An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
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1,597 citations
Cites background or methods from "An Anisotropic Sparse Grid Stochast..."
...Their systematic choice can be based either on a priori or a posteriori information and is motivated by the regularity of the solution and the error estimates derived in [45]....
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...It is also shown in [45] that, under the same condition, the constant Ĉ(r, N) is uniformly bounded with respect to N ....
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...Anisotropic sparse stochastic collocation [45] combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective in reducing the curse of dimensionality for problems depending on random variables which weigh approximately equally in the solution, and the second is appropriate for solving highly anisotropic problems with relatively low effective dimension, as in the case where input random variables are Karhunen–Loève truncations of “smooth” random fields....
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...The actual construction of α can be based on either a priori or a posteriori information, as suggested in [45]....
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...The second result we state is taken from [45] and analyzes the rate of convergence for the anisotropic Smolyak formula....
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Cites methods from "An Anisotropic Sparse Grid Stochast..."
...The current work includes adaptive choice of polynomial basis [19, 79], adaptive element selection in multi-element gPC [80], and adaptive sparse grid collocation [20, 78]....
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...Althoughwith the fast growth of computing power and some newly developed adaptive algorithms, in both stochastic Galerkin and collocation methods [19, 20, 78–80], the difficulty is alleviated to some degree....
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484 citations
References
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Additional excerpts
...[7, 5])....
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7,158 citations
Additional excerpts
...That is, s = 1 and C(s;φ) = ‖φ‖H2(D); see, for example, [5]....
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6,236 citations
"An Anisotropic Sparse Grid Stochast..." refers background in this paper
...4 (Karhunen-Loève expansion) We recall that any second order random field g(ω, x), with continuous covariance function cov[g] : D ×D → R, can be represented as an infinite sum of random variables, by means, for instance, of a Karhunen-Loève expansion [21]....
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