Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Coarse fine splitting — where it appears

2 essays name this object, across one field. What follows is each of them, and 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

Named alongside it

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

Algebraic multigridGalerkin coarse operatorStrength of connectionAnisotropyFill-inGraph laplacianGrid complexityInterpolation weightsM matrixOperator complexityProlongationRotated anisotropy

All concepts