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Freezing Calder\'on: Robust Preconditioning for Maxwell Scattering under Shape Uncertainty

Paul Escapil-Inchauspé, Carlos Jerez-Hanckes

arXiv:2609.30559v1math.NAcs.NA

TL;DR

EFIE Calderón preconditioning can be costly because assembling its dual barycentric operator can dominate total assembly. The paper freezes that operator at a nominal shape and transports the pairing across connectivity-preserving perturbations, obtaining uniform conditioning bounds and substantial per-realization savings.

  • Problem

    EFIE discretizations suffer mesh-dependent conditioning, while assembling the dual barycentric operator for Calderón preconditioning can dominate computational cost.

    Source: §1 · pp. 1–2

  • Method

    The method assembles the dual-mesh operator at a nominal geometry, reuses it across connectivity-preserving shape perturbations, and transports the duality pairing by an exact degree-of-freedom permutation.

    Source: §1 · p. 2

  • Results

    29–31× per new realization in the dense implementation and roughly twofold against published compressed-assembly timings; the spectral-conditioning bound is uniform in mesh size and stochastic approximation dimension.

    Source: §6 · p. 29

  • Takeaways & Limitations

    Freezing amortizes the dual barycentric assembly across realizations, although the reported speedups depend on implementation.

    Source: §6 · p. 29

    Wavenumber freezing is outside the geometric theorem and is characterized only numerically; at κsolve = 0.03125, frozen and fresh solutions agree to 2.8×10−5 relative ℓ2 norm against 10−8.

    Source: §5.6 · p. 22

Abstract

from arXiv · show

Calder\'on preconditioning restores mesh-independent conditioning for boundary-element discretizations of the Maxwell electric field integral equation, but assembling the dual barycentric operator can dominate the computational cost. We show that this preconditioner can be assembled once at a nominal geometry and reused across connectivity-preserving shape perturbations. Exact combinatorial alignment of the RWG-BC degrees of freedom transports the duality pairing, leaving only the dual-mesh operator frozen. Under uniform discrete-stability and geometric operator-continuity assumptions, a bi-parametric operator-preconditioning analysis yields a spectral-condition-number bound uniform in the mesh size and, after tensorization, in the stochastic-Galerkin dimension. Numerical experiments on smooth and non-smooth scatterers show that the frozen preconditioner closely tracks freshly assembled Calder\'on preconditioners over substantial shape variations. A Monte Carlo campaign on a NASA almond, whose admissibility is screened before each shape is constructed so that no realization is rejected, separates the two strategies by less than the variation between realizations, and mesh refinement there leaves the frozen iteration count unchanged. Reuse yields roughly 30x measured speedup per new realization in our dense implementation; a cost-model estimate based on published compressed-assembly timings gives roughly 2x.

1 Introduction

The paper asks whether an expensive Calderón preconditioner can be reused across geometrically perturbed EFIE systems, rather than rebuilt for every query. It freezes the dual-mesh operator at a nominal geometry, transports the duality pairing by an exact degree-of-freedom permutation, and analyzes and tests reuse under shape perturbations.

1 Introduction

  • Calderón preconditioning addresses EFIE systems whose condition numbers grow under mesh refinement, while assembling the dual BC operator can cost several times more than assembling the primal matrix.The BC dual discretization uses a barycentric refinement, which has six times as many triangles as the primal mesh.Source: §1 · pp. 1–2
  • Earlier updating and Krylov-recycling methods target algebraically changing matrices on a fixed discrete space, whereas this work reuses an operator across different surfaces via an exact degree-of-freedom correspondence.Prior cost-reduction approaches instead lower the per-solve cost for a single fixed geometry and wavenumber.Source: §1 · p. 2
  • The bi-parametric framework bounds degradation from the nominal preconditioner by a factor (1 + ν)/(1 − ν), provided the geometric family admits the required operator-norm estimate.The paper focuses on Calderón preconditioning because it is expensive to assemble but performs well in iteration count and spectral properties.Source: §1 · pp. 2–3

