Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Operator complexity — where it appears

The total nonzeros over all levels of a multigrid hierarchy divided by the fine matrix's, which is what an optimal method's cost is measured in. It is the number that says whether a multigrid method is actually optimal, since a hierarchy whose coarse levels are dense costs more than the fine solve it replaced.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

01234567110¹10²levelstored entries per rowalgebraicgeometrictwo complexities, one hierarchygrid complexity3.1operator complexity18geometric, for comparison1.5a ring plus random chords — nothing is anywherelevel 6 is 100% dense

A hierarchy with no grid behind it

On a graph Laplacian the algebraic V-cycle converges at 0.199 a cycle, its grid complexity is an unremarkable 3.05, and its operator complexity is 17.7 — one level of forty-one unknowns is entirely dense. The number people quote is the one that does not measure the work.

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 -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 10⁻¹100.20.40.60.8strength threshold θresidual reduction per cycleθ = εsemi-coarseningkept whole rowsfull coarseningone parameter, two methodsbest factor above ε0.053best factor below ε0.1the ratio across the switch231×31 anisotropic operatora switch, not a dial

The switch does not know which side is better

The strength threshold moves the coarsening from full to semi at θ = ε exactly, at every anisotropy. Which of the two converges faster is a separate question with a separate answer, and it changes sign between ε = 0.33 and ε = 0.34 — where nothing whatever happens to the switch.

iterative · Algebraic multigrid

Named alongside it

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

Strength of connectionAlgebraic multigridAnisotropyRotated anisotropyCoarse fine splittingConvergence rateGalerkin coarse operatorSemi-coarseningSmoothed aggregationDiscretisationFill-inGraph laplacian

All concepts