Efficient function approximation on general bounded domains using splines on a Cartesian grid

Abstract

Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a simple, regular grid that is defined on a bounding box. This approach allows the use of high order and highly structured splines as a basis for piecewise polynomials. The methodology is analogous to that of Fourier extensions, using Fourier series on a bounding box, which leads to spectral accuracy for smooth functions. However, Fourier extension approximations involve solving a highly illconditioned linear system, and this is an expensive step. The computational complexity of recent algorithms is O N log 2 (N ) in 1 dimension and O N 2 log 2 (N ) in two dimensions. We show that the compact support of B-splines enables improved complexity for multivariate approximations, namely O(N ) in 1-D, O N 3 2 in 2-D and more generally O N 3(d−1) d in d-D with d > 1. This comes at the cost of achieving only algebraic rates of convergence. Our statements are corroborated with numerical experiments and Julia code is available.

The AZ algorithm for least squares systems with a known incomplete generalized inverse

“…In that section we emphasized the connection to continuous and discrete dual bases for the selection of Z. This connection is explored in more detail for extensions based on B-spline bases in [6]. Since B-splines are compactly supported, the collocation matrix A is highly sparse.…”

“…Since B-splines are compactly supported, the collocation matrix A is highly sparse. B-splines are not orthogonal, but several different dual bases can be identified and it is shown in [6] that each of these leads to a suitable Z matrix. Some choices of dual bases lead to a sparse Z and even a sparse matrix A − AZ * A, with corresponding advantages for speed and efficiency.…”

“…Some choices of dual bases lead to a sparse Z and even a sparse matrix A − AZ * A, with corresponding advantages for speed and efficiency. We refer to [6] for the analysis and examples.…”