Uncertainty Quantification: Bayesian Multi-Fidelity Schemes
Most current uncertainty quantification schemes cannot be used in combination with large-scale nonlinear models with high stochastic dimension. On the one hand, sampling based schemes such as Monte Carlo or Latin Hyper Cube sampling can cope with high stochastic dimensionality, but require a large number of model evaluations and as a result the computational costs associated with these schemes are often extreme. On the other hand, typical surrogate model techniques based on, e.g., the polynomial chaos expansion or Gaussian processes suffer from the curse of dimensionality and thus cannot be applied effectively to models where uncertain parameters are modeled as three-dimensional random fields.
Therefore, our group investigates Bayesian multi-fidelity schemes, which rigorously incorporate information from low-fidelity versions of the computational models and thus can reduce the computational burden dramatically.
It is important to note that the employed low-fidelity models need not to be accurate in a deterministic sense; merely a similar stochastic structure needs to be retained by the low-fidelity version of the model. This very weak requirement opens up various ways to construct these low-fidelity models including, e.g., coarsening of the discretization, looser solver tolerances, and simpler physical models to name a few.
The sampling is done in large part on the low-fidelity model. Only very few evaluations of the high-fidelity model are required to construct a Bayesian regression model, which, in combination with the stochastic information from sampling the low-fidelity model, can provide accurate response statistics about the respective quantity of interest.
Furthermore, efficient parameter continuation schemes are employed to further reduce the computational effort. The implementation of the described uncertainty quantification scheme in our software framework with nested-parallelism allows us to exploit modern high performance computing platforms and enables us to perform uncertainty quantification for large-scale nonlinear finite element models with very high stochastic dimensionality