Contributions

  • A nominal Calderón preconditioner can be reused across shape perturbations by permuting the degree-of-freedom correspondence, which transports the pairing exactly while the dual-mesh operator remains frozen.This transport is exact for admissible perturbations that preserve mesh connectivity; the frozen dual-mesh operator itself does not equal a freshly assembled one.Source: §1 · p. 2, Contributions · p. 3
  • The analysis treats geometric perturbations as (h, ν)-perturbations and bounds the spectral condition number uniformly in mesh size and, after tensorization, stochastic approximation dimension.The uniformity relies on discrete-stability and geometric operator-continuity assumptions; GMRES convergence additionally requires a coercivity-type hypothesis not established here.Source: §1 · p. 2, Contributions · p. 3
  • Tests cover smooth sphere and ellipsoid geometries, a Fichera corner, and a 100-sample NASA almond campaign; perturbing a reentrant singularity is identified as difficult.Freezing across wavenumber rather than shape is outside Theorem 8, and its behavior is characterized only numerically.Source: Contributions · p. 3

2 Mathematical framework

The mathematical framework begins with the EFIE’s mesh-dependent conditioning, then describes Calderón preconditioning and its RWG–BC discretization. It applies bi-parametric operator-preconditioning results to geometric perturbations, under uniform discrete-stability and operator-continuity assumptions.

2.1 The electric field integral equation

  • RWG Galerkin EFIE matrices are dense and indefinite, with condition numbers classically growing as (κh)^−2 as h → 0; the nominal wavenumber is assumed not to be an interior Maxwell resonance.Uniform discrete inf-sup constants are a separate standing assumption, not a consequence of nominal non-resonance alone.Source: §2.1 · p. 4

2.2 Calder´on preconditioning

  • Calderón preconditioning uses the EFIE operator with a dual BC discretization to obtain a stable pairing; the dual operator’s barycentric assembly can dominate total assembly cost.For the sphere test at h = 0.1 and κ = 2, the dual operator accounts for 430 s of 442 s total assembly time, or 97%.Source: §2.2 · pp. 4–5

2.3 Bi-parametric operator preconditioning

  • The framework bounds a spectral condition number, but that bound alone does not guarantee GMRES convergence; the separate linear-convergence result requires a coercivity-type hypothesis.The spectral condition number agrees with the Euclidean condition number only for normal matrices, and the paper does not claim its stated mesh-uniform bound for the Euclidean counterpart.Source: §2.3 · p. 5
  • The spectral-condition-number bound depends on nominal and dual discrete stability, which the paper assumes uniformly across mesh levels rather than proving in this Petrov–Galerkin configuration.The RWG–BC pairing’s h-uniformity is attributed to the dual finite element complex on the barycentric refinement.Source: §2.3 · p. 6
  • The bi-parametric framework represents geometric changes as (h, ν)-perturbations of the EFIE form, with ν proportional to perturbation amplitude and constants uniform in h.It applies to a parametric family when the required operator-norm estimate is available.Source: §2.3 · pp. 5–6

3 Robustness of the frozen preconditioner to shape perturbation

This section establishes robustness of the frozen Calderón preconditioner under connectivity-preserving shape perturbations, using exact RWG-BC alignment to transport the duality pairing. Under uniform discrete-stability and geometric operator-continuity assumptions, tensorization preserves stability bounds uniform in mesh size and stochastic dimension.

