Reconstruction Plans and Efficient Rank-1 Lattice Construction for Chebyshev Expansions Over Lower Sets
Abdelqoddous Moussa, Moulay Abdellah Chkifa
TL;DR
This work addresses exact integration and reconstruction for functions supported on finite lower sets $\Lambda$ using rank-1 lattices by selecting an optimal generating vector $\boldsymbol{z}$ and point count $n$. It establishes sharp lower bounds on the minimal $n$ required and introduces a heuristic component-by-component CBC algorithm that efficiently identifies admissible $(n,\boldsymbol{z})$, leveraging the lower-set structure to reduce memory usage and computation time. The authors show equivalence between reconstruction plans under certain conditions for lower sets and formalize admissibility via sets $P_0(\Lambda)$, $P_A(\Lambda)$, $P_B(\Lambda)$, and $P_C(\Lambda)$, with extensions to Fourier, cosine, and Chebyshev spaces through rank-1 lattice and tent-transformed lattice rules. The results offer a scalable framework for exact reconstruction and integration in high-dimensional Chebyshev expansions, with practical impact for efficient approximation in numerical analysis and related computational applications.
Abstract
This study focuses on constructing efficient rank-1 lattices that enable the exact integration and reconstruction of functions within Chebyshev spaces, based on finite lower index sets. We establish the equivalence of different reconstruction plans under specific conditions for certain lower sets. Furthermore, we introduce a heuristic component-by-component (CBC) algorithm that efficiently identifies admissible generating vectors and suitable numbers of nodes $n$, optimizing both memory usage and computational time.