Reduced-wall order study

Reduced-wall order study

The wall eigenmode basis (Wall eigenmodes, #473) can represent the vessel current in any number of modes. This page is about how many, and which — the representation order M_repr = (M_1, …, M_G) that VEST-Tokamak/vfit#10 asks for, implemented in vaft.validation.wall_reduction (#494).

Three layers, kept apart on purpose:

Metrics only, no verdicts: tolerances are the study’s and are reported with the result.

The reduced circuit

With the projected operators of the basis the reduced wall obeys

L_r da/dt + R_r a = -M_r dI_coil/dt,        I_w = V a

integrated by the same routine as the full wall (vaft.process.electromagnetics.solve_eddy_currents), so the only difference between the two solutions is the retained subspace. Keeping every mode reproduces the 950-loop solve to 1e-12; the error of any truncation is the basis span, not the closure.

from vaft.process import wall_modes as wm
from vaft.validation import wall_reduction as wr

system = wr.wall_system(ods)                # R, L, source coupling, basis, PF drive
keep = wm.select_by_score(system["basis"], wm.mode_scores(system["basis"], system["R_mat"],
                          system["M_mat"], system["L_mat"], drive=system["drive"],
                          time=system["time"])["output_weight"], 76)
ops = wm.reduced_operators(system["basis"], system["R_mat"], system["M_mat"], system["L_mat"], keep)
a, I_w = wm.solve_reduced_eddy(ops, system["drive"], system["time"], V=system["basis"].V(keep))

Which modes: rankings, not the spectrum

mode_scores returns one value per mode under four rankings — tau (the decay time), drive_gain (quasi-static amplitude a unit source ramp excites, times observability; needs no drive), response_energy (rms projected amplitude under a drive) and output_weight (energy times observability). select_by_score keeps the top M across segments; allocate_per_segment grows a per-segment allocation greedily until a dissipation or output tolerance is met, adding modes where the remaining error is largest.

On the packaged wall (shot 39915, its PF programme, 62 probes and 11 flux loops), relative error of the wall term at the probes:

Decay time alone is a poor guide: the driven response is not sparse in eigen coordinates, and a whole-wall eigenbasis behaves the same as the segment-wise one, so the penalty is intrinsic to eigen coordinates rather than to the segmentation.

Moment patterns: the enrichment the contract does not yet include

The response to the coils lives in the block Krylov space span{R⁻¹M, (R⁻¹L)R⁻¹M, …} — the coil-controllable subspace — and moment_patterns returns an R-orthonormal basis of it, block by block, built as a block Arnoldi iteration in the R inner product. The first block is the resistive limit, the wall current a constant ramp settles into; each further block is the next inductive correction (the transfer function’s moments at zero frequency).

Ten drive-independent vectors do what 76 eigenmodes do, and 76 do what no eigen truncation reaches. These patterns are global, not segment-tagged, so they are outside the reduced-wall contract (vfit #8); the study reports them as the alternative the downstream EFIT issues can decide on. combined_operators projects the wall onto any R-orthonormal basis, eigen or not.

Running the study

observation = wr.observation_set(ods, n_coils=system["n_coils"])   # probes, loops, boundary ring, grid
rows = wr.order_convergence(system, observation=observation)        # every (rule, order, drive)
order = wr.representation_order(rows, {"probe": 0.01, "flux_loop": 0.01, "grid_psi": 0.01})

order_convergence scores the drive-dependent rankings on the PF programme and then tests every selection on three drives — the programme, the loudest coil alone, and a step on it — so a row for step says how a selection made for the programme transfers to a transient it never saw. Rows carry the Euclidean and dissipation-weighted current error, the error per segment, the relative error at each observation class, the probe peak error and the cost.

Two plots draw the rows: plot_passive_structure_overview_wall_reduction (error against retained order, one panel per metric) and plot_passive_structure_field_wall_reduction (the wall’s flux on the equilibrium region — full, reduced or their difference).

Against the measurements

result = wr.experimental_comparison(ods, {"tau_19": wm.select_slowest(system["basis"], 19),
                                          "moments_30": wm.moment_patterns(system["R_mat"],
                                                        system["M_mat"], system["L_mat"], 3)})

This runs the #190 plasma-free benchmark for the full PF-only wall and for each reduced wall over the same window, and reports two blocks per model: measurement (residual against the data) and reduction (distance from the full model at the same channels). On shot 39915 the full wall improves the coil-only residual by a median 78 % over the 73 usable channels, and a converged reduction improves it by the same 78 % to within a tenth of a point (moments_30, 0.5 % from the full wall term at the probes): the residual that remains is the vessel model’s, not the truncation’s. The nineteen slowest modes are the counter-example rather than a second instance of it: they reach only 64 %, and sit 34 % from the full wall term at the probes over this benchmark’s 25 ms window — the 28 % in the table above is the same rule and the same metric scored over the whole record instead, so the two are not in conflict. The separation is a claim about converged reductions, not about reduction in general.

find_plasma_free_shots classifies database shots as plasma_free, plasma, undriven or daq_missing from their plasma-current and coil peaks, with the evidence, so dedicated vacuum shots can be brought into the comparison; the plasma-free interval of a mapped shot is judged by vaft.validation.vacuum_benchmark.plasma_free_interval as for any other case.