Rational krylov — where it appears
Named by 7 essays across one field — each of them below, with the objects they name alongside it.
Exact at the points that were named
Balanced truncation asks for nothing and bounds everything, at a cost no large model can pay. The other kind of reduction asks for r numbers, costs r solves, is exact at every one of them — and bounds nothing anywhere else. That trade is the whole of large-scale model reduction.
A basis that is the same subspace and not the same thing
The interpolation conditions are conditions on a subspace, so any basis of it will do. The one a derivation writes down reaches a condition number of 7.7·10⁹ by its eighth vector, and the rate at which it gets there is set by a number the user chose with no information.
Interpolating at the model’s own poles
One choice of interpolation points is not arbitrary — the mirrored poles of the model about to be built. It is a fixed point rather than a guess, and when it is reached it beats a method costing O(n³) — by 0.4 per cent, which is the honest size of the whole contest.
A model that cannot be run
A stable system, reduced by matching its transfer function at three points exactly, comes back with a pole in the right half plane at four of nineteen placements — and matches at all three points to 1.3·10⁻¹⁴ while doing it. The construction did what it promised.
Half the conditions and a certificate
A two-sided interpolatory reduction of a stable convection–diffusion system came back unstable at four placements of nineteen. The one-sided reduction built from the same kind of solves came back stable at all 152 placements measured across eight Péclet numbers, and not by luck: every pole of a Galerkin model lies inside the numerical range of the operator, whose edge here is exactly the diffusion term's largest eigenvalue, −0.986 at Péclet 10. The price is the slope conditions, and spent as six one-sided shifts instead of three two-sided ones it beats the stable two-sided model at seven placements of ten.
The inner product the mesh already computed
A Galerkin reduced model is certified stable when the operator's numerical range sits in the left half plane, and the certificate belongs to the coordinates. The coordinates that broke it before were a constructed rescaling. Grade a convection–diffusion mesh towards its outflow boundary layer — the grading anyone resolving the layer would choose — and nodal values break it too: the range's edge is past zero at a ratio of ten and reaches +8.9 at a thousand, and at Péclet 100 seven reduced models come back unstable. Weight the projection by the cell sizes the discretisation already computed and every model at every grading is stable, with the range's edge back at −0.098.
The smaller cell downstream
Nodal values on a convection–diffusion mesh graded towards its outflow layer put the numerical range in the right half plane, and Galerkin reduced models then fail. A code that grades towards the layer usually also upwinds, and upwinding adds diffusion. It does not give the certificate back: at Péclet 100 and a grading of a thousand the range's edge falls from 8.94 to 6.78 and stays positive, and 18 reduced models fail where 23 did. Each row of the upwind stencil's symmetric part sums to half the difference between the inverse widths of the cell after the node and the cell before it, so wherever the cells shrink downstream the dissipation upwinding adds is outweighed by the coupling to the smaller cell next door.
Named alongside it
The objects these essays reach for when they reach for this one.
Transfer functionMoment matchingPetrov–GalerkinBalanced truncationConvection diffusionNon-normalityReduced stabilityGalerkin projectionNumerical rangeCondition numberFlop countHermite interpolation