ORTHOCUB: integral and differential cubature rules by orthogonal moments

TL;DR

ORTHOCUB is a computational package that efficiently computes integral and differential functionals on multivariate polynomial spaces using orthogonal moments, avoiding matrix inversion issues.

Contribution

It introduces a numerical package that leverages orthogonal polynomial moments and algebraic cubature for stable, matrix-free computation of multivariate integral and differential functionals.

Findings

No conditioning issues due to matrix inversion

Efficient computation via moments and small dense matrix-vector products

Open-source Matlab and Python implementations

Abstract

We discuss a numerical package, named ORTHOCUB, for the computation of linear functionals of both integral and differential type on multivariate polynomial spaces. The weighted sums corresponding to such integral and differential cubatures are implemented via orthogonal polynomial moments and auxiliary near-minimal algebraic cubature in a bounding box, with no conditioning issue since no matrix inversion or factorization is needed. The whole computational process indeed reduces to moment computation and dense matrix-vector products of relatively small size. The Matlab and Python codes are freely available, to be used as building blocks for integral and differential problems.