In numerical analysis, Lebedev quadrature, named after Vyacheslav Ivanovich Lebedev, is an approximation to the surface integral of a function over a three-dimensional sphere. The grid is constructed so to have octahedral rotation and inversion symmetry. The number and location of the grid points together with a corresponding set of integration weights are determined by enforcing the exact integration of polynomials (or equivalently, spherical harmonics) up to a given order, leading to a sequence of increasingly dense grids analogous to the one-dimensional Gauss–Legendre scheme. The Lebedev grid is often employed in the numerical evaluation of volume integrals in the spherical coordinate system, where it is combined with a one-dimensional integration scheme for the radial coordinate. Applications of the grid are found in fields such as computational chemistry and neutron transport.
Angular integrals The surface integral of a function over the unit sphere,
I [ f ] = ∫ d Ω f ( Ω ) = ∫ 0 π sin ( θ ) d θ ∫ 0 2 π d φ f ( θ , φ ) , {\displaystyle I[f]=\int \mathrm {d} \Omega \,f(\Omega )=\int _{0}^{\pi }\sin(\theta )\,\mathrm {d} \theta \int _{0}^{2\pi }\mathrm {d} \varphi \,f(\theta ,\varphi ),}
is approximated in the Lebedev scheme as
I ~ [ f ] = 4 π ∑ i = 1 N w i f ( θ i , φ i ) , {\displaystyle {\tilde {I}}[f]=4\pi \sum _{i=1}^{N}w_{i}f(\theta _{i},\varphi _{i}),}
where the particular grid points and grid weights are to be determined. The use of a single sum, rather than two one-dimensional schemes from discretizing the θ and φ integrals individually, leads to more efficient procedure: fewer total grid points are required to obtain similar accuracy. A competing factor is the computational speedup available when using the direct product of two one-dimensional grids. Despite this, the Lebedev grid still outperforms product grids. However, the use of two one-dimensional integration better allows for fine tuning of the grids and simplifies the use of any symmetry of the integrand to remove symmetry-equivalent grid points.
Construction
The Lebedev grid points are constructed so as to lie on the surface of the three-dimensional unit sphere and to be invariant under the octahedral rotation group with inversion. For any point on the sphere, there are either five, seven, eleven, twenty-three, or forty-seven equivalent points with respect to the octahedral group, all of which are included in the grid. Further, all points equivalent under the rotational and inversion group share the same weights. The smallest such set of points is constructed from all six permutations of (±1, 0, 0) (collectively denoted as a1), leading to an integration scheme
I ~ 6 [ f ] = A 1 ∑ i = 1 6 f ( a i 1 ) , {\displaystyle {\tilde {I}}_{6}[f]=A_{1}\sum _{i=1}^{6}f(a_{i}^{1}),}
where the grid weight is A1. Geometrically these points correspond to the vertices of a regular octahedron when aligned with the Cartesian axes. Two more sets of points, corresponding to the centers and vertices of the octahedron, are all eight uncorrelated permutations of 1 / 3 ( ± 1 , ± 1 , ± 1 ) {\displaystyle 1/{\sqrt {3}}(\pm 1,\pm 1,\pm 1)} (denoted as a3), and all twelve permutations of 1 / 2 ( ± 1 , ± 1 , 0 ) {\displaystyle 1/{\sqrt {2}}(\pm 1,\pm 1,0)} (denoted as a2). This selection of grid points gives rise to the scheme
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