smoothing-gain
At its defaults it draws the same aggregates, with and without the prolongator smoothing (ε = 1). Relative residual against V-cycle on a logarithmic vertical axis. Both hierarchies are built from the identical aggregation of the same matrix. Without the smoothing sweep the convergence factor is 0.8133; with it the factor is 0.3529, reached in 18 cycles. The smoothed hierarchy stores 1.34 times the fine matrix against 1.26.
smoothing-gain is one function in lib/figures/aggregation.js —
smoothed aggregation — the sweep that is the method, and the angle that defeats it. 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.
Relative residual against V-cycle on a logarithmic vertical axis. Both hierarchies are built from the identical aggregation of the same matrix. Without the smoothing sweep the convergence factor is 0.8133; with it the factor is 0.3529, reached in 18 cycles. The smoothed hierarchy stores 1.34 times the fine matrix against 1.26.
k: 31
The arguments are the ones A hierarchy with no grid behind it 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.
Relative residual against V-cycle on a logarithmic vertical axis. Both hierarchies are built from the identical aggregation of the same matrix. Without the smoothing sweep the convergence factor is 0.8133; with it the factor is 0.3529, reached in 18 cycles. The smoothed hierarchy stores 1.34 times the fine matrix against 1.26.
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.
4 distinct claims across 2 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.
a grid the hierarchy has room to coarsen on
an anisotropy the operator is defined at
and costs a denser hierarchy
the smoothing more than halves the convergence factor
Against the rule
It draws a decomposition and prints its residual. It calls
saSolve,
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 66
of 131 generators —
51 print a residual and
15 are exempt with a published reason;
65 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 hierarchy with no grid behind it
On a graph Laplacian the algebraic V-cycle converges at 0.199 a cycle, its grid complexity is an unremarkable 3.05, and its operator complexity is 17.7 — one level of forty-one unknowns is entirely dense. The number people quote is the one that does not measure the work.
Iterating, instead of factorisingAggregating what the matrix calls strong
The depth phase measured every method it had on the 45°-rotated anisotropic operator — 0.784, 0.883, 0.844 — and diagnosed the failure as being in the discretisation rather than in the hierarchy. Smoothed aggregation is the standard answer to anisotropy. It returns 0.789.
Iterating, instead of factorisingThe 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.
Iterating, instead of factorisingThe 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.
Sparsity, and what elimination costsThe factor is not sparse
A sparse matrix has a factor that is not sparse, and the gap between them is the entire reason iterative methods exist. The entries elimination creates can be counted before any arithmetic runs, from the graph alone.