The complex Helmholtz equation $(\Delta + k^2)u=f$ (where $k\in{\mathbb R},u(\cdot),f(\cdot)\in{\mathbb C}$) is a mainstay of computational wave simulation. Despite its apparent simplicity, efficient numerical methods are challenging to design and, in some applications, regarded as an open problem. Two sources of difficulty are the large number of degrees of freedom and the indefiniteness of the matrices arising after discretisation. Seeking to meet them within the novel framework of probabilistic domain decomposition, we set out to rewrite the Helmholtz equation into a form amenable to the Feynman-Kac formula for elliptic boundary value problems. We consider two typical scenarios, the scattering of a plane wave and the propagation inside a cavity, and recast them as a sequence of Poisson equations. By means of stochastic arguments, we find a sufficient and simulatable condition for the convergence of the iterations. Upon discretisation a necessary condition for convergence can be derived by adding up the iterates using the harmonic series for the matrix inverse -- we illustrate the procedure in the case of finite differences. From a practical point of view, our results are ultimately of limited scope. Nonetheless, this unexpected -- even paradoxical -- new direction of attack on the Helmholtz equation proposed by this work offers a fresh perspective on this classical and difficult problem. Our results show that there indeed exists a predictable range $k<k_{max}$ in which this new ansatz works with $k_{max}$ being far below the challenging situation.

翻译：复亥姆霍兹方程$(\Delta + k^2)u=f$（其中$k\in{\mathbb R},u(\cdot),f(\cdot)\in{\mathbb C}$）是计算波动模拟的核心工具。尽管形式上看似简单，但其高效数值方法的设计仍面临挑战，在某些应用领域甚至被视为未解难题。两大困难源于离散化后产生的大规模自由度与矩阵的不定性。为在概率域分解这一新型框架中应对这些挑战，我们致力于将亥姆霍兹方程转化为适用于椭圆边值问题费曼-卡茨公式的形式。本文考虑两种典型场景——平面波散射与空腔内部传播，并将其重构为泊松方程序列。通过随机分析论证，我们找到了迭代收敛的充分且可模拟条件。在离散化阶段，利用矩阵逆的调和级数对迭代项求和可推得收敛的必要条件——此处以有限差分法为例说明该过程。从实践角度看，本研究结果最终适用范围有限。尽管如此，本文提出的这种针对亥姆霍兹方程的新颖（甚至具有悖论色彩）攻関思路，为这个经典难题提供了全新视角。结果表明，确实存在可预测的适用范围$k<k_{max}$，其中$k_{max}$远低于具有挑战性的情形。