Approximate Duals of B-splines for the Exact Representation of Splines on Coarse Knot Vectors
Joachim Stöckler
TL;DR
This work advances spline-based approximation by constructing enhanced approximate duals for B-splines that reproduce all spline functions on a coarse knot vector, not just polynomials. It introduces a refined kernel ${\mathcal L}$, extending the original quasi-projection kernel ${\mathcal K}$ with a correction term involving $U_m$ so that ${\mathcal L}$ reproduces the full spline space ${\mathcal S}_m(\Theta_0)$, enabling optimal rates in bent Sobolev spaces ${\mathcal{H}}^s([a,b];\Theta_0)$. The authors provide a general solvability framework via the matrix equation $A U_m = B$ and present explicit enhanced dual formulations for $m=2$ and $m=3$, including practical computation cues. Numerical experiments illustrate that the enhanced quasi-projection achieves the expected $O(h^m)$ convergence on bent Sobolev data, outperforming the original kernel $K$ which yields only $O(h^s)$ in these settings. These results have direct implications for isogeometric analysis and domain-decomposition methods, where accurate, locally supported representations on coarse knot vectors improve robustness and efficiency.
Abstract
Approximate duals of B-splines were first used by Chui et al. (2004) for the purpose of constructing tight wavelet frames on bounded intervals. They are splines with local support, whose inner product with a polynomial in the spline space provides the exact coefficient in the representation of the same polynomial in the B-spline basis. This implies that the associated integral operator is a quasi-projection in the sense of Jia (2004). Moreover, the approximation of smooth functions in Sobolev spaces by this quasi-projection yields the optimal approximation order. More recently, for applications in isogeometric analysis, the optimal approximation order should also be obtained for functions in a slightly larger space, the so-called bent Sobolev space defined in \cite{Bazilevsetal2006,daVeigaetal2014}. This requires the construction of enhanced approximate duals, whose corresponding integral operator provides the exact representation of all spline functions with respect to a coarse knot vector, using only few interior knots of the given knot vector. Our analysis provides an explicit construction and hints for an efficient computation of enhanced approximate duals. Explicit representations as linear combinations of B-splines are provided for $m=2$ and $m=3$, and two numerical examples for splines of order $3\le m\le 6$ demonstrate the optimal approximation order for functions in bent Sobolev spaces.