Where a ringing mode stops being a pair

A mode that rings is counted twice found balanced truncation’s bound — twice the sum of the discarded Hankel singular values — loose on lightly damped oscillators by factors up to five, where on a heat model it had been attained exactly. The reason was in the σ themselves. A mode whose poles are $-\zeta\omega \pm i\omega\sqrt{1 - \zeta^2}$ contributes two Hankel singular values that agree to about ζ, and removing the mode costs about what removing one of them would. So the essay proposed a reading in place of the bound: at an order that removes whole modes, charge only the first pair cut, at twice its larger σ.

The reading was within two per cent of the error at every even order at damping 0.01, within five at 0.03, and at 0.1 it “falls apart”: 1.24, 1.13, 0.54, 1.17 and 1.14 times the error at orders two to ten. The essay’s open question was where between those dampings the reading stops working — “whether the crossing can be predicted from the σ alone, from how closely consecutive values agree.”

Eleven dampings, five orders, three readings #

The system is the same: six oscillators in parallel at frequencies spread geometrically from one to thirty, each driven and observed through seeded weights, at a common damping ratio ζ. The sweep takes ζ at eleven values from 0.005 to 0.5 and reduces to two, four, six, eight and ten states, each of which removes whole modes when the σ pair.

Fifty-five reductions need fifty-five infinity norms, and each is computed by sweeping the error’s frequency response for its largest value. The full system’s response comes from its two-by-two blocks in closed form, the reduced model’s from a complex solve of at most ten unknowns, and the grid starts at zero frequency and puts forty points inside every resonance’s half-power width before refining the best of them. It agrees with the general routine the earlier essays used to eight digits on every case checked. It needed one repair on the way, and the repair is part of the result: the first version of the grid began at a frequency of 0.03 and reported, at order six and damping 0.1, an error of 3.59, where the true largest value — 7.13 — sits at zero frequency, in a part of the axis the grid did not visit. The earlier essay’s 0.54 at that order was right, and the reason for it is the second half of this page.

Three readings of the error are taken from the first dropped pair, $\sigma_{r+1}$ and $\sigma_{r+2}$: twice the larger, the earlier essay’s reading; twice the smaller; and the two added.

A pair costs its two σ #

At light damping the error of a reduction is one resonance. The reduction to two states keeps the slowest mode and drops five; at damping 0.01 the error’s frequency response is a row of sharp peaks, one at each dropped mode, and the tallest is at the second mode’s frequency, 1.97, at a height of 125.7. The dropped pair there is 63.9 and 62.7. Their sum is 126.6, within seven tenths of a per cent; twice the larger is 127.8, within 1.7 per cent; twice the smaller is 125.4, within three tenths.

Across the whole sweep the pattern is the same. Twice the larger σ overstates the error, by up to ten per cent at damping 0.05 — 1.098 at order two — and twice the smaller understates it by about as much, 0.879 at order four. Turn the dial to the sum and the reading is within 5.3 per cent of the error at every order and every damping up to 0.05, on twenty-five reductions whose bound over error runs from 2.0 to 5.1. The figure at the top of the page is that dial position.

This is the earlier essay’s other finding generalised. It measured the last mode alone and found that removing it cost almost exactly its two σ added — bound over error 2.02 and 2.00 at the lightest dampings, the bound being twice that sum. The sum reading says the same thing about whichever mode the cut removes first: a lightly damped mode costs the sum of its pair, and the larger σ doubled is a slight overstatement of it that grows with the damping.

That growth has a simple size. For an isolated mode the two σ of its pair differ by twice its damping: at order ten, which drops only the fastest mode, the split over twice the damping is 1.01, 1.01, 1.01, 1.00 and 0.96 at dampings from 0.005 to 0.05. The same holds at orders two and eight to within a few per cent. So a pair of Hankel singular values carries its mode’s damping in the ratio of its two members, and twice the larger σ overstates the pair’s sum by exactly half that ratio less one — by ζ. The earlier reading was a reading of the sum with an error the size of the damping built into it.

Order four is the exception in this figure, and it is the first sign of what the σ cannot say. From damping 0.03 its “pair” is not one mode’s: the third and fourth modes contribute four σ within a few per cent of each other at light damping — 24.4, 23.9, 23.2 and 22.9 at 0.01 — and as the damping rises the cut at four falls between values belonging to different modes, whose ratio has nothing to do with either mode’s damping. Its split reaches 0.43 at damping 0.1.

