Near-algebraic Tchakaloff-like quadrature on spherical triangles

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Abstract

We present a numerical code for the computation of nodes and weights of a low-cardinality positive quadrature formula on spherical triangles, nearly exact for polynomials of a given degree. The algorithm is based on subperiodic trigonometric Gaussian quadrature for planar elliptical sectors and on Caratheodory–Tchakaloff quadrature compression via NNLS.

Introduction

In this note we develop an algorithm for the computation of nodes and positive weights of a quadrature formula on spherical triangles, which is nearly exact for algebraic polynomials of a given degree n on the 2-sphere, and whose cardinality does not exceed the dimension of the corresponding polynomial space, namely dim(Pn3(S2))=(n+1)2. One of our main goals is also to provide easily usable numerical codes.

Indeed, despite of the relevance of spherical triangles in the field of Geomathematics, the topic of numerical quadrature on spherical triangles, starting from a classical paper by K. Atkinson in the ’80s [1], has received some attention in the literature of the last decades, with however a substantial lack of easily available numerical software (at least to our knowledge); cf. [2], [3], [4], [5] and [6, §7.2] for an overview. Some of the methods have been developed in the framework of numerical PDEs on the sphere, cf. e.g. among others [7], [8]. Notable exceptions are [9], [10], where however the algorithms are not tailored to polynomial spaces, the problem being discussed in the framework of local RBF interpolation on scattered data.

In Section 2 we describe our algorithm (implemented in Matlab) for the construction of the quadrature formula, based on subperiodic trigonometric Gaussian quadrature for planar elliptical sectors [11] and on Caratheodory–Tchakaloff quadrature compression via NNLS [12], and in Section 3 we present some numerical tests.

Section snippets

Quadrature on spherical triangles

We shall concentrate on spherical triangles T=ABC with centroid (A+B+C)A+B+C2 at the north pole (with no loss of generality, up to a suitable rotation) that are not “too large” in the sense that they are completely contained in the northern hemisphere, and do not touch the equator. Then we can write the surface integral of a continuous function f(x,y,z) on such a spherical triangle in Cartesian coordinates (cf. e.g. [13]) Tf(x,y,z)dσ=Ωf(x,y,g(x,y))1g(x,y)dxdy,where g(x,y)=1x2y2 and Ω is

Numerical examples

In this section we present several numerical tests, in order to assess the quality of our compressed near-algebraic quadrature formulas on three spherical triangles with different extension (the key parameter being ρ in (4)). In particular, the triangle of Table 3 is the sphere octant with vertices (1,0,0),(0,1,0),(0,0,1). In Table 1, Table 2, Table 3 we show the cardinalities of the basic and of the compressed formula, the corresponding compression ratio and the CPU time needed for the

Acknowledgments

Work partially supported by the DOR funds and the biennial project BIRD 192932 of the University of Padova, Italy, and by the INdAM-GNCS . This research has been accomplished within the RITA “Research ITalian network on Approximation” and the UMI Group TAA “Approximation Theory and Applications”.

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