Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Near null space — where it appears

The directions an operator barely changes, which a coarse space has to represent because a smoother cannot reduce them.

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

081624324000.250.50.7511.25angle of the strong direction (degrees)ratio, and convergence factorenergy ratio of oneenergy ratioclassicalminimiserwith the constraintwhere the formula is optimalenergy ratio at 0°1energy ratio at 45°1.2classical rate at 45°0.35the minimiser's rate there0.28a ratio of exactly one while the assumption holdsand a diagnostic when it stops

The formula that was already optimal

Ask for the interpolation that minimises the energy of its own columns and the answer is the classical AMG formula — to zero at every row of the one-dimensional Laplacian, and to four digits in two dimensions. On the operator rotated to 45° the two part company, and the gap between them is a diagnostic that needs no reference solution.

iterative · algebraic multigrid 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

Named alongside it

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

Algebraic multigridAnisotropyStrength of connectionA normCoarse grid correctionConstrained minimisationEnergy minimisationGalerkin operatorInterpolationInterpolation weightsOperator complexityRotated anisotropy

All concepts