Utilizing Adjoint-based Error Estimates to Adaptively Resolve Response Surface Approximations (Tim Wildey, Sandia National Lab., USA)

Uncertainty and error are ubiquitous in predictive modeling andsimulation due to unknown model parameters, boundary conditions andvarious sources of numerical error. Consequently, there isconsiderable interest in developing efficient and accurate methods to quantify the uncertainty in the outputs of a computational model. Monte Carlo techniques are the standard approach due to their relativeease of implementation and the fact that they effectively circumventthe curse of dimensionality. Unfortunately, the number of samplesrequired to accurately estimate certain probabilistic quantities,especially the probability of high-risk, low-probability events, maybe prohibitively large for high-fidelity computational models.

A number of recently developed methods for uncertainty quantificationhave focused on constructing response surface approximations of theinput-to-output mapping using only a limited number of high-fidelity model evaluations. The fact that a very large number of samples can be efficiently evaluated using the response surface effectively reduces the statistical component of the error in the probabilistic quantity of interest. However, the deterministic component of the error may be quite large for each sample due to the standard sources of discretization error as well as the interpolation of the response surface approximation. The accumulation of these deterministic errors may significantly affect the accuracy of the probabilistic quantity of interest.

In this presentation, we show how adjoint-based techniques can be usedto efficiently estimate the error in a quantity of interest computedfrom a sample of a response surface approximation. We then show howthese error estimates can be used to provide enhanced convergence, newadaptive strategies, and a means to avoid over-adapting the responsesurface beyond the accuracy of the spatial discretization. Finally, wedemonstrate that these a posteriori estimates can also be used toguide adaptive improvement of a response surface approximation withthe specific goal of accurately and efficiently estimating probabilities of events.