Interpolation — where it appears
Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.
A model with no matrices behind it
Twenty-four numbers — values of a transfer function at twenty-four points — produce a sixth-order model of a forty-state system that passes through every sample to 10⁻¹² and matches the function it never saw to 10⁻⁹. Noise of 10⁻¹⁰ on those numbers takes the rank decision's gap from 10⁸ to 4.
The formula that was already optimal
Ask for the interpolation that minimises the energy of its own columns and the answer is the classical AMG formula — to zero at every row of the one-dimensional Laplacian, and to four digits in two dimensions. On the operator rotated to 45° the two part company, and the gap between them is a diagnostic that needs no reference solution.
An error estimate made of samples
A Loewner model built from samples of a transfer function comes with no error bound, and the number every code reads to choose its order — the pencil's next singular value — is the obvious stand-in. On four systems sampled at 48 points it is the error of the order-k model for the first two orders, within a factor of 2.5, and then falls away from it by about a decade an order: by the fifth it is optimistic by 109, 181 and 457, and on one system by 6,700 at the sixth. An AAA fit of the same degree to the same samples has an error within a factor of six of the Loewner model's at every order, so the shortfall is not the projection's. The model's misfit at its own samples is optimistic the same way. What works is more samples: the error at the 47 midpoints between the samples is within a factor of 2.2 of the true error at every order on every system, and the four midpoints nearest the poles alone within a factor of three.
A pole between the samples
A Loewner model needs a check on its error, and the one predicted to be cheap is four points chosen where the model disagrees most with the model one order higher. With exact samples it is better than predicted: on four systems the true error at those four points is the model's error within two per cent at every order, and the disagreement alone, with no sample at all, within three per cent at every order but the last. With noisy samples the checks keep working on every model that has no pole between the samples, and seventeen of thirty-two models do. Past the order the noise allows, the projection puts a real pole inside the sampled interval, so the error is infinite at one point and small a few per cent of a gap away, and every finite set of checks reports a number between 130 and 49,000 times too small. The test that catches every such model costs nothing: the model's own poles.
Named alongside it
The objects these essays reach for when they reach for this one.
Data-driven realisationLoewner matrixTransfer functionMcMillan degreeNoise floorRational approximationA normAdaptive interpolationAlgebraic multigridAnisotropyCoarse grid correctionConstrained minimisation