Haiyan Cheng
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.
- L. G. Crespo and S. P. Kenny. Robust control design for systems with probabilistic uncertainty. Technical report, NASA, LRC, 2005.Google Scholar
- 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.Google ScholarCross Ref
- G. Fishman. Monte Carlo: Concepts, Algorithms and Applications, Springer-Verlag, New York, 1996.Google Scholar
- R. G. Ghanem and P. Spanos. Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York, 1991. Google ScholarDigital Library
- J. M. Hammersley. Monte Carlo methods for solving multivariables problems. Ann. New York Acad. Sci., 86:844--874, 1960.Google ScholarCross Ref
- B. A. P. James. Variance reduction techniques. J. Operations Research Society, 36(6):525, 1985.Google ScholarCross Ref
- O. M. Knio and O. P. Le Maitre. Uncertainty propogation in cfd using polynomial chaos decomposition. Fluid Dynamics Research, 38:616--640, 2006.Google ScholarCross Ref
- J. S. Liu. Monte Carlo Strategies in Scientific Computing. Springer-Verlag, 2001. Google ScholarDigital Library
- 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.Google Scholar
- 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.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- S. Smolyak. Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl., 4:240--243, 1963.Google Scholar
- N. Wiener. The homogeneous chaos. Amer. J. Math., 60:897--936, 1938.Google ScholarCross Ref
- D. Xiu and G. Karniadakis. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput., 24:619--644, 2002. Google ScholarDigital Library
- 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.Google ScholarCross Ref
Index Terms
- Numerical study of uncertainty quantification techniques for implicit stiff systems
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