values
Sebastian Krumscheid | RWTH Aachen | Germany
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Authors:
Sebastian Krumscheid | RWTH Aachen | Germany
Fabio Nobile | Ecole Polytechnique Fédérale de Lausanne (EPFL) | Switzerland
Many applications across sciences and technologies require a careful quantification of nondeterministic effects to a system output, for example when evaluating the system’s reliability or when gearing it towards more robust conditions. These considerations rely on an accurate yet efficient characterisation of uncertain system outputs. For the approximation of moments of said outputs, the multilevel Monte Carlo (MLMC) method has been established as a computationally efficient sampling method that is applicable to a wide range of applications. In this talk we will review recent advances in MLMC techniques for a characterisation of an uncertain system output’s distribution. Specifically, we will first introduce efficient methods for accurately estimating higher order moments and showcase their use in aeronautics problems. We will then discuss MLMC techniques for approximating general parametric expectations, i.e. expectations that depend on a parameter, uniformly on some interval. The resulting MLMC estimators of such functions enable to derive efficient approximations of various means to characterise a system output’s distribution, e.g. to the characteristic function or to the cumulative distribution function. A further important consequence of these results is that they allow to construct MLMC for various robustness indicators, such as for quantiles (value-at-risk) or for the conditional value-at-risk, which we will also exemplify.
14:30
Multifidelity uncertainty quantification from a parametric Bayesian viewpoint
Alex Gorodetsky | University of Michigan | United States
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Authors:
Alex Gorodetsky | University of Michigan | United States
John Jakeman | Sandia National Laboratories | United States
Gianluca Geraci | Sandia National Laboratories | United States
Michael Eldred | Sandia National Laboratories | United States
Multifidelity uncertainty quantification uses cheap-to-evaluate low-fidelity models to accelerate the convergence of the statistics of computationally expensive simulation models. The approach to synthesize low-fidelity and high-fidelity data is predominantly approached from the frequentist viewpoint. Sampling approaches are predominantly variance reduction strategies that seek to reduce the variance of some estimator (e.g., Monte Carlo), and surrogate approaches typically recursively approximate discrepancies between models with the aim of minimizing L2 error. In this talk we discuss strategies from a Bayesian perspective where prior information about model relationships are encoded via conditional independence assumptions, and Bayesian inversion is used to update this prior information to obtain posteriors. We show how this approach compares to traditional frequentist estimators and discuss how it can be made computationally feasible by exploiting the intrinsic natural structure. Examples and benefits of the proposed approach are provided for wide varying applications.
15:00
- CANCELED - An Improved Multilevel Particle Filter
Marco Ballesio | KAUST | Saudi Arabia
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Authors:
Marco Ballesio | KAUST | Saudi Arabia
Ajay Jasra | KAUST | Saudi Arabia
Erik von Schwerin | KAUST | Saudi Arabia
Raul F. Tempone | KAUST | Saudi Arabia
Hidden Markov models have a wide range of applications in engineering, finance and
weather forecasting. We consider filtering problems connected to partially observed diffusions that are regularly observed at discrete times. In many cases of practical interest, a particle filter (PF) is used to evaluate the filtering distribution. A multilevel particle filter (MLPF) uses a hierarchy of discretizations to reduce the cost of the PF. It is essential for MLPFs that the resamplings, which are performed in PFs when the effective sample size becomes too small, do not destroy the coupling between the particles on coarse-fine discretization pairs. We propose an alternative resampling method, based on optimal Wasserstein coupling, that outperforms existing MLPFs in certain cases. In many applications of the MLPFs, a change of measure technique, is used to overcome the instability of the underlying diffusion process.
15:30
Unbiased Filtering for Partially Observed Diffusions
Fangyuan Yu | KAUST | Saudi Arabia
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Authors:
Fangyuan Yu | KAUST | Saudi Arabia
Kody Law | University of Manchester | United Kingdom
Ajay Jasra | KAUST | Saudi Arabia
We consider a Monte Carlo-based method to filter partially observed diffusions observed
at regular and discrete times. We present a new procedure, which, given only access to Euler discretizations of the diffusion process, can return online estimates of the filtering distribution which possess no discretization bias and further have finite variance. Our approach is based upon a novel double application of randomization methods along with a multilevel particle filter (MLPF) approach. We provide, in addition, a numerical comparison of our new approach with the MLPF and show that similar errors are possible for a mild increase in computational cost.