Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Iterative regularisation — where it appears

Using the number of iterations of a Krylov method as the regularisation parameter, so the knob is an integer rather than a real number.

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

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

015304560759010512010⁻²10⁻¹110¹steprelative sizeleast error: 20discrepancy stop: 7errorresidualthe knob is an integerleast error, at step20error there0.14error at step 1206the residual falls at every stepthe error turns and keeps rising

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.

combination · iterative regularisation 00.250.50.75110⁻¹110¹10²10³fraction of the method's own rangerelative errorfloor 0.141truncation KTikhonov λCGLS steprandomised rankfour methods, one floortruncation K0.14Tikhonov λ0.14CGLS step0.14randomised rank0.14four knobs from four fieldsand one obstruction underneath them

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.

combination · parameter choice

Named alongside it

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

Filter factorsSemi convergenceTikhonov regularisationConjugate gradientsDiscrepancy principleIll posed problemKrylov subspaceParameter choiceRandomised SVDReorthogonalisationRitz valuesSingular values

All concepts