40 research outputs found
A sparse-grid isogeometric solver
Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS
as a basis for the approximation of the solution of PDEs. In this work, we
investigate to which extent IGA solvers can benefit from the so-called
sparse-grids construction in its combination technique form, which was first
introduced in the early 90s in the context of the approximation of
high-dimensional PDEs. The tests that we report show that, in accordance to the
literature, a sparse-grid construction can indeed be useful if the solution of
the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the
case of non-smooth solutions when some a-priori knowledge on the location of
the singularities of the solution can be exploited to devise suitable
non-equispaced meshes. Finally, we remark that sparse grids can be seen as a
simple way to parallelize pre-existing serial IGA solvers in a straightforward
fashion, which can be beneficial in many practical situations.Comment: updated version after revie
Sparse grid quadrature on products of spheres
We examine sparse grid quadrature on weighted tensor products (WTP) of
reproducing kernel Hilbert spaces on products of the unit sphere, in the case
of worst case quadrature error for rules with arbitrary quadrature weights. We
describe a dimension adaptive quadrature algorithm based on an algorithm of
Hegland (2003), and also formulate a version of Wasilkowski and Wozniakowski's
WTP algorithm (1999), here called the WW algorithm. We prove that the dimension
adaptive algorithm is optimal in the sense of Dantzig (1957) and therefore no
greater in cost than the WW algorithm. Both algorithms therefore have the
optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and
Wozniakowski (1999). A numerical example shows that, even though the asymptotic
convergence rate is optimal, if the dimension weights decay slowly enough, and
the dimensionality of the problem is large enough, the initial convergence of
the dimension adaptive algorithm can be slow.Comment: 34 pages, 6 figures. Accepted 7 January 2015 for publication in
Numerical Algorithms. Revised at page proof stage to (1) update email
address; (2) correct the accent on "Wozniakowski" on p. 7; (3) update
reference 2; (4) correct references 3, 18 and 2