## Uncertainty quantification in kinetic and hyperbolic problems

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

Tim Barth

NASA Ames

Abstract:

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.