A weighted l1-minimization approach for sparse polynomial chaos expansions

@article{Peng2014AWL,
  title={A weighted l1-minimization approach for sparse polynomial chaos expansions},
  author={Ji-Gen Peng and Jerrad Hampton and Alireza Doostan},
  journal={J. Comput. Phys.},
  year={2014},
  volume={267},
  pages={92-111}
}
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References

SHOWING 1-10 OF 72 REFERENCES
STOCHASTIC COLLOCATION ALGORITHMS USING 𝓁 1 -MINIMIZATION
TLDR
The analysis suggests that using the Chebyshev measure to precondition the ‘1-minimization, which has been shown to be numerically advantageous in one dimension in the literature, may in fact become less efficient in high dimensions.
Adaptive sparse polynomial chaos expansion based on least angle regression
A least-squares approximation of partial differential equations with high-dimensional random inputs
ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS
TLDR
This work uses the Stochastic Collocation method, and the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that features better convergence properties compared to standard Smolyak or tensor product grids.
A non-adapted sparse approximation of PDEs with stochastic inputs
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
TLDR
It is proved that replacing the usual quadratic regularizing penalties by weighted 𝓁p‐penalized penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem.
Sparse Pseudospectral Approximation Method
Sparse Tensor Galerkin Discretization of Parametric and Random Parabolic PDEs - Analytic Regularity and Generalized Polynomial Chaos Approximation
TLDR
For initial boundary value problems of linear parabolic partial differential equations with random coefficients, analyticity of the solution with respect to the parameters is shown and regularity result of the parametric solution is proved for both compatible as well as incompatible initial data and source terms.
...
...