Semi convergence — where it appears
Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.
A parameter that counts steps
The regularisation field's knob is a positive real number chosen by one of three rules. The iterative field's is an integer nobody called a knob — where to stop. On the same problem the best step is 20 and the best λ is 0.025, and they reach 0.1426 and 0.1406.
Four knobs and one floor
A truncation, a Tikhonov parameter, a step count and a randomised rank, on one problem with an answer that is known. Their best errors are 0.1445, 0.1406, 0.1426 and 0.1449 — a spread of 3% across four methods that share no arithmetic.
The basis decides what a filter is
The vocabulary of regularisation is spectral — a method keeps a component or discards it, and the weights are a function of the singular value. Row-normalising a symmetric blur so that it preserves a constant makes it 8.6% asymmetric, and that is enough to move GMRES's weights from 7·10⁻¹⁴ off a function of σ to 4.4·10⁻².
The step that stops mattering
Regularise the problem the iteration has built rather than the problem it was given, and the error curve stops turning. The unregularised run ends 1,127 times above its own best; the same run with a penalty inside it ends 1.000000000003 times above.
Named alongside it
The objects these essays reach for when they reach for this one.
Filter factorsIll posed problemIterative regularisationTikhonov regularisationKrylov subspaceReorthogonalisationRitz valuesStopping criterionArnoldi iterationConjugate gradientsDiscrepancy principleEigenvectors