Tools for Uncertainty Quantification
Models of physical systems typically involve inputs/parameters that are determined from empirical measurements, and therefore exhibit a certain degree of uncertainty. Estimating the propagation of this uncertainty into computational model output predictions is crucial for purposes of model validation, hypothesis testing, model optimization, and decision support.
FASTMath researchers are involved in the development of probabilistic methods and software for efficient uncertainty quantification (UQ) in computational models. These efforts are organized on a number of fronts, addressing a range of algorithmic and software developments, targeting key challenges in UQ. These are highlighted in the following.
Low rank sparse methods: Low rank sparse representations are of utility in the integration of high-dimensional functions. We are working on the deployment of advanced adaptive methods for functional low-rank and sparse decompositions in mixed temporal, spatial, and stochastic spaces. We are targeting well-conditioned sampling strategies for high-dimensional approximation of functions, with both continuous and discrete parameters, over both the full space and lower-dimensional manifolds.
Manifold and basis adaptation methods: Identication of underlying low-dimensional manifolds, and basis adaptation for functional representations, also provide effective means of representation of high-dimensional functions. We are working on integration of manifold sampling with state-of-the-art eigensolvers, and on deploying parallel stochastic differential equation solvers for sampling on manifolds, with requisite information pooling approaches. We are also working on iterative strategies for fast convergence of basis adaptation, on efficient parametrization and interpolation approaches in basis adaptation, and on effective software deployments of adaptive basis constructions.
Statistical inversion in high dimensions: We rely on Bayesian inference for the formulation of statistical inverse problems in parameter estimation and model calibration, and employ a range of methods for efficient solution of these systems in high-dimensional spaces. We are working on software deployment of state-of-the-art likelihood informed subspace (LIS) Markov chain Monte Carlo (MCMC) methods for Bayesian inference in high-dimension. We are working on parallelized LIS implementations in the context of MCMC, and will be demonstrating the effectiveness of these approaches relative to other state-of-the-art methods.
Surrogate models: Efficient and accurate model surrogates are necessary for various UQ purposes, whether in forward propagation of uncertainty or in statistical inversion and model calibration. We are working on deployment and evaluation of a range of surrogate construction methods, including adaptive quadrature and sparse regression with low rank tensor representations. We will also develop and deploy surrogate strategies relying on LIS/data-driven dimension reduction, adaptive local surrogate methods, as well as regression/classication-based surrogate construction methods.
Model error: Methods for estimation of model discrepancy, or model error, in the process of parameter estimation, are of importance in the calibration of physical models. Successful disambiguation of model and data errors requires suitable error-modeling constructions that are also consistent with the physical system constraints. We are working on deployment of effective model error strategies that employ informative priors on error-model parameterizations, and on advanced strategies for model and variable selection in model error estimation.
Multilevel and multidelity methods: Multilevel-multidelity (MLMF) methods, utilizing computations at different degrees of mesh resolution (multi-level) and model complexity (multi-fidelity) for informing UQ estimation, are very effective for efficient utilization of computational resources. We are working on the deployment of MLMF UQ methods with sparse low-rank estimators employing compressed sensing and functional sensor trains. We are leveraging multilevel capabilities for both forward and inverse UQ, along with advanced MCMC methods, and deploying a unified approach for multi-dimensional model hierarchies.
Optimization under uncertainty: Optimization and uncertainty are ubiquitous across a broad range of applications of interest, hence the importance of optimization under uncertainty (OUU) strategies that employ suitable means of handling uncertainty in operating conditions and model parameters, along with utilization of requisite risk-based objective functions, in the identication of optimal design parameters. We will advance our existing OUU software tools, leveraging recursive trust-region model management to support optimization across deep model hierarchies. We will adaptively rene regression-based constructions to accurately resolve failure boundaries for probability of failure estimation, and will deploy reliability-based OUU for design in the presence of rare events.