energy-gap
At its defaults it draws the classical interpolation's energy against the minimum, as the operator turns. Four quantities against the angle of the strong direction. The ratio of the classical interpolation's energy to the least any interpolation on its own sparsity pattern can have is 1.0000 on the aligned operator and 1.2029 at 45°. The three convergence factors below it are what each interpolation achieves.
energy-gap is one function in lib/figures/emin.js —
interpolation as a minimisation — where the classical formula already was the answer. 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.
Four quantities against the angle of the strong direction. The ratio of the classical interpolation's energy to the least any interpolation on its own sparsity pattern can have is 1.0000 on the aligned operator and 1.2029 at 45°. The three convergence factors below it are what each interpolation achieves.
eps: 0.001
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.
Four quantities against the angle of the strong direction. The ratio of the classical interpolation's energy to the least any interpolation on its own sparsity pattern can have is 1.0000 on the aligned operator and 1.2029 at 45°. The three convergence factors below it are what each interpolation achieves.
eps: 0.1
The arguments are the ones A rate that does not notice the size 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.
Four quantities against the angle of the strong direction. The ratio of the classical interpolation's energy to the least any interpolation on its own sparsity pattern can have is 1.0097 on the aligned operator and 1.2230 at 45°. The three convergence factors below it are what each interpolation achieves.
eps: 0.05
The arguments are the ones A rate that is known in advance 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.
Four quantities against the angle of the strong direction. The ratio of the classical interpolation's energy to the least any interpolation on its own sparsity pattern can have is 1.0029 on the aligned operator and 1.2129 at 45°. The three convergence factors below it are what each interpolation achieves.
eps: 0.01
The arguments are the ones The best approximation there is 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.
Four quantities against the angle of the strong direction. The ratio of the classical interpolation's energy to the least any interpolation on its own sparsity pattern can have is 1.0001 on the aligned operator and 1.2047 at 45°. The three convergence factors below it are what each interpolation achieves.
eps: 0.02
The arguments are the ones The same problem on a coarser grid 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.
Four quantities against the angle of the strong direction. The ratio of the classical interpolation's energy to the least any interpolation on its own sparsity pattern can have is 1.0005 on the aligned operator and 1.2068 at 45°. The three convergence factors below it are what each interpolation achieves.
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.
7 distinct claims across 6 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 large enough for the rotated case to separate and small enough to solve densely
a sweep that starts aligned and leaves the grid
an anisotropy between a ratio and none
and is above the minimum once the operator is rotated
LU is for square matrices
the classical interpolation is the minimiser on the aligned operator
with the minimiser converging faster there
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 70
of 151 generators —
55 print a residual and
15 are exempt with a published reason;
81 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 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 factorisingA rate that is known in advance
On the model problem, Jacobi contracts by cos(π/(n+1)) per step, Gauss–Seidel by its square, and optimally relaxed SOR by a number given in closed form. Three rates, all known before anything runs, and all measurable against what runs.
Eigenvalues, singular values, rankThe best approximation there is
The error of the best rank-k approximation is not bounded by the next singular value. It is equal to it. That is an unusually sharp theorem, and it makes the theorem itself usable as an independent check on the computation.
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 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 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.
Iterating, instead of factorisingThe formula that was already optimal
Ask for the interpolation that minimises the energy of its own columns and the answer is the classical AMG formula — to zero at every row of the one-dimensional Laplacian, and to four digits in two dimensions. On the operator rotated to 45° the two part company, and the gap between them is a diagnostic that needs no reference solution.
Iterating, instead of factorisingThe 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.