Numerical cubature and hyperinterpolation over Spherical Polygons

Abstract

The purpose of this work is to introduce a strategy for determining the nodes and weights of a low-cardinality positive cubature formula nearly exact for polynomials of a given degree over spherical polygons. In the numerical section we report the results about numerical cubature over a spherical polygon $\cal P$ approximating Australia and reconstruction of functions over such $\cal P$, also affected by perturbations, via hyperinterpolation and some of its variants. The open-source Matlab software used in the numerical tests is available at the author's homepage.

Figures (7)

• Figure 1: cubature nodes (and weights) on a spherical triangle lifted from the projected elliptical triangle.
• Figure 2: The 380544 nodes of a rule of PI-type with ADE=8, on a spherical polygon approximating South and North America, before Caratheodory-Tchakaloff compression. The computation of this formula requires about 10 seconds.
• Figure 3: Triangulation of a spherical polygon (967 spherical triangles) and cubature rule of PI-type with ADE=8, 81 points, after Caratheodory-Tchakaloff compression (magenta). The compression cputime requires approximatively 3.5 seconds.
• Figure 4: cubature nodes of a formula of PI-type having ADE equal to 10, on a coarse approximation of Australia using 169 points of ${\mathbb{S}}_2$. To this purpose, we determine a triangulation of such a spherical polygon in 167 spherical triangles, obtained a first rule of PI-type with 82413 nodes, and then a compressed one with 121, with moments error of $\approx 5\cdot 10^{-15}$. The computation of the Caratheodory-Tchakaloff formula took takes 6 seconds.
• Figure 5: cubature relative errors, in semilogarithmic scale, obtained by approximating $I_{{\cal{P}}}(f_k)$, for $k=1,2,\ldots,6$, by rules with ADE equal to $1,2,\ldots,16$.

Theorems & Definitions (2)

• Remark 1
• Remark 2