3.1 Admissible shape perturbations

  • Small connectivity-preserving deformations keep the mesh connectivity fixed, yielding an exact canonical correspondence between nominal and perturbed RWG-BC degrees of freedom without geometric interpolation or re-meshing.The perturbation framework is uniform over the parameter domain, and its shape maps are assumed to remain bi-Lipschitz diffeomorphisms.Source: §3.1 · pp. 6–7

3.2 A shape-perturbation bound for the EFIE bilinear form

  • Under Assumption G, the EFIE form perturbation is bounded by Cgeo ε ∥uh∥X∥vh∥Y uniformly in y, with a constant independent of h.For C1,1 geometries, Lemma 6 verifies Assumption G when the deformation fields’ W2,∞ norms are in ℓp for some 0 < p < 1.Source: §3.2 · pp. 8–9

3.3 Main theorem

  • With Assumptions S and G and ε < γa0/Cgeo, the frozen-preconditioned perturbed system has a spectral-condition-number bound uniform in h ≤ h0.The sharper estimate is mesh-specific; the second bound is h-independent. A weighted GMRES residual-factor bound is also given, but only under an additional coercivity-type hypothesis.Source: §3.3 · p. 10

3.4 Scope of the theoretical result

  • The condition-number result does not alone establish Krylov convergence, and its GMRES estimate is conditional rather than verified for Maxwell.The theory applies literally to conforming exact-surface discretizations, not the assembled polyhedral matrices; the paper does not provide the required lifting analysis. Its verified geometric-continuity range is C1,1 and its bound is at fixed wavenumber; merely Lipschitz boundaries remain outside that verified range.Source: §3.4 · pp. 10–11

3.5 Bochner-space and stochastic-Galerkin consequence

  • The stochastic-Galerkin robustness bound uses the same perturbation parameter as the deterministic result and is uniform in both mesh size and stochastic approximation dimension.The lifted frozen preconditioner is block-diagonal with repeated physical-space copies, so its dual-mesh ingredients are assembled once for all stochastic modes; a GMRES bound additionally requires a uniform coercivity-type hypothesis.Source: §3.5 · pp. 12–13
  • Tensorization preserves the deterministic stability constants uniformly in stochastic dimension and uses repeated nominal-geometry Calderón blocks, rather than a preconditioner for the mean operator.For nonlinear geometric perturbations, the nominal operator generally differs from the expected operator; extension to first-order sparse BEM moment systems is left for future work.Source: §3.5 · p. 13

3.6 Implementation: DOF alignment via mesh connectivity

  • The nominal-to-perturbed RWG-BC degree-of-freedom matching is constructed from mesh connectivity alone and verified as bijective for each perturbed mesh.RWG functions are matched by primal-edge vertex indices; BC functions use the primal vertices in their supports. The vertex-displacement family preserves connectivity by construction.Source: §3.6 · pp. 13–14
  • The transported duality pairing is an exact identity, independent of vertex positions, when the triangulations share connectivity and the RWG and BC functions satisfy the stated transport hypotheses.For the vertex-displacement perturbations, the element maps are affine bijections when no element degenerates; the RWG Piola transports and connectivity-determined BC coefficients then give the exact pairing transport.Source: §3.6 · pp. 14–15
  • At 30% perturbation, the transported nominal pairing matched the freshly assembled pairing to floating-point assembly accuracy, unlike the geometry-dependent TY Y.The exactness hypotheses hold for the default BEAST.jl RWG normalization and combinatorial BC weights, but not for the area-scaled BC variant.Source: §3.6 · p. 15

4 A practical cost model

The cost model compares fresh assembly and solves with a frozen strategy that amortizes nominal dual-mesh assembly across realizations. Its practical speedup depends on assembly costs and can be smaller with compressed assembly.

4 A practical cost model

  • The frozen strategy assembles the nominal dual-mesh blocks once, then reuses them; its per-query cost retains the primal assembly, pairing assembly, and solve.The measured per-iteration cost is about 3.6 times higher for frozen runs because of storage, not conditioning; lazy permutation removes that artifact.Source: §4 · pp. 15–16
  • The model amortizes nominal assembly across N frozen realizations, while fresh assembly pays the dual-mesh costs for every query.For dense assembly, the dual-mesh operator accounts for 430 of 442 seconds at the stated sphere configuration.Source: §4 · p. 16, Table 1 · p. 16
  • 29–31× is the measured dense-implementation speedup per new realization as the number of frozen realizations grows.Removing the storage artifact raises the estimate to about 36×; neglecting solve cost gives a ceiling near 38×.Source: §4 · p. 16
  • 2.0–2.3× is the cost-model speedup estimated from published H-matrix timings, compared with 4.2–5.9× when solve cost is neglected.Under compression, solving is no longer negligible relative to assembly.Source: §4 · p. 16

