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(n3) to
O(S3n3). The S3-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