Grundmann-Möller n-dimensional Simplex Quadrature

Description

This package calculates integrations points and weights for numerical quadrature (i.e. integrating a function) over an n-dimensional simplex. It supports arbitrary (odd) degrees of accuracy, and supports arbitrary floating-point types.

Sample usage

julia> using GrundmannMoeller

julia> # Obtain quadrature scheme (2 dimensions, degree 5; degree must be odd)

julia> scheme = grundmann_moeller(Float64, Val(2), 5);

julia> length(scheme.weights)
10

julia> # Apply scheme

julia> vertices = [[0,0], [1,0], [0,1]];

julia> f(x) = x[1]*x[2]
f (generic function with 1 method)

julia> q = integrate(f, scheme, vertices)
0.04166666666666668

julia> q ≈ 1/24
true

Reference

Create a quadrature scheme

    grundmann_moeller(::Type{T}, ::Val{D}, degree::Int)

• T: desired floating point type
• D: dimension
• degree: desired polynomial degree of accuracy (must be odd)

Evaluate an integral (apply the scheme)

    integrate(fun, scheme, vertices::AbstractVector)
    integrate(fun, scheme, vertices::AbstractMatrix)

• fun: integrand, should accept an SVector as argument
• scheme: quadrature scheme
• vertices: vertices of the simplex

The vertices need to be passed either as a vector-of-vectors or as a matrix. In the first case, there need to be D+1 points with D coordinates each. In the second case, the matrix needs to have size D×D+1.

Provenance of this package

This package is a Julia translation of one of the algorithms of the Python quadpy package by Nico Schlömer.

The algorithm itself was published by

• A. Grundmann, H. M. Möller, "Invariant integration formulas for the n-simplex by combinatorial methods", SIAM J. Numer. Anal. 15, 282-290 (1978), DOI 10.1137/0715019.

Related work

See SimplexQuad.jl for a simplex quadrature package that uses a different algorithm.