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
Orthogonal Tensor Decompositions
SIAM Journal on Matrix Analysis and Applications
Multi-resolution analysis of wiener-type uncertainty propagation schemes
Journal of Computational Physics
Algorithms for Numerical Analysis in High Dimensions
SIAM Journal on Scientific Computing
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
Journal of Computational Physics
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Multivariate Regression and Machine Learning with Sums of Separable Functions
SIAM Journal on Scientific Computing
A non-adapted sparse approximation of PDEs with stochastic inputs
Journal of Computational Physics
A probabilistic graphical model approach to stochastic multiscale partial differential equations
Journal of Computational Physics
Basis adaptation in homogeneous chaos spaces
Journal of Computational Physics
Hi-index | 31.46 |
Uncertainty quantification schemes based on stochastic Galerkin projections, with global or local basis functions, and also stochastic collocation methods in their conventional form, suffer from the so called curse of dimensionality: the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem. In this paper, to overcome the curse of dimensionality, a low-rank separated approximation of the solution of a stochastic partial differential (SPDE) with high-dimensional random input data is obtained using an alternating least-squares (ALS) scheme. It will be shown that, in theory, the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system. For the case of an elliptic SPDE, an a priori error analysis of the algorithm is derived. Finally, different aspects of the proposed methodology are explored through its application to some numerical experiments.