核心概念
The authors address challenges in computing far-field patterns induced by polygonal obstacles, focusing on numerical errors and coefficient selection.
要約
The paper introduces an efficient method for calculating far-field patterns induced by polygonal obstacles. It addresses challenges related to numerical errors and coefficient selection through oversampling strategies. The approach involves reformulating embedding formulas and using computational complex analysis techniques.
Key points include the introduction of embedding formulas, theoretical results on rational polygons, and the sensitivity of numerical approximations. The paper discusses the implications of coalescence points, contour integrals, and residue calculations in finite precision arithmetic. Strategies for selecting coefficients and addressing ill-conditioning are explored through numerical experiments.
The study emphasizes the importance of oversampling to increase the chances of finding suitable coefficient vectors while minimizing errors. By considering multiple canonical incident angles, the authors aim to improve efficiency in far-field pattern computations.
統計
Only by overcoming these practical issues can embedding formulae provide a highly efficient approach to computing the far-field pattern induced by a large number of incident angles.
For standard bases of polynomials or trigonometric functions, unisolvence theorems tell us that an M-dimensional function can be reconstructed uniquely from a set of M distinct points.
The system is under-determined when multiple solutions exist to (3.1), allowing for pseudo-inverse solutions with minimal norm.
引用
"The beauty of Theorem 1.2 is that given far-field patterns D(θ, αm) for distinct α1, . . . , αM, the embedding formula provides an exact expression for D(θ, α), valid for all (θ, α) ∈ T."
"By representing the integral as (2.4), we address pole-zero pairs and avoid catastrophic cancellation in finite precision arithmetic."
"In practice, we must work with an approximation to the matrix and right-hand side of (1.9)."