Discretisation — where it appears
Named by 15 essays across 3 fields — each of them below, with the objects they name alongside it.
The size the rank does not notice
Sample a kernel block at 32, 64, 128 and 256 points a side and it needs five columns, five, five and five. Sample the touching block next to it at the same four sizes and it needs nine, eleven, twelve and thirteen. Same kernel, same accuracy, one number and a logarithm.
The grid was the first filter
A continuous deconvolution discretised on n points and solved with no regularisation at all is not unregularised. Its error against the continuous signal is least at 24, 26 and 34 points for noise of 1%, 0.1% and 0.01% per sample — beside best truncations of 24, 28 and 32 components on a 64-point grid — and within 4 to 16 per cent of their error. The grid's own filter factors sum to n exactly and fall through a half at k = n. Choosing the grid was choosing a truncation, before anybody chose a λ.
A better discretisation is a weaker filter
A coarse grid's error has two sources — how well the discrete operator approximates the integral, and how well the grid's function represents the answer — and the grid essay could not separate them. Changed one at a time they separate: integrating the kernel against the hat functions takes a fifth off the 12-point error, reading the answer as a cubic spline takes 15 per cent more, and both roughly double the condition number on every grid. At 0.1% noise the spline discretisation's unregularised solve on 26 points reaches the best truncation of a 64-point grid to 0.3%. At 1% it is worse than the crude grid.
Where the grid hands over to λ
An unregularised solve on a coarse grid comes within a tenth of the best Tikhonov answer on a fine one, and the pair of a grid and a λ was left unmeasured. Measured, the two do not trade. On every grid up to the best unregularised one no λ helps at all. On every grid of 40 points and more the best λ is the same to within a quarter of a decade — 3.2·10⁻² at 1% noise per sample, 10⁻³ at 0.01% — and the 96-point grid with it beats the best coarse grid by 7, 9 and 13 per cent. The grids between the two, given their own λ, land between them.
The grid on which the discretisation stops mattering
Without regularisation, integrating the blur's kernel against cubic splines beat sampling it at 0.1% noise and lost to it at 1%. Give each discretisation its own best λ on every grid and the difference shrinks to nothing where grids are fine — 0.17, 0.28 and 0.20 per cent apart on 96 points at the three noise levels, with every discretisation choosing the same λ — and stays at 17 to 18 per cent on 16 points. The choice between them is a choice of how coarse a grid can be: at 0.1% noise the integrated discretisations reach the fine-grid answer on 26 points and the sampled one needs 40.
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.
A corner the penalty can afford
Every smooth reading of the deconvolution's grid needed about forty points and then stopped improving, and the step was the suspect. Give the step one coefficient of its own and forty-eight points reach an error of 0.0070 at 0.1% noise, against 0.118 for the best smooth reading on ninety-six — the step was most of the error. But the same step given two coefficients recovers half as well, and given a doubled node at each edge it recovers worse than no breakpoint at all, while representing the signal to 0.07%. What decides is what the penalty is charged for the corner, and whether the data can say where it is.
The data count their dimensions, not the step's
Every grid in the deconvolution essays was chosen with the answer in hand, and so was every λ. From the data alone, the discrepancy principle's worst draw is within 16 per cent of the oracle on every grid from 16 points to 96; generalised cross-validation is better on the median draw and, on grids of thirty points and more, has draws thousands of times worse. And the data can say how many dimensions they carry — about 20, 25 and 29 at three noise levels, one number once the grid exceeds it — but not how many more the step needs: the grid that count chooses is 14 to 19 per cent worse than forty points at the lower two.
The minimum on the right
Generalised cross-validation's worst draws on a fine grid were all one mistake: a second dip in its function at λ near zero, deeper than the real minimum. The proposed repair was a residual threshold, one number, refusing any λ whose residual falls too far below the real minimum's. Measured over 528 draws, a threshold of one half still lets two hundredfold misses through; only the extreme value, which is no threshold at all but the rule 'take the rightmost local minimum', removes all eleven. It costs nothing on the coarse grids where the dip is the right answer, and on the collection's own problem over five thousand draws it turns 243 tenfold misses into 17.
More samples take the floor and leave the dip
GCV's catastrophic misses on fine grids were blamed on squareness: on an n × n system the residual and n − t both reach zero as λ does, and their ratio can dip there. With more samples than unknowns neither reaches zero, and the prediction was that the dip would be gone by construction. Half of it is. The minimum at the floor of the scale, 21 draws in 240 on square systems, is gone at every ratio. The interior dip is not — 28, 20, 13, 8 and 4 draws at one to four samples per unknown — and at four per unknown one draw still misses the oracle by 877 times.
An estimate that shares the dip's luck
GCV's interior dips form where the residual per remaining degree of freedom is low, and the proposal was a guard that estimates the noise from the least-squares residual and refuses a minimum whose residual is implausibly small. Over 960 draws holding 45 dips it refuses none, and picks plain GCV's minimum on every draw. At the dip ρ is 0.84 with the true noise and 0.98 with the estimate, because the estimate is made from the residual at the dip's own end of the scale and inherits its luck. Told the true noise instead, the guard can refuse 14 dips only by refusing 78 real minima. Handed the same estimate, the discrepancy principle misses by 570,000 times.
An estimate that shares nothing with the fit
GCV's guard failed because the noise estimate it read came from the residual its dips live in, and the proposal was to estimate the noise from samples held out of the fit instead. Over the same 960 draws the held-out guard refuses none of the 45 dips at the 1% threshold — fewer than the guard told the true noise, which refuses three — because a quarter of the samples gives its threshold too few degrees of freedom to refuse anything. Loosened to 10%, it refuses sixteen dips and 170 real minima. But the discrepancy principle, which missed by 570,000 times when told the least-squares estimate, never misses by more than eighteen told the held-out one, at every ratio and not only where the held-out set is the larger. The held-out estimate is the worse estimate, biased upward by up to half. What makes it the better input is that it is independent of the residual it is compared with.
A different equation on every grid
Upwinding is the exact discretisation of a convection–diffusion problem with diffusion ε + h/2, entry for entry, at a relative difference of between 0 and 1.26·10⁻¹⁶ on every mesh from 15 points to 511. The equation it is exact for is chosen by the mesh and not by ε — the added diffusion is 0.01563 on a 31-point grid whether ε is 0.2 or 0.001.
A parameter that is also a price
ξ = coth(Pe) − 1/Pe is the fraction of h/2 that makes a boundary-layer solution exact at every node. On a problem with no layer in it, the error the same scheme commits is ξ times upwinding's — 0.2511 against a ξ of 0.2504, 0.7461 against 0.7448 — so the number that buys the exactness is also the invoice.
How much direction there was to lose
At 45° the nine-point stencil hands smoothed aggregation the same wrong hierarchy at every anisotropy — six strong neighbours per interior point, 121 aggregates, the identical partition from ε = 10⁻⁴ to 0.099. The convergence factor that one hierarchy produces runs from 0.802 to 0.581 over the same range.
Named alongside it
The objects these essays reach for when they reach for this one.
RegularisationDeconvolutionTikhonov regularisationDiscrepancy principleParameter choiceFilter factorsGeneralised cross-validationIll-posed problemCondition numberArtificial diffusionConvection diffusionDiscretisation error