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Comparison of Clenshaw–Curtis and Leja Quasi-Optimal Sparse Grids for the Approximation of Random PDEs

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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014

Abstract

In this work we compare different families of nested quadrature points, i.e. the classic Clenshaw–Curtis and various kinds of Leja points, in the context of the quasi-optimal sparse grid approximation of random elliptic PDEs. Numerical evidence suggests that both families perform comparably within such framework.

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Notes

  1. 1.

    Also known as admissible sets or lower sets, i.e. such that \(\forall \,\mathbf{i} \in \mathcal{I}(\mathrm{w})\) and \(\forall \,\mathbf{j} \in \mathbb{N}_{+}^{N}\) s.t. j ≤ i, there holds \(\mathbf{j} \in \mathcal{I}(\mathrm{w})\), where the inequality is to be understood component-wise.

  2. 2.

    When 2m + 1 p-Disk Leja points are computed, they coincide with the Clenshaw–Curtis points.

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Acknowledgements

F. Nobile and L. Tamellini have been partially supported by the Swiss National Science Foundation under the Project No. 140574 “Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media” and by the Center for ADvanced MOdeling Science (CADMOS). R. Tempone is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.

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Correspondence to Lorenzo Tamellini .

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Nobile, F., Tamellini, L., Tempone, R. (2015). Comparison of Clenshaw–Curtis and Leja Quasi-Optimal Sparse Grids for the Approximation of Random PDEs. In: Kirby, R., Berzins, M., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham. https://doi.org/10.1007/978-3-319-19800-2_44

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