5 Numerical experiments

The experiments use RWG/BC boundary elements and GMRES to evaluate the preconditioners on several geometries and perturbation families. They report solver tolerances, reproducibility details, and the residual convention used for the preconditioned runs.

5 Numerical experiments

  • All experiments discretize (4) with RWG/BC elements; the outer solver is flexible, unrestarted GMRES with absolute and relative residual tolerances of 10^-8 and a 1500-iteration maximum.Inner unrestarted GMRES uses the same fixed absolute and relative 10^-8 tolerances and 1500-iteration maximum, without adaptive variation; Section 5.5 is the one exception.Source: §5 · pp. 16–17
  • The test geometries include a sphere, a triaxial ellipsoid, a Fichera corner, and a NASA almond, with perturbations including radial and localized families.Figure 2 illustrates selected perturbed meshes against their nominal outlines; the almond example has a largest displacement of 13.30 mm and a surface mean of 9.75 mm.Source: §5 · p. 17, Figure 2 · p. 18
  • The NASA-almond meshes, field coefficients, amplitude ladder, seeds, timings, and iteration counts are archived for exact reproducibility.Timings depend on the stated software and hardware configuration; iteration counts and residuals are described as insensitive to it apart from floating-point-library and threading effects.Source: §5 · pp. 17–18
  • A 10^-8 stopping tolerance applies to the preconditioned residual, while reported true residuals are at most 1.5 × 10^-6.Except for the smallest-wavenumber mismatch sweep, reported true residuals are at most 6.3 × 10^-7.Source: §5 · p. 17

5.1 Sphere: amplitude sweep

On the sphere, the frozen Calderón preconditioner stays close to a freshly assembled one across the tested amplitude sweep, even as plain GMRES requires far more iterations.

5.1 Sphere: amplitude sweep

  • 11–15 frozen iterations track 10–13 fresh Calderón iterations across nominal amplitudes of 3%–50%, with a gap of at most two iterations at 50%.The maximum realized radial displacement reaches ±17.1% (about 1.7h), and all meshes remain valid.Source: §5.1 · p. 18
  • 272–290 iterations are required by plain GMRES across the same sweep.The table caption identifies the sphere, κ = 2, h = 0.1, and a single realization.Source: §5.1 · p. 18, Table 3 · p. 19

5.2 Sphere: statistical campaign (N = 20 realizations)

Across 20 sphere realizations, frozen iteration counts remain close to fresh counts; separate tests examine perturbation scaling and the mesh-uniform conditioning quantity. The direct tests show linear first-order dependence and closely aligned conditioning curves across mesh levels.

5.2 Sphere: statistical campaign (N = 20 realizations)

  • Across 20 realizations at ε = 3% and 30%, frozen iteration counts differ from fresh by at most one on average and two per realization, versus 272–282 plain iterations.All runs converged, and frozen and fresh solutions agree with the plain-GMRES solution within 2 × 10^-7.Source: §5.2 · p. 19

5.3 Direct test of the perturbation estimate

  • 1.001, 0.999, and 0.999 are the fitted slopes of the perturbation-matrix norm against amplitude at h = 0.2, 0.15, and 0.1, respectively.The logarithmic-scale fit has R^2 > 1 − 10^-6, supporting linear dependence to within 0.2%.Source: §5.3 · p. 19, Figure 3 · p. 20
  • The frozen-preconditioned spectral condition number rises from 1.79 to 2.33, while mesh-level curves agree within 0.8% across the tested amplitudes.This tests the quantity bounded in (25); the coefficient-space plot does not establish the h-uniform trace-space estimate.Source: §5.3 · p. 20, Figure 3 · p. 20

5.4 Mesh refinement at fixed perturbation

At a fixed 30% sphere deformation and κ = 2, frozen-preconditioner iterations remain flat under mesh refinement, unlike plain GMRES, and match freshly assembled preconditioning within one iteration.

