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IAMCS Workshop in Large-Scale Inverse Problems and Uncertainty Quantification

Bruce Fryxell, University of Michigan
Calibration and Prediction in a Radiative Shock Experiment


  • Bruce Fryxell
  • Members of the CRASH Team


The CRASH experiment uses a laser to generate a strong shock in a Be disk μm thick. The shock breaks out of the disk after about 400 ps into a Xe-filled tube and produces sufficient radiation to modify the shock structure. The shock location is predicted using two simulation codes, Hyades and CRASH. Hyades models the laser-plasma interaction at times less than 1.1 ns and can predict the shock breakout time. The CRASH code is initialized at 1.1 ns and is used to predict the shock location at later times for comparison to experiment.

We use the simulation tools and experiments conducted in one region of input space to predict in a new region where no prior experiments exist. Two data sets exist on which to base predictions: shock break time data, and shock location data at 13, 14, and 16 ns, and we wish to predict shock locations at 20 and 26 ns to compare to subsequent experiments. We use two models of the Kennedy-O'Hagen form to combine experiments with simulations, using one to inform the other, and interpret the discrepancy in these models in a way that allows us to gain some understanding of model error separately from parameter tuning.

Shock breakout times are modeled by constructing \(t=\eta_{BO}\left(x, \theta\right)+\delta_{BO}\left(x\right)+\epsilon_{BO}\) that jointly fits the field measurements \(T\) of shock breakout time \(t\) and a set of 1024 Hyades simulations over a 6 dimensional input space (4 experimental variables \(x\) and 2 calibration parameters \(\theta\)). This model provides posterior distributions for the calibration parameters \(\pi\left(\theta \mid T\right)\), as well as for the parameters in Gaussian Process (GP) models of the emulator \(\eta_{BO}\), the discrepancy function \(\delta_{BO}\), and for the replication error \(\epsilon_{BO}\). If the discrepancy function is significant compared to measurement uncertainty, we would call this process "tuning," but if the discrepancy is small (as in our case), we refer to this as calibration.

Next, we use the shock location field data at 13-16 ns along with shock locations from 1024 CRASH simulations to construct a model of the form \(z=\eta_{SL}\left(x,\theta\right)+\delta_{SL}\left(x\right)+\epsilon_{SL}\) , with \(\theta\) now treated as an experimental, rather than a calibration parameter, drawn from the posterior constructed in the previous step, so \(\theta \sim \eta\left(\theta \mid T\right)\). The \(x\) are drawn from distributions representing uncertainties in the experimental parameters. This second model is used to construct the emulator \(\eta_{SL}\), its discrepancy \(\delta_{SL}\), and a best estimate of the replication error \(\epsilon_{SL}\). The discrepancy can be studied to understand the defects of the physics model. The result shows that our model tends to under predict shock location.

Finally we can use \(\eta_{SL}\left(x, \theta\right)+\delta_{SL}\left(x\right)+\epsilon_{SL}\) to predict shock location at 20 and 26 ns, times at which we had simulations but no previous measurements. In doing so we can separate the code prediction \(\eta_{SL}\left(x, \theta\right)\) and the uncertainty due to this prediction (caused by uncertainty in \(\theta\), \(x\), and in the GP modeling parameters) from the uncertainty due to discrepancy \(\delta_{SL}\left(x\right)\). The uncertainty in discrepancy is of course large, because we are extrapolating the discrepancy to a new region of input space. The uncertainty in the emulator \(\eta_{SL}\left(x, \theta\right)\) is significantly smaller because there were simulation data in this region. Finally, comparison of the predictions with field measurements at 20 and 26 ns show that even the smaller predictive interval from the emulator alone contains the actual field measurements.