Efficient uncertainty quantification with the polynomial chaos method for stiff systems

Abstract

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.

Real-Time Parameter Estimation Study for Inertia Properties of Ground Vehicles

“…Table 1 shows the variation in the parameters for each simulation, and figures (3)(4)(5) show the parameter estimations for each simulation.…”

“…The methods employed are an application of Bayesian statistics, and blending of the Extended Kalman Filter to the mathematical technique of Generalized Polynomial Chaos (gPC). The Generalized Polynomial Chaos technique gives a computationally efficient method for quantifying the uncertainty in the parameters [4,12,23,24].…”

Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems

“…introduced the least-squares collocation method (LSCM) and used the roots of the associated orthogonal polynomials in selecting the sampling points. Cheng and Sandu showed the LSCM maintains the exponential convergence of GPM yet was superior in computational speed in [28]; where the Hammersley LDS data set was the preferred method in selecting collocation points. Cheng and Sandu also presented a modified time stepping mechanism where an approximate Jacobian was used when solving stiff systems.…”

“…Sandu and coworkers introduced its application to multibody dynamical systems in [27,28,[36][37][38][39][40]. Significant work has been done applying it as a foundational element in parameter [23][24][25][26][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59] and state estimation [60,61], as well as system identification [62].…”

“…al. [27,28] showed that the collocation method solves formulations such as (13) by solving (1) at a set of points, ∈ ℝ , ݇ = 1 … ݊ , selected from the ݊ dimensional domain of the random variables ∈ ℝ . Meaning, at any given instance in time, the random variables' domain is sampled and solved ݊ times with = (updating the approximations of all sources of uncertainty for each solve), then the uncertain coefficients can be determined at that given time instance.…”