Efficient Uncertainty Quantification with the Polynomial Chaos Method for Stiff Systems
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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 $S \times n$, 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 $O(S^3n^3)$. 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.