5.4 Mesh refinement at fixed perturbation

  • 13 frozen-preconditioner iterations persist from h = 0.2 to 0.075, matching fresh preconditioning within one iteration at every mesh level.The sweep covers 1224 to 8172 degrees of freedom at a fixed sphere deformation of 30%, realization 1, and κ = 2.Source: §5.4 · p. 20, Table 4 · p. 21
  • Plain GMRES iterations rise from 168 to 346 as the mesh is refined, while frozen-preconditioner iterations stay at 13.The table caption describes frozen counts as flat in h and plain GMRES as deteriorating.Source: §5.4 · p. 20, Table 4 · p. 21
  • The flat frozen counts are consistent with the h-uniform bound of Theorem 8.Table 4 also reports converged true relative GMRES residuals in its last three columns.Source: Table 4 · p. 21

5.5 Stochastic Galerkin p-refinement

For the sphere’s stochastic Galerkin system, the block-diagonal frozen preconditioner maintains a constant iteration count as polynomial degree and system size increase, while plain GMRES takes more iterations.

5.5 Stochastic Galerkin p-refinement

  • The three random modes use (l, m) = (2, −2), (2, −1), and (2, 0) coefficients under a product uniform measure.Total-degree Legendre chaos spaces use p = 1, . . . , 4.Source: §5.5 · p. 21
  • The block-diagonal frozen preconditioner I ⊗P0 requires exactly 12 iterations at every chaos degree as unknowns grow from 4 896 to 42 840.Table 5 reports flat iterations across stochastic Galerkin p-refinement.Source: §5.5 · p. 21
  • Plain GMRES iterations rise from 230 to 279 as the coupled stochastic Galerkin system grows.The experiment uses the sphere at h = 0.2, κ = 2, ε = 20%, and d = 3 random modes.Source: §5.5 · p. 21

5.6 Wavenumber: higher frequency and mismatch

At κ = 4, frozen preconditioning remains within one iteration of fresh preconditioning across the amplitude sweep; wavenumber mismatch behaves differently, with asymmetric penalties and low-frequency accuracy loss.

5.6 Wavenumber: higher frequency and mismatch

  • At κ = 4, frozen iterations are 22–34 versus 21–33 for fresh preconditioning, with a gap of at most one throughout the sweep.Plain GMRES requires 260–317 iterations; the freezing penalty does not scale with solve difficulty across the two wavenumbers κ = 2 and κ = 4.Source: §5.6 · p. 21
  • A preconditioner frozen at κprecond = 2 incurs an asymmetric mismatch penalty: it reaches 6.5× upward at a ratio of 4 and saturates near 4.6× downward.The downward plateau is attributed to EFIE low-frequency breakdown; no corresponding limit is observed as κ grows.Source: §5.6 · p. 22
  • At κsolve = 0.03125, frozen and fresh solutions agree only to 2.8×10^-5 in relative ℓ2 norm, versus 10^-8 in the mid-range.The authors draw no conclusion there about attainable solution accuracy and characterize the mismatch behavior only numerically.Source: §5.6 · p. 22

5.7 Ellipsoid: a non-spherical geometry

On a triaxial ellipsoid, frozen and fresh Calderón preconditioning remain closely matched across the amplitude sweep, indicating the sphere results are not a consequence of spherical symmetry.

