Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Discretisation error — where it appears

The part of a computed answer's error that comes from replacing a continuous problem by a finite one, before any noise or rounding enters. It shrinks as the grid is refined, and on an ill-posed problem the refinement that shrinks it also admits components the data cannot support.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

1216202428323640444810⁻¹110¹10²grid points nrelative error against the continuous signalbest truncation, 0.117best grid: n = 26best K = 28with noisenoise-freeno λ anywhere in this curvebest grid, error0.13best truncation, error0.12κ on the best grid61grid where it has doubled34every point is an unregularised solvethe grid chose the truncation

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.

regularisation · Regularisation 00.250.50.75100.51xuexact & tunedupwindcentralone of these is exacttuned, worst nodal error2.4·10⁻¹⁷upwind, worst nodal error0.14central, points outside [0, 1]16exact at every nodeand only at the nodes

The diffusion that makes the answer exact

Upwinding adds h/2 of artificial diffusion. Central differencing adds none. Add ε·ξ·Pe with ξ = coth(Pe) − 1/Pe and the computed solution is the exact one at every grid point, to 2.4·10⁻¹⁷ — at every Péclet number, on the problem it was derived from and on no other.

iterative · Convection

Named alongside it

The objects these essays reach for when they reach for this one.

Artificial diffusionBoundary layerCondition numberConvection diffusionDeconvolutionDiscretisationFilter factorsIll-posed problemM-matrixManufactured solutionPeclet numberQuadrature

All concepts