Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Prolongation — where it appears

The operator carrying a correction from a coarse level of a hierarchy up to a fine one. Its weights are where a multigrid method's convergence rate is mostly decided, and its transpose is usually taken as the restriction.

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

15731pointsfinest gridone unknown — the recursion bottoms out in a divisionthe coarse operator, two ways‖RAₕP − A₂ₕ‖/‖A₂ₕ‖10⁻¹⁸unknowns / finest grid1.7cycle cost, in fine sweeps12each coarse point reaches three fine ones½, 1, ½ — and the restriction is its transpose

The same problem on a coarser grid

Restriction, the coarse operator and interpolation are three matrices with nine distinct entries between them. Two of the three are each other's transpose, and their product with the fine operator is the coarse discretisation exactly — not approximately, entry for entry, at every level.

iterative · Multigrid -0.0833-0.167-0.0833-0.1671-0.167-0.0833-0.167-0.0833R · A · P, normalised to a unit centrestored entries per rowlevel 0 · 31×314.87/rowlevel 1 · 15×158.22/rowlevel 2 · 7×77.37/rowlevel 3 · 3×35.44/rowlevel 4 · 1×11.00/rowstill a stencil, still annihilates a constantentries in an interior row9weight outside the 3×30row sum0the isotropic model problemnine, at every level below the first

The coarse problem is a different problem

In one dimension the Galerkin coarse operator is the coarse discretisation, entry for entry — this site asserted it. In two dimensions a five-point operator produces a nine-point coarse one, so the recursion solves a different discretisation at every level below the first, and converges at 0.20 a cycle regardless.

iterative · Multigrid point · strong in y0.951point · strong in x0.956y-line · strong in y0.037y-line · strong in x0.967semi-y · strong in y0.107semi-y · strong in x0.9661.00 — no convergenceone problem, seen from two sidesy-line, aimed0.037y-line, turned sideways0.97the ratio2631×31 grid, twelve V-cyclesthe direction is in the code, not in the problem

Coarsening in one direction only

Leave the smoother alone and halve only the strong direction, and the smoothing factor is 0.3340 — identical to line relaxation's, at every anisotropy and every weight, to twelve digits. The convergence factors are then a factor of three apart, and at 45° both repairs fail outright.

iterative · Anisotropy 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

Named alongside it

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

Galerkin coarse operatorRestrictionThe V-cycleAnisotropyFourier modesM-matrixSemi-coarseningSmoothing factorAlgebraic multigridCoarse fine splittingCondition numberFive-point stencil

All concepts