Efficient uncertainty quantification with the polynomial chaos method for stiff systems

Haiyan Cheng and Adrian Sandu

Mathematics and Computers in Simulation (MATCOM), 2009, vol. 79, issue 11, 3278-3295

Abstract: The polynomial chaos (PC) method has been widely adopted as a computationally feasible approach for uncertainty quantification (UQ). Most studies to date have focused on non-stiff systems. When stiff systems are considered, implicit numerical integration requires the solution of a non-linear 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 PC 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 computation prohibitive. This paper explores computationally efficient UQ techniques for stiff systems using the PC Galerkin, collocation, and collocation least-squares (LS) formulations. 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 no negative 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 UQ has similar accuracy with the Galerkin approach, is more efficient, and does not require any modification of the original code.

Keywords: Uncertainty quantification; Polynomial chaos; Least-squares collocation; Smolyak algorithm; Low-discrepancy data sets (search for similar items in EconPapers)
Date: 2009
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Citations: View citations in EconPapers (3)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:79:y:2009:i:11:p:3278-3295

DOI: 10.1016/j.matcom.2009.05.002

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