Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Anisotropy — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    rung 1 · iterative
  2. 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.

    rung 2 · iterative
  3. 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.

    rung 3 · iterative

All ladders