Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Smoothed aggregation — where it appears

A multigrid hierarchy built by partitioning the unknowns into aggregates, then smoothing the interpolation that partition defines with one Jacobi sweep. The aggregation is read from the same strength graph classical algebraic multigrid uses, and the sweep is what turns a piecewise-constant coarse space into a multigrid method.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

75 aggregates over 225 unknownsthe matrix chose thismean extent along y3mean extent along x1points adopted by pass two15no coordinate enters the methodand the shape follows the coupling

Aggregating what the matrix calls strong

The depth phase measured every method it had on the 45°-rotated anisotropic operator — 0.784, 0.883, 0.844 — and diagnosed the failure as being in the discretisation rather than in the hierarchy. Smoothed aggregation is the standard answer to anisotropy. It returns 0.789.

iterative · Anisotropy -213284300.250.50.751rotation of the anisotropy (degrees)factor / couplingusableconvergence factoraxis couplingdiagonal couplingthe standard answer, and the anglefactor at 0°0.19factor at 45°0.8axis ÷ diagonal coupling at 45°2the hierarchy reads the matrixand the matrix lost the direction

How much direction there was to lose

At 45° the nine-point stencil hands smoothed aggregation the same wrong hierarchy at every anisotropy — six strong neighbours per interior point, 121 aggregates, the identical partition from ε = 10⁻⁴ to 0.099. The convergence factor that one hierarchy produces runs from 0.802 to 0.581 over the same range.

iterative · Anisotropy

Named alongside it

The objects these essays reach for when they reach for this one.

AnisotropyOperator complexityRotated anisotropyStrength of connectionAlgebraic multigridConvergence rateDiscretisationGrid alignmentInterpolation weightsNear-null spaceSemi-coarseningWeighted jacobi

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