Uncertainty quantification in kinetic and hyperbolic problems

Combined Uncertainty and A-Posteriori Error Bound Estimates for General CFD Calculations

Tim Barth



Hydrodynamic realizations often contain numerical error arising from finite-dimensional approximation (e.g. numerical methods using grids, basis functions, particles, etc) and statistical uncertainty arising from incomplete information and/or statistical characterization of model parameters and random fields. In this presentation, we posit that a general framework for uncertainty and error quanitification should include the combined effects of statistical uncertainty and numerical error so that uncertainty statistics with a-posteriori error bound estimates are provided. For problems containing no sources of uncertainty, a standard error bound estimate is obtained. For problems containing no numerical error, a standard uncertainty estimate is obtained. Specifically, we consider error bounds for moment statistics given realizations containing finite-dimensional numerical error (Barth, 2013). The error in computed output statistics contains contributions from both realization error and the error resulting from the calculation of statistics integrals using a numerical method. We then devise computable a-posteriori error bounds by numerically approximating all terms arising in the error bound estimates for a variety of standard UQ methods * Dense tensorization basis methods (Tatang,1994) and a subscale recovery variant (Barth,2013) for piecewise smooth data, * Sparse tensorization basis methods (Smolyak,1963) utilizing node-nested hierarchies, * Multi-level sampling methods (Mishra,2010) for high-dimensional random variable spaces. We have developed a general software package that provides the necessary tools and graphical user interface (GUI) for rapidly posing uncertainty quantification problems to a CFD method and calculating uncertainty statistics with error bounds for output quantities of interest. Example CFD calculations are shown to demonstrate features of the general framework. T.J. Barth,``Non-intrusive Uncertainty with Error Bounds for Conservation Laws Containing Discontinuities,'' ewblock Springer-Verlag Publishing, LNCSE, Vol 92, 2013. S. Smolyak,``Quadrature and Interpolation Formulas for Tensor Products of Centain Classes of Functions," Dok. Akad. Nauk SSSR, Vol. 4, 1963. M.A. Tatang,``Direct Incorporation of Uncertainty in Chemical and Environmental Engineering Systems,'" MIT, Dept. Chem. Engrg, 1994. S. Mishra and C. Schwab,``Sparse Tensor Multi-Level Monte Carlo Finite Volume Methods for Hyperbolic Conservation Laws,'', ETH Zurich, SAM Report 2010-24, 2010.