The split cannot say when the reading fails #

So the earlier essay’s question can be put to the data directly: does a wider split mean a worse reading? It does not. At damping 0.1 order four has the widest split of any order, 0.43, and its sum reading is 3.6 per cent from the error; order two has a split of 0.15 and its reading is 15.9 per cent off. The points in the figure climb with the damping, because both the split and the departure grow with it, but at a given damping the split does not order them, and across dampings it does not separate the readings that work from the ones that do not.

The reason is that the split measures one mode and the departure is about the others. A pair’s split is set by its own damping; whether the pair reads the error depends on what else is at the frequency where the error peaks.

What the reading misses is the background under the peak #

At the peak frequency the error is the dropped mode’s resonance plus the off-resonance response of every other dropped mode — their tails. At light damping a resonance is a hundred times taller than its neighbours’ tails at the same frequency, because its height grows as $1/\zeta$ and their tails do not. As the damping rises the resonance falls and the tails do not, so the background under the peak grows roughly in proportion to ζ. At order two it is 1.1, 2.2, 4.4, 6.8, 12, 17 and 28 per cent of the error at dampings from 0.005 to 0.1.

The figure puts the reading’s departure against that background, and on all thirty cases measured the departure is the smaller: the sum reading is never further from the error than the other modes’ response under the peak. The tails can add to the peak or partly cancel it, depending on the signs of the modes’ weights, so the departure is often much smaller than the background — at order four it stays under five per cent while its background reaches 37 — but it never exceeds it.

That is a usable statement and it is not one the σ can supply. The background is a property of the frequency response at one frequency, and the Hankel singular values are a property of the Gramians, which integrate over all frequencies. A reading of the error from the σ is accurate to within the background, and to know the background the response has to be evaluated, at which point the error could be measured directly.

The failure that is not gradual #

One case in the sweep departs abruptly rather than gradually. At order six the sum reading is 1.022 at damping 0.05, and then 0.518 at 0.07 and 0.523 at 0.1 — the reading falls to half the error between two neighbouring dampings. The earlier essay’s 0.54 at this order was the same case.

The figure shows what happened. At damping 0.1 the error’s response has the usual row of resonances, and the tallest of them, at the fourth mode’s frequency, is 3.6 — close to the pair’s sum of 3.73, as the reading says. But at zero frequency the error is 7.13, and it is not a resonance of the full system: it is a spike narrower than the plotted axis can show, about $10^{-4}$ wide, sitting on a shoulder of 3.2. At $10^{-4}$ the error is 5.9; at $10^{-3}$ it is 3.3.

A spike that narrow is a pole that close. Every pole of the full system is at least ζ from the imaginary axis — 0.1 at this damping. Every reduced model in the sweep has its slowest pole near that line except one: the reduction to six states at dampings 0.07 and 0.1, whose slowest poles are $1.9 \cdot 10^{-4}$ and $1.2 \cdot 10^{-4}$ from the axis, four hundred to eight hundred times closer than anything in the system it was reduced from. The reduced model is stable, as balanced truncation guarantees, and it is almost an integrator.

The σ explain this one. Balanced truncation’s stability guarantee needs the kept and the dropped singular values to differ, $\sigma_r > \sigma_{r+1}$, and it says nothing about how stable the result is when they nearly do not. At order six the cut falls between values that agree to within 3.4 per cent at damping 0.005 and to 0.2, 0.4, 0.7 and 0.8 per cent at 0.05, 0.07, 0.1 and 0.15. A cut through a near-tie leaves a reduced model whose state basis is nearly arbitrary in the tied pair’s span, and here the arbitrariness put a pole near the origin. It is the earlier essay’s second open question — modes in a cluster — met from the other side: the cluster is the third and fourth modes, whose σ nearly coincide, and the order that cuts between them is the one that fails.

But the gap at the cut is a warning, not a predictor. At damping 0.05 the gap is the smallest of all, 0.24 per cent, and the reduced model’s slowest pole is 0.030 — a little slow, nowhere near the axis — and the reading there is 1.022. At 0.15 the gap is 0.8 per cent and the pole is back at 0.17. What the σ can say is that a cut through a near-tie is a cut at which the reduced model is not well determined, so that a near-integrator is possible; they cannot say whether one appears.

What a near-integrator does to a simulation #

A pole $1.2 \cdot 10^{-4}$ from the axis is not an inaccuracy that shows up in a frequency plot unless someone looks at the bottom of the axis. It shows up in time. A mode with that real part decays with a time constant of about 8,300 time units, against a slowest time constant of ten in the system being modelled: after a step input, the reduced model keeps drifting towards its steady state for a thousand times longer than the real system takes to settle, and the steady state it drifts to is wrong by the 7.13 the spike reports. A simulation run for a hundred time units — ten of the system’s own slowest time constants — sees the reduced model still moving and calls it a transient.

