Probabilistic finite elements for nonlinear structural dynamics
Computer Methods in Applied Mechanics and Engineering
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Matrix computations (3rd ed.)
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Modeling uncertainty in flow simulations via generalized polynomial chaos
Journal of Computational Physics
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Efficient stochastic Galerkin methods for random diffusion equations
Journal of Computational Physics
Multi-element probabilistic collocation method in high dimensions
Journal of Computational Physics
Journal of Computational and Applied Mathematics
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This work is concerned with scalar transport equations with random transport velocity. We first give some sufficient conditions that can guarantee the solution to be in appropriate random spaces. Then a Galerkin method using bi-orthogonal polynomials is proposed, which decouples the equation in the random spaces, yielding a sequence of uncoupled equations. Under the assumption that the random wave field has a structure of the truncated KL expansion, a principle on how to choose the orders of the approximated polynomial spaces is given based on the sensitivity analysis in the random spaces. By doing this, the total degree of freedom can be reduced significantly. Numerical experiments are carried out to illustrate the efficiency of the proposed method.