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arXiv · 2606.16192

Infinite-lattice discrete Calderón projection via the lattice Green's function for active noise shielding and confinement

Qing Xia
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Abstract

We construct an infinite-lattice discrete Calderón projection for the Helmholtz equation by convolution with the lattice Green's function (LGF), and apply it to active noise shielding and confinement on Cartesian grids with arbitrary geometry. The LGF fixes the outgoing radiation condition and removes the geometry-dependent auxiliary Helmholtz problem and artificial outer boundary from the projection and control synthesis; its finite numerical tabulation depends only on $(h,k)$ and is reusable across geometries. We prove idempotence, characterize the range as the trace space of interior lattice-Helmholtz solutions, and establish range equivalence with a well-posed Tsynkov-type projection. The two projectors coincide as operators when the auxiliary problem reproduces the exact lattice radiation condition. A capacity-matrix realization yields closed-form shielding and confinement densities supported on the exterior and interior sublayers of a single lattice boundary strip, respectively. For pure-noise shielding, exterior-sublayer measurements suffice under explicit invertibility assumptions; preservation of an unknown wanted interior field requires the full strip trace. Experiments on circular, L-shaped, and star-shaped regions verify machine-precision cancellation for LGF-consistent sources and near-second-order convergence for analytic plane waves and point sources. Conditioning and measurement noise tests quantify the configuration dependence of the reconstruction.

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Qing Xia. 2026-06-15. Infinite-lattice discrete Calderón projection via the lattice Green's function for active noise shielding and confinement. https://arxiv.org/abs/2606.16192

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