vcycle-2d
At its defaults it draws v-cycle convergence factor against problem size, ε = 1. Three flat curves of convergence factor against the number of unknowns on a logarithmic axis. At small anisotropy one sits near one and the others near a tenth.
vcycle-2d is one function in lib/figures/grid2d.js —
two dimensions — where the galerkin identity stops being an identity. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
Three flat curves of convergence factor against the number of unknowns on a logarithmic axis. At small anisotropy one sits near one and the others near a tenth.
eps: 0.001
The arguments are the ones A direction the smoother cannot see passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Three flat curves of convergence factor against the number of unknowns on a logarithmic axis. At small anisotropy one sits near one and the others near a tenth.
eps: 1
The arguments are the ones The coarse problem is a different problem passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Three flat curves of convergence factor against the number of unknowns on a logarithmic axis. At small anisotropy one sits near one and the others near a tenth.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
9 distinct claims across 3 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
every method returns a factor at k = 15 — asserted 3 times
line relaxation converges at k = 15 whatever the anisotropy — asserted 3 times
and at a real anisotropy the point smoother is far behind it
the grid size is one less than a power of two
the repaired factor does not move with the grid size
Against the rule
It draws a decomposition and prints its residual. It calls
solve2d,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
Across the library: the rule bites on 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
Iterating, instead of factorisingA 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.
Iterating, instead of factorisingCoarsening 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.
Iterating, instead of factorisingSmoothing 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.
Iterating, instead of factorisingThe 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.
Iterating, instead of factorisingThe rate the condition number predicts
Conjugate gradients converge at a rate governed by the square root of the condition number. That is a bound rather than an estimate, it is provable, and it is loose enough that provisioning iterations from it wastes nine out of ten.