Numerical study of uncertainty quantification techniques for implicit stiff systems

ABSTRACT

Galerkin polynomial chaos and collocation methods have been widely adopted for uncertainty quantification purpose. However, when the stiff system is involved, the computational cost can be prohibitive, since stiff numerical integration requires the solution of a nonlinear system of equations at every time step. Applying the Galerkin polynomial chaos to stiff system will cause a computational cost increase from O(n³) to O(S³n³). This paper explores uncertainty quantification techniques for stiff chemical systems using Galerkin polynomial chaos, collocation and collocation least-square approaches. We propose a modification in the implicit time stepping process. The numerical test results show that with the modified approach, the run time of the Galerkin polynomial chaos is reduced. We also explore different methods of choosing collocation points in collocation implementations and propose a collocation least-square approach. We conclude that the collocation least-square for uncertainty quantification is at least as accurate as the Galerkin approach, and is more efficient with a well-chosen set of collocation points.

REFERENCES

• 1: L. G. Crespo and S. P. Kenny. Robust control design for systems with probabilistic uncertainty. Technical report, NASA, LRC, 2005.
• 2: V. Damian, A. Sandu, M. Damian, F. Potra, and G. Carmichael. The Kinetic PreProcessor KPP - a software environment for solving chemical kinetics. Comput. Chem. Eng., 26:1567--1579, 2002.
• 3: G. Fishman. Monte Carlo: Concepts, Algorithms and Applications, Springer-Verlag, New York, 1996.
• 4: Roger G. Ghanem , Pol D. Spanos, Stochastic finite elements: a spectral approach, Springer-Verlag New York, Inc., New York, NY, 1991
• 5: J. M. Hammersley. Monte Carlo methods for solving multivariables problems. Ann. New York Acad. Sci., 86:844--874, 1960.
• 6: B. A. P. James. Variance reduction techniques. J. Operations Research Society, 36(6):525, 1985.
• 7: O. M. Knio and O. P. Le Maitre. Uncertainty propogation in cfd using polynomial chaos decomposition. Fluid Dynamics Research, 38:616--640, 2006.
• 8: J. S. Liu. Monte Carlo Strategies in Scientific Computing. Springer-Verlag, 2001.
• 9: H. N. Najm, M. T. Reagan, O. M. Knio, R. G. Ghanem, and O. P. Le Maitre. Uncertainty quantification on reacting flow modeling. Technical Report 2003-8598, SANDIA, 2003.
• 10: A. Sandu and R. Sander. Technical notes: Simulating chemical system in Fortran90 and Matlab with the Kinetic PreProcessor KPP-2.1. Atmos. Chem. Phys., 6:187--195, 2006.
• 11: A. Sandu, J. G. Verwer, J. G. Blom, E. J. Spee, G. R. Carmichael, and F. A. Potra. Stiff ODE solvers for atmospheric chemistry problems II: Rosenbrock solvers. Atmos. Environ., 31:3459--3472, 1997.
• 12: S. Smolyak. Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl., 4:240--243, 1963.
• 13: N. Wiener. The homogeneous chaos. Amer. J. Math., 60:897--936, 1938.
• 14: Dongbin Xiu , George Em Karniadakis, The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations, SIAM Journal on Scientific Computing, v.24 n.2, p.619-644, 2002 [doi>10.1137/S1064827501387826]
• 15: D. Xiu and G. E. Karniadakis. Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg., 191(43):4927--4948, 2002.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.6 SIMULATION AND MODELING

Keywords:
least-square collocation, polynomial chaos, uncertainty quantification