This paper introduces a new pseudodifferential preconditioner for the Helmholtz equation in variable media with absorption. The pseudodifferential operator is associated with the multiplicative inverse to the symbol of the Helmholtz operator. This approach is well-suited for the intermediate and high-frequency regimes. The main novel idea for the fast evaluation of the preconditioner is to interpolate its symbol, not as a function of the (high-dimensional) phase-space variables, but as a function of the wave speed itself. Since the wave speed is a real-valued function, this approach allows us to interpolate in a univariate setting even when the original problem is posed in a multidimensional physical space. As a result, the needed number of interpolation points is small, and the interpolation coefficients can be computed using the fast Fourier transform. The overall computational complexity is log-linear with respect to the degrees of freedom as inherited from the fast Fourier transform. We present some numerical experiments to illustrate the effectiveness of the preconditioner to solve the discrete Helmholtz equation using the GMRES iterative method. The implementation of an absorbing layer for scattering problems using a complex-valued wave speed is also developed. Limitations and possible extensions are also discussed.

翻译：本文针对含吸收项的变介质亥姆霍兹方程提出了一种新的伪微分预条件子。该伪微分算子与亥姆霍兹算子符号的乘法逆相关联。此方法特别适用于中高频区域。实现预条件子快速计算的核心创新思路在于：对其符号进行插值时，不将其视为（高维）相空间变量的函数，而是视为波速本身的函数。由于波速是实值函数，即使原问题建立在多维物理空间中，该方法也允许我们在单变量设定下进行插值。因此，所需的插值点数量较少，且插值系数可通过快速傅里叶变换计算。整体计算复杂度继承快速傅里叶变换的特性，相对于自由度呈对数线性关系。我们通过数值实验展示了该预条件子结合GMRES迭代法求解离散亥姆霍兹方程的有效性。同时发展了利用复数值波速实现散射问题吸收层的实施方案。文中亦讨论了该方法的局限性及可能的扩展方向。