Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Polynomial chaos for multirate partial differential algebraic equations with random parameters
Applied Numerical Mathematics
Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms
Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms
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We consider boundary value problems of systems of ordinary differential equations, where uncertainties are present in physical parameters of the systems. We introduce random variables to describe the uncertainties. The resulting stochastic model is resolved by the strategy of the polynomial chaos. On the one hand, a non-intrusive approach requires the solution of a large number of nonlinear systems with relatively small dimension. On the other hand, an intrusive approach yields just a single nonlinear system with a relatively high dimension. Alternatively, we present a non-intrusive method, which still exhibits a single large nonlinear system. Consequently, the convergence of a single Newton iteration has to be ensured only to solve the boundary value problem, while many initial value problems of the original ordinary differential equations are involved.