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Simulations of large-scale physical systems are often affected by the uncertainties in data, in model parameters, and by incomplete knowledge of the underlying physics. The traditional deterministic simulations do not account for such uncertainties. It is of interest to extend simulation results with "error bars" that quantify the degree of uncertainty. This added information provides a confidence level for the simulation result. For example, the air quality forecast with an associated uncertainty information is very useful for making policy decisions regarding environmental protection. Techniques such as Monte Carlo (MC) and response surface are popular for uncertainty quantification, but accurate results require a large number of runs. This incurs a high computational cost, which maybe prohibitive for large-scale models. The polynomial chaos (PC) method was proposed as a practical and efficient approach for uncertainty quantification, and has been successfully applied in many engineering fields. Polynomial chaos uses a spectral representation of uncertainty. It has the ability to handle both linear and nonlinear problems with either Gaussian or non-Gaussian uncertainties.This work extends the functionality of the polynomial chaos method to Source Uncertainty Apportionment (SUA), i.e., we use the polynomial chaos approach to attribute the uncertainty in model results to different sources of uncertainty. The uncertainty quantification and source apportionment are implemented in the Sulfur Transport Eulerian Model (STEM-III). It allows us to assess the combined effects of different sources of uncertainty to the ozone forecast. It also enables to quantify the contribution of each source to the total uncertainty in the predicted ozone levels.

The polynomial chaos method has been widely adopted as a computationally feasible approach for uncertainty quantification. Most studies to date have focused on non-stiff systems. When stiff systems are considered, implicit numerical integration requires the solution of a nonlinear system of equations at every time step. Using the Galerkin approach, the size of the system state increases from n to S × n, where S is the number of the polynomial chaos basis functions. Solving such systems with full linear algebra causes the computational cost to increase from O(n 3 ) to O(S 3 n 3 ). The S 3 -fold increase can make the computational cost prohibitive. This paper explores computationally efficient uncertainty quantification techniques for stiff systems using the Galerkin, collocation and collocation least-squares formulations of polynomial chaos. In the Galerkin approach, we propose a modification in the implicit time stepping process using an approximation of the Jacobian matrix to reduce the computational cost. The numerical results show a run time reduction with a small impact on accuracy. In the stochastic collocation formulation, we propose a least-squares approach based on collocation at a low-discrepancy set of points. Numerical experiments illustrate that the collocation least-squares approach for uncertainty quantification has similar accuracy with the Galerkin approach, is more efficient, and does not require any modifications of the original code.

The polynomial chaos (PC) method has been used in many engineering applications to replace the traditional Monte Carlo (MC) approach for uncertainty quantification (UQ) due to its better convergence properties. Many researchers seek to further improve the efficiency of PC, especially in higher dimensional space with more uncertainties. The intrusive PC Galerkin approach requires the modification of the deterministic system, which leads to a stochastic system with a much bigger size. The non-intrusive collocation approach imposes the system to be satisfied at a set of collocation points to form and solve the linear system equations. Compared with the intrusive approach, the collocation method is easy to implement, however, choosing an optimal set of the collocation points is still an open problem. In this paper, we first propose using the low-discrepancy Hammersley/Halton dataset and Smolyak datasets as the collocation points, then propose a least-squares (LS) collocation approach to use more collocation points than the required minimum to solve for the system coefficients. We prove that the PC coefficients computed with the collocation LS approach converges to the optimal coefficients. The numerical tests on a simple 2-dimensional problem show that PC collocation LS results using the Hammersley/Halton points approach to optimal result.

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