Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Galerkin projection — where it appears

Reducing a matrix or a model by restricting it to a subspace and testing against the same subspace, so the reduced matrix is VᵀAV for an orthonormal basis V. It keeps whatever the numerical range guarantees and gives up the extra conditions a second subspace would buy.

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

Also named here as numerical range — the same set of essays touches all of them, so they are one junction rather than several.

10⁻¹110¹10²s, the smallest interpolation pointrightmost pole of the reduced model−10−10+1+10unstable above zeronumerical range edge, -0.986system's abscissa, -3.49two-sided, order 3one-sided, order 3one-sided, order 6Pe 10, nineteen placementssystem's rightmost eigenvalue-3.5edge of the numerical range-0.99two-sided, order 3: unstable placements4one-sided, order 3: unstable placements0one-sided, order 6: unstable placements0signed logarithmic axisshaded: poles in the right half plane

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.

reduction · Reduced stability edge of the numerical rangespectrum, -3.49zero37.9-0.986-3.496.59unstable one-sided models, of nineteen placements051015-3-2-1.5-1-0.500.511.522.53cond 1.4·10⁶cond 112t, the rescaling exponent — the k-th state is multiplied by γ to the power tk; t = 1 symmetrisesorder 1order 2order 3order 6the transfer function is the same at every tonly the inner product changes

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.

reduction · Reduced stability

Named alongside it

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

Convection diffusionNon-normalityNumerical rangePetrov–GalerkinReduced stabilityTransfer functionBalanced truncationLyapunov equationMoment matchingRational krylovSimilarity transformation

All concepts