Sparse Grids and Applications - Miami 2016 by Jochen Garcke Dirk Pflüger Clayton G. Webster & Guannan Zhang

Author:Jochen Garcke, Dirk Pflüger, Clayton G. Webster & Guannan Zhang
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(18)

where γ 1 is defined as in Theorem 2 , and

(19)

with the constant .

Theorem 5

For some 0 < ε ≤ 0.8, let f be analytic in , and let g 2 be the conformal mapping (9) truncated at degree M = 4. Then the sparse quadrature (14) built from transformed Gauss–Legendre quadrature rules satisfies the following error bound in terms of the number of quadrature nodes:

(20)

with γ 2 as in Theorem 3 , and ξ(d) as in (19).

Sketch of Proof

From the one dimensional results of Theorems 2 and 3, resp., the proof of the results above follows from well-known sparse grid analysis techniques and estimates on the number of quadrature nodes [22]. Specifically, we may follow along the lines of the proof of [22, Theorem 3.19], with the one-dimensional convergence estimates [22, p. 2230] replaced by (16) and (17), resp, and noting that, e.g.,

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