Quasi-Monte Carlo quadratures for multivariate smooth functions

Numerical solution of the Poisson equation over hypercubes using reduced Chebyshev polynomial bases

“…This means that the reduced basis is less sensitive to the smoothness of the functions than the usual tensor product Chebyshev interpolation. This has been already observed in [1] and as our basis is also a lot less sensitive to the dimensional effect, we only keep it for higher dimensions. Table 4 Numerical approximation sparse basis Q = 4.…”

“…γ (1) k,j = −s l 1 (k),l 1 (j) s l 3 (k),l 3 (j) (−1) l 2 (k)+l 2 (j)+1 l 2 (j) 2 + l 2 (k) 2 , γ (2) k,j = s l 1 (k),l 1 (j) s l 3 (k),l 3 (j) l 2 (j) 2 + l 2 (k) 2 , γ (3) k,j = s l 2 (k),l 2 (j) s l 3 (k),l 3 (j) l 1 (j) 2 + l 1 (k) 2 , γ (4) k,j = −s l 2 (k),l 2 (j) s l 3 (k),l 3 (j) (−1) l 1 (k)+l 1 (j)+1 l 1 (j) 2 + l 1 (k) 2 , γ (5) k,j = s l 1 (k),l 1 (j) s l 2 (k),l 2 (j) l 3 (j) 2 + l 3 (k) 2 , γ (6) k,j = −s l 1 (k),l 1 (j) s l 2 (k),l 2 (j) (−1) l 3 (k)+l 3 (j)+1 l 3 (j) 2 + l 3 (k) 2 and finally…”

“…We describe quickly the least-square method developed in [1] to compute the coefficients b k . For the sake of normalization, we use a least-square problem weighted by the norms of the basis functions…”

“…The use of Chebyshev polynomials combined with random drawings associated to the Chebyshev weight were crucial in these works. We have finally replaced the algorithm by the least-square problem of fitting our Chebyshev polynomial approximation model to some random, quasi-random points or quantified points [1]. This has led for instance to high accurate quadrature formulae, but our concern here is numerical approximations.…”

“…The aim of this paper is to use a sparse Chebyshev polynomial basis [1] to obtain, at a reasonable computational cost, very accurate approximations of the solution of the Poisson equation over hypercubes. These approximations are computed by means of a hybrid variational formulation [2].…”