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 address challenges (i) and (ii), supporting our theory with numerical experiments. Challenge (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. Challenge (ii) is addressed using techniques from numerical linear algebra: we oversample, considering more canonical incident angles than are necessary, thus expanding the set of valid coefficient vectors. The coefficient vector 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模式表示为相对较小（与频率无关）的典型入射角集合对应的远场模式函数。尽管这些非凡的公式在理论上严格成立，我们在此证明：(i) 实际计算中它们对数值误差高度敏感；(ii) 即使采用精确算术运算，对特定典型入射角集合也可能无法直接计算公式中的系数。只有克服这些实际问题，嵌入公式才能为计算大量入射角激发的远场模式提供高效方法。本文通过数值实验验证理论，解决挑战(i)和(ii)。针对挑战(i)，采用计算复分析技术：将嵌入公式重新表述为复围道积分，并证明该形式对数值误差的敏感度大幅降低。实际计算中该围道积分可通过留数定理高效求值。针对挑战(ii)，采用数值线性代数技术：通过过采样引入更多典型入射角，扩展有效系数向量空间。随后可采用最小二乘法或列子集选择法确定系数向量。