That is the sense in which the reduction failed. Its frequency response is right to the pair’s sum everywhere except a band of width $10^{-4}$ around zero, and a model that cannot be run is the same failure at its extreme: a reduction of a stable system that interpolates its data to $10^{-14}$ and comes back with a pole in the right half plane. Balanced truncation cannot go that far — its stability is a theorem — but the theorem is about the sign of a real part and not its size, and a stability margin of $10^{-4}$ is a model that behaves in time almost as badly as one with none. Half the conditions and a certificate found the kind of certificate that gives a margin rather than a sign: a numerical range in the left half plane bounds every reduced pole away from the axis by the range’s own distance from it. The Gramians give no such bound, and the near-tie is where the absence shows.

The Gramians themselves are not to blame either. They are computed here by a Kronecker solve and agree with their defining equations to the rounding level; the σ that tie are tied in the system, not in the arithmetic. A condition number that is not the model’s found that the Gramians’ condition number says nothing about how reducible a model is, and here the nearest thing to a warning the Gramians carry is not their conditioning but the ratio of two consecutive σ — which is a statement about the system, read off a computation that is accurate.

What the earlier question gets for an answer #

The question was whether the crossing — the damping past which a pair stops reading the error — can be predicted from the σ alone. The answer has three parts and only one of them is yes.

The reading itself is better than the one proposed. The pair’s two σ added read the error to within six per cent at every order up to damping 0.05, and twice the larger was that reading plus the damping. That is a reading a code can use where the heat model’s formula could not be: exact at the points that were named set balanced truncation’s cost against interpolation’s, and a reading accurate to six per cent is enough to choose an order from a tolerance before any reduced model is built. The bound remains what it was, a factor of two to five above the error on these systems, and the bound that is known in advance remains the only number here that is guaranteed.

The gradual failure is not in the σ. Why a Gramian can be truncated at all derived how fast the σ fall from the spectrum alone, and that derivation is about how many states matter, not about what any one of them costs at any one frequency. The failure is the background the other dropped modes put under the error’s peak, it grows with the damping, and it bounds the reading’s departure on every case measured. The σ do not contain it, because they summarise the Gramians and the background is a value of the response at one frequency. The state that is removed is not a mode found the same separation from the other direction: a balanced state is not a mode, and a σ is not a resonance, even when at light damping the two line up.

The abrupt failure has a warning in the σ and no prediction. A cut through a near-tie can leave a reduced model with a pole near the axis, and then the largest error is at a frequency no dropped mode has. Every cut in the sweep with a gap above ten per cent kept its slowest pole within a factor of three of the full model’s. The two cuts that did not were both at gaps under one per cent — and so was one cut that was fine.

What one system does not show #

Six modes, one geometric spacing of their frequencies, one seeded set of weights and one damping common to every mode. A system whose modes are damped differently would split its pairs differently and put its backgrounds in different places; the statements above about the split are about one mode’s own damping and should survive that, while the ones about the background are about the spacing and the weights. The frequencies here are a factor of about two apart; closer modes put more background under every peak and would move the gradual failure to lighter damping. The near-integrator at order six is one event at two dampings, and how often a cut through a near-tie produces one, rather than whether it can, is not measured. And every number here is an infinity norm, the error at the worst frequency; an error averaged over frequency would weight the zero-frequency spike by its width, which is tiny.

Still open: the tie’s other side, a reading with its background, and closer modes #

Which side of a tie to cut. At order six the near-integrator came from cutting between two nearly equal σ. Cutting at five or seven instead keeps or drops both members of the tie. Whether either of those reductions keeps its poles near the full model’s at every damping — and what the extra or missing state costs in error — is one more pair of orders on the same sweep, and would turn the warning into a rule: never cut inside a tie.

A reading that carries its own background. The background bounded the reading on every case, and it is one evaluation of the response at the peak frequency of the dropped pair’s mode — a frequency the σ do not give but the mode does. A reading that adds the dropped modes’ response at that one frequency to the pair’s sum might be within a per cent where the sum alone is within sixteen, at the cost of one evaluation per reduction.

Modes in a cluster, on purpose. The third and fourth modes here are a cluster by accident of their weights. A system built with several modes at nearly the same frequency, as a periodic structure has, would make every cut a cut near a tie. The prediction with a sign is that the near-integrators become common there, not rare, and that the bound’s over-statement — which on this system never fell below two at any order that removes a whole mode — falls towards one, because a cluster’s σ no longer pair.