Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Coarse fine splitting — where it appears

The choice of which unknowns survive to the next level of an algebraic hierarchy. It is made from the matrix entries through a strength test, and everything the hierarchy later does inherits whatever that test got wrong.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

the grid the operator came froman edge is a coupling the matrix calls strongkeptinterpolatedwhat the entries decidedcoupling ratio, x against y0.001strong couplings across x0strong couplings along y110rows kept or dropped whole1111×11 grid, θ = 0.25the coarse grid, from the matrix alone

The coarse grid the matrix chooses

Given a tridiagonal matrix and no information about a grid, the coarsening keeps every other point and derives the weights ½, 1, ½ — the operators the geometric method was handed. Given the anisotropic operator, it discovers semi-coarsening, in the right direction, without a coordinate.

iterative · Algebraic multigrid 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 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.

Algebraic multigridGalerkin coarse operatorStrength of connectionAnisotropyOperator complexitySemi-coarseningConvergence rateFill-inGraph laplacianGrid complexityInterpolation weightsM-matrix

All concepts