5.7 Ellipsoid: a non-spherical geometry

  • On the triaxial ellipsoid, frozen and fresh Calderón preconditioning require 16–17 iterations across the sweep, with a gap of at most one and zero at ε = 20%.Plain GMRES requires 322–325 iterations; all Table 7 meshes are valid throughout.Source: §5.7 · p. 22, Table 7 · p. 23
  • The ellipsoid has semi-axes (1.5, 1.0, 0.7), with no residual rotational symmetry beyond the discrete sign-flip group.The experiment uses the deformation family of Section 5.Source: §5.7 · p. 22
  • At ε = 30%, realized displacement reaches 10.28% of the local ellipsoid radius, about 1.5h and λ/20.Table 7 uses h = 0.1 and a single realization with seed 1001.Source: §5.7 · p. 22, Table 7 · p. 23

5.8 Fichera corner geometry: beyond the verified regime

The Fichera-corner tests distinguish perturbations away from the reentrant singularity from perturbations that move it: freezing tracks fresh preconditioning in the first case but degrades in the second. The corner-moving experiment lies outside the verified geometric regime, and mesh degradation complicates its larger-amplitude results.

5.8 Fichera corner geometry: beyond the verified regime

  • 26 frozen and 26 fresh iterations persist at every tested amplitude when deformation is confined to two flat faces, with 197 plain iterations.The sweep covers 2%–15% of the cube’s half-width; the largest displacement is 0.48h, and refinement to 4170 unknowns leaves the 26-iteration preconditioned count unchanged.Source: §5.8 · p. 24
  • The corner-moving deformation displaces the three notch faces, moves the reentrant corner by 3ε, and preserves mesh connectivity; the flat-face control leaves the corner untouched.The Fichera solid is non-convex and non-smooth, so the C1,1 boundary hypothesis and smooth-surface Calderón identity do not apply; the perturbation-away-from-corner test uses two other faces.Source: §5.8 · pp. 23–24
  • 1, 4, 13, then 23 iterations separate frozen from fresh as the reentrant corner moves; the frozen-to-fresh ratio rises from 1.04 to 1.51.At matched displacement λ/18, the corner-moving case has a 13-iteration gap, versus at most two for the sphere; Table 8 reports the corner sweep.Source: §5.8 · p. 24, Table 8 · p. 25
  • At 5% corner displacement, the frozen gap is already four despite only mild mesh-quality changes and an unchanged fresh-preconditioned count.The shape-regularity ratio changes from 2.29 to 3.05, the smallest angle from 42.7° to 31.8°, plain GMRES from 296 to 291, and fresh GMRES remains 25.Source: §5.8 · p. 24
  • At 10%–15%, severe angle collapse and worsening fresh solves confound mesh degradation with operator perturbation, while the frozen-to-fresh ratio rises from 1.04 to 1.51.The minimum angle falls to 10.0° and then 2.8°; the authors identify moving-corner failure of a small operator-norm perturbation bound as consistent with the deterioration.Source: §5.8 · p. 24
  • The experiments suggest that moving a reentrant singularity, rather than non-smoothness alone, marks the observed boundary of robust reuse; theory for Lipschitz or polyhedral Maxwell boundaries remains open.The flat-face perturbation remains robust even outside the verified regime, whereas the corner-moving sweep degrades monotonically.Source: §5.8 · p. 25

5.9 NASA almond: a Monte Carlo shape-uncertainty campaign

The NASA almond campaign tests frozen Calderón preconditioning on an electrically large, irregular scatterer under screened random shape perturbations. Across the sampled fields, frozen and fresh iteration counts remain close, while refinement preserves the frozen count and reuse reduces assembly cost.

