Fast Multipole Method with Complex Coordinates

TL;DR

This paper develops a complex-coordinate fast multipole method (FMM) to efficiently evaluate boundary layer potentials when source/target locations lie in complex spaces $oldsymbol{C}^2$ and $oldsymbol{C}^3$, which arise from complex scaling used to truncate unbounded domains. The authors maintain the core FMM structure, constructing a real-part–based hierarchical tree and extending multipole/local expansions and translation operators under Lipschitz conditions on the imaginary parts (with constant $L<1$), achieving near-linear time complexity comparable to the classical FMM. They provide rigorous truncation error estimates for the 2-D and 3-D Laplace/Helmholtz kernels under these complex-geometry assumptions, and develop practical implementations that include rotations and z-axis translations via a “point-and-shoot” approach to keep the cost at $O(P^3)$ per level. Numerical results demonstrate linear-time scaling and successful application to large-scale time-harmonic water-wave problems and Helmholtz transmission problems in unbounded domains, highlighting the method’s potential for efficient wave scattering simulations. The work also discusses limitations such as stability for high frequencies and the need for sharper Helmholtz translation error analyses, outlining avenues for further theoretical and algorithmic improvements.

Abstract

In this work we present a variant of the fast multipole method (FMM) for efficiently evaluating standard layer potentials on geometries with complex coordinates in two and three dimensions. The complex scaled boundary integral method for the efficient solution of scattering problems on unbounded domains results in complex point locations upon discretization. Classical real-coordinate FMMs are no longer applicable, hindering the use of this approach for large-scale problems. Here we develop the complex-coordinate FMM based on the analytic continuation of certain special function identities used in the construction of the classical FMM. To achieve the same linear time complexity as the classical FMM, we construct a hierarchical tree based solely on the real parts of the complex point locations, and derive convergence rates for truncated expansions when the imaginary parts of the locations are a Lipschitz function of the corresponding real parts. We demonstrate the efficiency of our approach through several numerical examples and illustrate its application for solving large-scale time-harmonic water wave problems and Helmholtz transmission problems.

Figures (7)

• Figure 1: The domain $\Omega$, shown in yellow is bounded below by the infinitely long boundary $\Gamma$. The boundary $\Gamma$ is flat outside the interval when $|x_1|> L$. The blue dashed line represents $\psi(x_1)$, the imaginary part of the $x_1$-component of the deformed complex boundary $\tilde{\Gamma}$. The short red lines indicate the truncation limits of the computational domain used for solving the BIE on $\Gamma$, while the long green lines indicate the smaller truncation limits used for solving the BIE on $\tilde{\Gamma}.$
• Figure 2: (A) Real parts of a box $b$ and its Lists 1-4 for $k=1$. (B) Real parts of a box $b$ and its Lists 1-4 for $k=2$.
• Figure 3: (A) The 2-D interface $\tilde{\Gamma}$ used in \ref{['sec:time_comparison']}. The black solid line shows the real interface $\Gamma$. The blue dashed line shows the imaginary part of the $x_1$-component of the deformed interface $\tilde{\Gamma}$ as a function of $\Re x_1$. (B) Log-log plot of the evaluation time for the single layer potentials of 2-D Laplace and Helmholtz kernels using the direct methods and the complex-coordinate FMMs with relative accuracies $10^{-12}$ and $10^{-6}$, over varying numbers $N$ of complex source and target locations on $\tilde{\Gamma}$ defined by \ref{['eqn:wobble_2D_cplx']}.
• Figure 4: (A) The interface $\Gamma$ and the imaginary parts of $x_1$ and $x_2$ coordinates of the complex deformation $\tilde{\Gamma}$. The red and blue transparent sheets represent $\Im x_1$ and $\Im x_2$ of the surface, respectively. (B) Log-log plot of the evaluation time for the single layer potentials for 3-D Laplace and Helmholtz kernels using the direct methods and the complex-coordinate FMMs with relative accuracies $10^{-12}$ and $10^{-6}$, over varying numbers $N$ of complex source and target locations on $\tilde{\Gamma}$ defined by \ref{['eqn:complex_wobble']}.
• Figure 5: (A) The geometry for the 3-D water wave problem in Section \ref{['sec:water_wave']}. The blue flat interface represents $\Gamma_t$ and the Gaussian bump below represents $\Gamma_b$. (B) The real interface $\Gamma_b$ and the imaginary part of the $x_1,x_2$ components of the complexified interfaces $\tilde{\Gamma}_t$ and $\tilde{\Gamma}_b$. The red sheet represents $\Im x_1$ and the blue sheet represents $\Im x_2$.

Theorems & Definitions (42)

• Definition 3.1.1: Complex polar coordinates
• Remark
• Definition 3.1.2: Complex spherical coordinate
• Theorem 3.2.1: Graf's Addition Formula abramowitz+stegun
• proof
• Theorem 3.2.2
• proof
• Definition 3.2.1: Spherical harmonics
• Lemma 3.2.1
• Lemma 3.2.2