Abstract: This paper develops a Smolyak-type sparse-grid stochastic collocation method (SGSCM) for uncertainty quantification of nonlinear stochastic dynamic equations. The solution obtained by the method is a linear combination of tensor product formulas for multivariate polynomial interpolation. By choosing the collocation point sets to coincide with cubature point sets of quadrature rules, we derive quadrature formulas to estimate the expectations of the solution. The method does not suffer from the curse of dimensionality in the sense that the computational cost does not increase exponentially with the number of input random variables. Numerical analysis of a nonlinear elastic oscillator subjected to a discretized band-limited white noise process demonstrates the computational efficiency and accuracy of the developed method.

Key words: sparse grid| Smolyak algorithm| stochastic dynamic equation| uncertainty quantification

摘要： This paper develops a Smolyak-type sparse-grid stochastic collocation method (SGSCM) for uncertainty quantification of nonlinear stochastic dynamic equations. The solution obtained by the method is a linear combination of tensor product formulas for multivariate polynomial interpolation. By choosing the collocation point sets to coincide with cubature point sets of quadrature rules, we derive quadrature formulas to estimate the expectations of the solution. The method does not suffer from the curse of dimensionality in the sense that the computational cost does not increase exponentially with the number of input random variables. Numerical analysis of a nonlinear elastic oscillator subjected to a discretized band-limited white noise process demonstrates the computational efficiency and accuracy of the developed method.

关键词: sparse grid| Smolyak algorithm| stochastic dynamic equation| uncertainty quantification