Reduced stability — where it appears
Named by 9 essays across one field — each of them below, with the objects they name alongside it.
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.
A certificate written in coordinates
Every one-sided reduced model of the convection–diffusion system was stable, because its poles cannot leave the numerical range and the range sat in the left half plane. Rescale the state by a diagonal matrix and nothing about the system changes — not its eigenvalues, not its transfer function, not one pole of any two-sided model — but the numerical range moves. At Péclet 40 a rescaling with condition number 260 pushes its edge to +6.6 and ten placements of nineteen return unstable one-sided models. Projecting in the inner product of a Lyapunov solution puts the certificate back in any coordinates.
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.
Where a ringing mode stops being a pair
A lightly damped mode puts two nearly equal Hankel singular values into balanced truncation, and the reduction error was read as twice the larger of the first pair cut — right to two per cent at damping 0.01, wrong by half at 0.1. Read instead as the pair's two values added, the error is right to within six per cent at every order up to damping 0.05, and the pair's split turns out to be twice the damping. What it cannot do is say when the reading fails. The split predicts nothing: the widest split at damping 0.1 has the best reading. The failures come from somewhere the pair does not look — the other dropped modes' response under the peak, and a cut between two singular values that agree to under one per cent, which left a reduced model with a pole a thousand times closer to the axis than any pole of the system it came from.
A tie is a gap smaller than its neighbours
A balanced truncation that cut between two Hankel singular values agreeing to under one per cent left a reduced model with a pole near the axis and twice the error the σ promised, and the warning drawn from it was a rule: never cut inside a tie. Over 1,452 reductions of twelve oscillators, all ten doubled errors are at ties — but three of them have gaps of about two per cent, which a one-per-cent rule lets through, and what they share is a gap under a fifth of the gaps beside it. Moving every such cut one order down never doubles and halves each doubled error, for a median cost of one per cent. It is not free everywhere: at damping 0.5 it costs up to two and a half times. And the near-integrator was never the harm: 32 reductions have one and 22 of them read their σ correctly. The cheapest test of all is one evaluation of the error at zero frequency, which picks out the ten doubled cuts and nothing else.
The edge sits where the cells meet the flow
On a convection–diffusion mesh graded towards its outflow, the nodal numerical range reaches into the right half plane, and Galerkin reduced models fail. The account was that the edge lives in the cells where diffusion and convection are comparable, and drawing the eigenvector that attains it says the account is right: on every operator whose mesh has a cell Péclet number above one, the eigenvector peaks where that number is between a half and one and a quarter, just downstream of where it crosses one, and the outflow layer's five smallest cells hold under four per cent of it. The other question has a short answer too. A one-sided stencil scaled by a cell width is dissipative in nodal coordinates only if its weights do not grow downstream, and a consistent one must grow there: the only dissipative weight is the constant one, which is wrong by up to four units of the derivative.
A weight is not a location
On a mesh graded towards a convection–diffusion operator's outflow, the edge of the numerical range in nodal coordinates lives in the cells where diffusion and convection are comparable, and weighting the inner product by cell size everywhere moves it into the left half plane. The prediction was that the weight is only needed where the edge lives: cell sizes over the nodes whose cell Péclet number is between a quarter and two, and one elsewhere, would make the range dissipative at every grading and repair every failed model. It does the opposite. From a grading of ten the band-only weight's edge is positive, and from thirty it is further right than the nodal edge it was meant to repair — 144 against 8.9 at Péclet 100. Its edge eigenvector leaves the transition and sits at the weight's own step, where it rises from 0.16 to 1; and its models fail on thirteen placements to the nodal twenty-three. The cell sizes work because they are the one diagonal weight that makes the centred convection stencil skew, row by row, and any other — stepped, ramped over half the mesh, or the reverse — leaves a symmetric part where it departs.
Named alongside it
The objects these essays reach for when they reach for this one.
Convection diffusionNon-normalityTransfer functionGalerkin projectionNumerical rangeBalanced truncationLyapunov equationMass matrixPetrov–GalerkinRational krylovA-priori boundHankel singular values