For problems of time-harmonic scattering by rational polygonal obstacles, embedding formulae express the far-field pattern induced by any incident plane wave in terms of the far-field patterns for a relatively small (frequency-independent) set of canonical incident angles. Although these remarkable formulae are exact in theory, here we demonstrate that: (i) they are highly sensitive to numerical errors in practice, and; (ii) direct calculation of the coefficients in these formulae may be impossible for particular sets of canonical incident angles, even in exact arithmetic. 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. Here we propose solutions for problems (i) and (ii), backed up by theory and numerical experiments. Problem (i) is solved using techniques from computational complex analysis: we reformulate the embedding formula as a complex contour integral and prove that this is much less sensitive to numerical errors. In practice, this contour integral can be efficiently evaluated by residue calculus. Problem (ii) is addressed using techniques from numerical linear algebra: we oversample, considering more canonical incident angles than are necessary, thus expanding the space of valid coefficients vectors. The coefficients vectors can then be selected using either a least squares approach or column subset selection.

翻译：针对有理多边形障碍物的时谐散射问题，嵌入公式将任意入射平面波激发的远场方向图表示为相对较小（频率无关）的典型入射角集合对应的远场方向图。尽管这些公式理论上精确，本文证明：（i）实践中对数值误差高度敏感；（ii）即使采用精确算术，特定典型入射角集合下的公式系数也可能无法直接计算。只有克服这些实际问题，嵌入公式才能高效计算大量入射角下的远场方向图。本文基于理论与数值实验提出解决上述问题（i）和（ii）的方案。问题（i）通过计算复分析技术解决：将嵌入公式重新表述为复围道积分，并证明其对数值误差的敏感性显著降低。实践中，该围道积分可通过留数计算高效求值。问题（ii）借助数值线性代数技术解决：采用过采样策略，引入比所需更多的典型入射角，从而扩展有效系数向量空间，随后通过最小二乘法或列子集选择法确定系数向量。