Efficient uncertainty quantification with the polynomial chaos method for stiff systems

  • Authors:

  • Haiyan Cheng;Adrian Sandu

  • Affiliations:

  • Computer Science Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA;Computer Science Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

  • Venue:
  • Mathematics and Computers in Simulation
  • Year:
  • 2009

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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 Sxn, where S is the number of PC 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 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.