Problem description

A good experimentalist always presents results with error bars. The latter are a good indication of the measurement uncertainty. The experimental curves and their bars are obtained by repeating the measurements under identical (controllable) conditions and recording the mean and the standard deviation of the results over an empirical statistical ensemble of data. Theoreticians and engineers, however, typically assume exact input parameters (boundary conditions, coefficients, source functions, etc.) in their models and produce smooth curves without any error bars. While a lot of effort goes into ensuring the validity and convergence of discretization procedures and numerical algorithms, not much has been achieved in the way of analyzing the uncertainty of numerical models in practice. To be sure, there are many fine theories about the uncertainty/sensitivity of mathematical/numerical models but the error bars are still missing. Similar to the actual experiments, multiple runs of a numerical simulation with different randomly perturbed inputs can be very expensive. In this Project the student will look into the ways of accelerating the uncertainty analysis in numerical models based on standard PDE’s, with the goal of making the computation of error bars easier and thus bringing the numerical experiments on a par with the actual ones