5.9 NASA almond: a Monte Carlo shape-uncertainty campaign

  • Screening ε sup |DV|² ≤ 1/2 before constructing each shape admits fields without rejection, avoiding the amplitude-correlated selection that post-hoc rejection could introduce.The random deformation uses analytic squared-exponential fields; case U moves the tip with its neighborhood, while case T applies a cutoff that fixes it.Source: §5.9 · pp. 25–26
  • At ℓ = d/4, the five tested amplitudes span 2.66–13.30 mm, or 0.38h–1.90h; shorter correlation length admits a lower amplitude ceiling.At equal amplitude, shortening ℓ to d/10 costs at most one additional iteration over the two levels that its screening admits.Source: §5.9 · pp. 26–27
  • 17.4× measured, 22.9× artifact-corrected, and 27.4× assembly-only are the three NASA-almond assembly speedups reported.The factors are smaller than for the sphere because the assembly-time ratio differs by geometry and mesh.Source: §5.9 · p. 26, Table 9 · p. 26
  • 0.92–1.04 is the frozen-to-fresh iteration-count ratio across 38 paired samples, with median 1.00 and no systematic amplitude trend.Frozen is cheaper on 20 of the 38; the six highest frozen counts are also the six highest plain counts, and all eight hardest paired draws favor frozen, with a worst case of 59 frozen versus 64 fresh iterations.Source: Table 10 · p. 26, §5.9 · p. 27, Figure 4 · p. 27
  • 51 is the median frozen GMRES count among 100 fields at ε = 13.3 mm; on 20 fresh-solved fields, the frozen-to-fresh ratio has median 1.02.The ratio spans 0.93–1.11, all fields converge to 10^-8, and no realization is discarded.Source: §5.9 · p. 27, Figure 5 · p. 28
  • At maximum amplitude, refinement from 2673 to 7269 unknowns leaves the frozen median nearly unchanged, 51 to 52, while the unpreconditioned median rises from 429 to 617.The frozen-to-fresh ratio is 0.98–1.06; the unpreconditioned-to-frozen iteration advantage grows from 8.4× to 11.9×.Source: §5.9 · pp. 27–28
  • Frozen and fresh solves give currents agreeing to relative error 1.9 × 10^-8, while perturbations shift monostatic radar cross section by a median 2.2 dB and up to 14 dB.Across 73 azimuths at both polarizations, the frozen and fresh iteration counts differ by at most one; bistatic far fields also separate by several dB for the nominal and perturbed shapes.Source: §5.9 · p. 28, Figure 6 · p. 28
  • 311 s for frozen solves plus assembly versus 327 s overall for fresh solves reaches break-even in the single-geometry, 73-angle sweep.The reported crossover is near 70 right-hand sides in the current implementation, or near 200 with the storage artifact removed; reuse therefore favors many geometries with few excitations each.Source: §5.9 · pp. 28–29

6 Conclusions and outlook

The paper concludes that a nominal Calderón preconditioner can be reused across admissible shape-perturbed EFIE problems by transporting its pairing exactly and freezing the dual-mesh operator. The analysis and experiments support mesh-robust reuse within stated assumptions, with limitations on geometry, frequency, and workloads.

6 Conclusions and outlook

  • Fixed connectivity makes the RWG-BC degree-of-freedom correspondence a permutation, transporting the duality pairing exactly while reusing the nominal dual-mesh operator.The spectral-conditioning bound is uniform in spatial discretization and, after tensorization, stochastic approximation dimension.Source: §6 · p. 29
  • 29–31× measured speedup per new realization in the dense implementation and roughly twofold against published compressed-assembly timings quantify the assembly-amortization benefit.The factors depend on implementation; the transferable result is amortizing dual barycentric assembly across realizations.Source: §6 · p. 29
  • Reuse requires mesh-morphing with fixed connectivity; topology changes, quality-driven remeshing, or deformations too large for the nominal connectivity fall outside the strategy.The admissible perturbation window also narrows with frequency, and assembly savings favor many geometries with few excitations; the multi-right-hand-side experiment reaches break-even for one geometry.Source: §6 · p. 29
  • Assumption G remains open for merely Lipschitz or polyhedral boundaries, and the paper leaves end-to-end savings in Monte Carlo, multilevel, and tensorized moment-system pipelines for future work.Further directions include extending the geometric analysis, other operator-preconditioning systems, and complementary reuse strategies for wavenumber sweeps.Source: §6 · pp. 29–30
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