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.
16:30
A PDE-Based Hierarchical Sampling Approach for Inputs to Multilevel Markov Chain Monte Carlo
Hillary Fairbanks | Lawrence Livermore National Laboratory | United States
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Authors:
Hillary Fairbanks | Lawrence Livermore National Laboratory | United States
Panayot Vassilevski | Lawrence Livermore National Laboratory | United States
Markov chain Monte Carlo (MCMC), while a standard approach for nonlinear Bayesian inference of high-dimensional problems, is not feasible for large-scale applications. With the increase in problem size and thus simulation cost, MCMC becomes intractable, and acceleration approaches become necessary. Of particular interest within the scope of acceleration methods is the recently developed hierarchical multilevel MCMC approach in Dodwell et al. 2015 based on the Karhunen-Loeve expansion (KLE) that performs MCMC on the coarse grid problem and adds MCMC estimates of multilevel correction terms to account for the error between solutions on adjacent grids. While it has been shown to reduce the cost in comparison to standard MCMC for model 2D problems, a drawback is that the KLE-based hierarchy is not well-suited (and not even feasible) for 3D problems, as the KLE-based hierarchical sampling approach does not scale with the size of the problem. In this work, we develop a new PDE-based hierarchical sampling formulation based on the hierarchical decomposition of white noise across multiple levels of discretization, and apply it to this existing multilevel MCMC framework. In particular, we form a scalable hierarchical sampling method for a 3D subsurface flow example, and investigate the cost reduction and scaling of this approach in comparison to standard MCMC.
17:00
Approximate control variate approaches for Bayesian inverse problems
Nicholas Galioto | University of Michigan | United States
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Authors:
Nicholas Galioto | University of Michigan | United States
Alex Gorodetsky | University of Michigan | United States
We consider embedding a newly developed framework - approximate control variates - for variance-reduction estimators within Bayesian inference algorithms. In this work, we consider leveraging multiple model fidelities - or an ensemble of models - to accelerate computation of the posterior of a single high-fidelity model using MCMC. A majority of existing literature for this problem uses the Multi-level Monte Carlo framework as the principle paradigm through which to enhance convergence of MCMC chains. Here, we explore an approach that breaks from the traditional "model hierarchy" notions and is based on a more holistic model fusion technique provided by control variates. In particular, we study this approach for joint parameter-state inference within dynamical systems.
17:30
Mufti-fidelity uncertainty quantification for coupled systems
Sam Friedman | Texas A&M University | United States
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Authors:
Sam Friedman | Texas A&M University | United States
Michael Eldred | Sandia National Laboratories | United States
John Jakeman | Sandia National Laboratories | United States
Quantifying the uncertainty in system models composed of coupled multi-physics models can be computationally intractable when the computational cost of running the system model is high. We address this challenge present by performing independent uncertainty quantification (UQ) on each component independently and efficiently combining the results. Specifically, we treat coupling variables between components as additional unknown variables to surrogate models of each component. If the the number of coupling variables is small and each model is only dependent on a subset of system variables the cost of building surrogates of each component can be orders of magnitudes smaller than the cost of a full system approximation. To further improve computational efficiency we greedily allocate resources to the evaluation of each component model in a manner that maximizes the reduction in error per unit cost. Finally we show that our approach can be used to, not only effectively allocate resources between components, but also assign work across a hierarchy of models used to simulate a given component.
18:00
Recent advancements in sampling approaches for multifidleity uncertainty quantification
Gianluca Geraci | Sandia National Laboratories | United States
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Authors:
Gianluca Geraci | Sandia National Laboratories | United States
Alex Gorodetsky | University of Michigan | United States
Michael Eldred | Sandia National Laboratories | United States
John Jakeman | Sandia National Laboratories | United States
In the last decades, the advancements in both computer hardware/architectures and scientific computing algorithms enabled engineers and scientists to more rapidly study and design complex systems by heavily relaying on numerical simulations. Uncertainty Quantification (UQ) evolved as a task within the most comprehensive Verification and Validation framework which aims at obtaining truly predictive numerical simulations. Despite the recent efforts and successes in advancing the algorithms’ efficiency, the combination of a large set of uncertainty parameters (often correlated to the complexity of the numerical/physical assumptions) and the lack of regularity of the system's response still represents a formidable challenge for UQ. One of the possible ways of circumventing these difficulties is to rely on multilevel/multifidelity sampling approaches. In this work we present several recent advancements in this area which include, but are not limited to, approximate control variate and multifidelity for models with dissimilar parametrization via Active Subspace and/or Adaptive Basis mapping. Several numerical examples will be presented ranging from simple verification cases up to more realistic engineering problems.