Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Smoothing factor — where it appears

How much of the high-frequency error a relaxation removes in one sweep. It is computed in closed form over the Fourier modes and it is what decides whether a two-grid method converges at a mesh-independent rate.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

00.250.50.751-1-0.500.51mode frequency θ / πdamping factor per sweeppredictedmeasured±0.333a coarse grid seestwo routes to one factorsmoothing factor, scanned0.33smoothing factor, closed form0.33worst mode disagreement4.4·10⁻¹⁶63 interior points, one sweepthe left-hand end is what the coarse grid is for

The error smoothing cannot reach

One weighted Jacobi sweep multiplies every mode of the error by a number, and the number is a sine. Half the modes are cut by three or better, and the other half come back at 0.999 — which is not a failure of the method but the fact the whole of multigrid is built 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 10⁻⁴10⁻³10⁻²10⁻¹100.250.50.751anisotropy εsmoothing factor μpoint + fully-line + fullpoint + semi-y⅓, the one-dimensional answertwo routes, three curvesgap between the repairs2.2·10⁻¹⁶scan against closed form2.2·10⁻¹⁶the dashed curve lies on the solid one beneath itone repair, written two ways

Smoothing a whole line at once

Solve every grid line in the strong direction exactly rather than sweeping over it, and the smoothing factor goes from 0.9993 back to 0.3340 — which is the one-dimensional answer, on a problem that is not one-dimensional. The repair replaces one ε in the closed form by a one.

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 00.250.50.75100.250.50.751xuexactcentral differencesupwindthe oscillation is exact‖Ax − b‖/‖b‖ for the central answer4.6·10⁻¹⁸values outside [0, 1]16worst excursion0.52the dashed lines are 0 and 1, which the equation guaranteesno solver was involved

The stencil that is not symmetric

Past a cell Péclet number of exactly one — measured by bisection at 1.0000000000000002 — the central-difference solution of a convection–diffusion problem oscillates from point to point and leaves the interval the equation guarantees, at 16 of 31 grid points. It is the exact solution of its own linear system, to 4.6·10⁻¹⁸. No solver was involved.

iterative · Convection 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.

Fourier modesThe V-cycleWeighted jacobiAnisotropyCondition numberM-matrixConvection diffusionFive-point stencilGalerkin coarse operatorLine relaxationPeclet numberProlongation

All concepts