Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

The V-cycle — where it appears

One pass down and up a multigrid hierarchy, smoothing at each level and correcting from the one below. Its convergence factor is a constant independent of the mesh on the model problem, which is what makes the whole method linear.

Named by 6 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 10¹10¹.³10¹.⁶10¹.⁹⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰¹10².²00.250.50.751grid points nresidual reduction per stepJacobiGauss–SeidelV-cycleV-cycle spread, 8× in size0.003Jacobi at n = 1270.99work exponent, fitted0.079the dashed curve is cos(πh), Jacobi's closed formthe flat line is the whole method

A rate that does not notice the size

The V-cycle reduces the residual by a factor of ten a cycle at fifteen points and at a hundred and twenty-seven. Jacobi on the same four problems goes from 0.981 to 0.9978, climbing towards one. One of those is a constant and the other is an exponent, and that is the whole distinction the field turns on.

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 θx (frequency across x)θy0π/2π0π/2πthe coarse grid'sunder 0.2under 0.40under 0.60under 0.80under 0.95under 1.01damping per sweeptwo routessmoothing factor, scanned1closed form130×30 frequency cellsthe marker is the mode nothing removes

A direction the smoother cannot see

Give the Laplacian a strong direction and multigrid stops working — from 0.2016 a cycle to 0.9565 — with every component unchanged and the condition number identical to twelve digits. The problem did not get harder. The link between the method's two halves broke.

iterative · Anisotropy 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 00.7853981.57082.356193.1415900.250.50.751frequency θdamping |g(θ)|the oscillatory half →⅓ — the symmetric optimumno convectionwith convectionits imaginary parta modulus, not a valuesmoothing factor at this ω0.71best over every ω0.71the symmetric operator's, at ω = 2/30.33the imaginary part does not depend on ωso no ω removes it

A smoother that stops being one

Weighted Jacobi's smoothing factor on the convection–diffusion operator is a function of the cell Péclet number and nothing else — identical to eight digits at five grid sizes at matched Pe. It is 0.3335 at Pe = 0.016, exactly 1/√2 at Pe = 1, and 5.2190 at Pe = 7.8, where the sweep amplifies the modes it exists to remove.

iterative · Multigrid

Named alongside it

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

Smoothing factorFourier modesGalerkin coarse operatorProlongationRestrictionAnisotropyCondition numberFive-point stencilJacobi iterationM-matrixWeighted jacobiBoundary layer

All concepts