Validation by Monte Carlo sampling of experimental observation functionals

We present and analyze a validation procedure for a given state estimate u⋆ of the true field utrue based on Monte Carlo sampling of experimental observation functionals. Our method provides, given a set of N possibly noisy local experimental observation functionals over the spatial domain Ω, confidence intervals for the L2(Ω) error in state and the error in L2(Ω) outputs. For L2(Ω) outputs, our approach also provides a confidence interval for the output itself, which can be used to improve the initial output estimate based on u⋆. Our approach implicitly takes advantage of variance reduction, through the proximity of u⋆ to utrue, to provide tight confidence intervals even for modest values of N. We present results for a synthetic model problem to illustrate the elements of the methodology and confirm the numerical properties suggested by the theory. Finally, we consider an experimental thermal patch configuration to demonstrate the applicability of our approach to real physical systems.