Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Moment matching — where it appears

Building a reduced model by projecting onto subspaces of shifted solves, so that it matches the original exactly at chosen points. It costs a few solves and bounds nothing between the points, which is the opposite trade from balanced truncation.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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

10⁻¹110¹10²10³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵s, on the real axis|H − Hᵣ| ÷ |H|exact where askedpoints4conditions bought8worst at a point5.3·10⁻¹⁶worst away from one8.4·10⁻⁴4 points, 8 conditionsand no bound in between

Exact at the points that were named

Balanced truncation asks for nothing and bounds everything, at a cost no large model can pay. The other kind of reduction asks for r numbers, costs r solves, is exact at every one of them — and bounds nothing anywhere else. That trade is the whole of large-scale model reduction.

reduction · moment matching 2345678110²10⁴10⁶10⁸10¹⁰vectors in the basisκ₂ of the basisas derivedsolves spread outone subspace, two spanning setseight moments at one point7.7·10⁹eight points, spread1growth per vector661/u4.5·10¹⁵the same subspaceand only one of them usable

A basis that is the same subspace and not the same thing

The interpolation conditions are conditions on a subspace, so any basis of it will do. The one a derivation writes down reaches a condition number of 7.7·10⁹ by its eighth vector, and the rate at which it gets there is set by a number the user chose with no information.

reduction · moment matching 01234567891011121310⁻¹110¹10²10³iterationinterpolation point σa fixed point that is a conditionorder4iterations13σ against −λ(Aᵣ)2.8·10⁻¹²H₂ error4.3·10⁻⁵it stopped movingand the condition holds there

Interpolating at the model’s own poles

One choice of interpolation points is not arbitrary — the mirrored poles of the model about to be built. It is a fixed point rather than a guess, and when it is reached it beats a method costing O(n³) — by 0.4 per cent, which is the honest size of the whole contest.

reduction · moment matching

Named alongside it

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

Rational krylovBalanced truncationCondition numberFlop countHermite interpolationKrylov subspacePetrov–GalerkinTransfer functionA-priori boundBasisFixed point iterationH2 norm

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