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 regular but oversampled 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 ill-conditioned linear system, and this is an expensive step. The computational complexity of recent algorithms is $\mathcal {O}\left (N\log ^{2}\left (N\right )\right )$ in 1-D and $\mathcal {O}\left (N^{2}\log ^{2}\left (N\right )\right )$ in 2-D. We show that, compared to Fourier extension, the compact support of B-splines enables improved complexity for multivariate approximations, namely $\mathcal {O}(N)$ in 1-D, $\mathcal {O}\left (N^{3/2}\right )$ in 2-D and more generally $\mathcal {O}\left (N^{3(d-1)/d}\right )$ in d-D with d > 1. By using a direct sparse QR solver for a related linear system, we also observe that the computational complexity can be nearly linear in practice. This comes at the cost of achieving only algebraic rates of convergence. Our statements are corroborated with numerical experiments and Julia code is available.