Computational Uncertainty Quantification (17w5072)

Arriving in Banff, Alberta Sunday, October 8 and departing Friday October 13, 2017

Organizers

(École Polytechnique de Montréal)

Roger Ghanem (University of Southern California)

(University of New Mexico)

Raúl Tempone (King Abdullah University of Sciences and Technology)

Objectives

The overall objective of the workshop will be to address the theoretical and computational foundations of UQ methods for complex systems arising in science and engineering. Specific objectives will be concerned with the following issues:
  1. How to characterize uncertainty, particularly when available measurements and data are
  2. scarce and lack accuracy?
  3. How to propagate uncertainty in problems where quantities of interest lack regularity with
  4. respect to the input parameters?
  5. How to treat inverse problems in settings that feature high-dimensional parameter spaces
  6. and computationally expensive forward models?
These questions describe challenging computational and analytical problems. Some results have been achieved with respect to Item 1 for certain classes of methods, but many open questions remain. Items 2 and 3 have been partially addressed by extending sparse methods to non-smooth quantities using adaptive refinement methods and employing reduced-order models and low rank approximations.

It is generally agreed that there are two major types of UQ problems that one may be interested in: 1) the forward problem, namely uncertainty propagation, where uncertainties in model input parameters are propagated through the mathematical model to give uncertainty in outputs; 2) the inverse problem, in which model and parameter uncertainties are estimated given noisy experimental measurements. Forward UQ focuses on the influence of uncertain input parameters on the outputs. Inverse UQ is used to quantify the model inadequacy (bias correction) and to estimate the values of unknown model parameters (calibration).

There has been a proliferation of research work on both types of UQ problems. The majority of the theories and methodologies have focused on forward uncertainty propagation, including Multi-Level Monte Carlo, Multi-Level Quasi Monte Carlo, Multi-Index Monte Carlo methods, adaptive sparse and Generalized Polynomial Chaos for Galerkin and collocation formulations, and tensor compression techniques. A number of approaches for inverse UQ problems have also been considered and proved to be useful for most small- to medium-scale problems. The Bayesian approach, for instance, recasts the inverse problem as one of statistical inference, incorporating uncertainties in the measurements, the forward model, and any prior information about the parameters. Special sampling techniques, such as Markov Chain Monte Carlo, have been proposed to exploit the structure of the forward model in order to reduce the dimension of both the parameter and state spaces in the inverse problem. Despite all these attempts, many issues remain unsolved. The major difficulty in these problems is that their computational complexity dramatically increases with the dimensionality of the problem, i.e. the number of input model parameters. For very high-dimensional problems, all proposed approaches become intractable. Moreover, in the case of inverse problems, multiple combinations of unknown parameters and discrepancy functions can yield the same experimental prediction. Hence different values of parameters cannot be distinguished or identified.

We expect the workshop to advance the field of UQ on those issues by bringing together computational scientists, numerical analysts, and statisticians to a five-day workshop at Banff. This workshop will be mutually beneficial to all participants. Numerical analysts and statisticians will be informed of new UQ methodologies developed by computational scientists and be exposed to problems and applications of engineering interest. Computational scientists will in turn be exposed to new mathematical tools and concepts recently developed by the analysts and statisticians. More importantly, the three groups of participants will be able to interact and exchange ideas during the workshop in order to determine a set of important problems for building theoretically sound and computationally feasible UQ approaches.