Fast wavefield evaluation method based on modified proxy-surface-accelerated interpolative decomposition for two-dimensional scattering problems

TL;DR

This work tackles efficient evaluation of wavefields in 2D scattering problems by avoiding kernel expansions required by traditional FMM. It introduces a fast wavefield evaluation method based on a modified proxy-surface-accelerated interpolative decomposition (ID), enabling accurate near-boundary computations and leveraging low-rank representations of layer potentials. The method uses unit-cell–level proxy surfaces and ID-based skeletonization to achieve substantial speedups while maintaining accuracy comparable to conventional boundary-integral approaches. The approach holds promise for rapid design and optimization tasks in wave scattering and can be extended to 3D or problems with nonstandard kernels.

Abstract

This paper presents a fast wavefield evaluation method for two-dimensional wave scattering problems. The proposed method is based on a modified version of proxy-surface-accelerated interpolative decomposition, making it effective even if the evaluation points are near the boundary. The commonly known fast multipole method requires the use of direct evaluations near the boundaries of scatterers because the analytical expansion of kernel functions does not converge. On the one hand, the proposed method does not require the analytical expansion of kernel functions. The validity and effectiveness of the proposed method are demonstrated using numerical examples.

Figures (4)

• Figure 1: Evaluation points and boundary. Each evaluation point is located in a unit cell (size: $1.0 \times 1.0$). The midpoints of the discretized boundary are plotted.
• Figure 2: Results of proxy surface method for $x$-side. In this study, this result was reused in all unit cells regardless of the distance to the boundary.
• Figure 3: Comparison of normalized elapsed time (s) and $l_{2}$ relative error for various methods. The elapsed time of Conv was 55.1 s. The maximum $l_{2}$ relative errors among those evaluated for each unit cell are shown.
• Figure 4: Relative error (color bar) of real part of total wavefield $u$ with respect to that of analytical solution at each evaluation point. The plotted domain corresponds to the region $x = \{ (x_1, x_2) \in \mathbb{R}^{2} \mid -3 < x_1 < 3, \,\, -3 < x_2 < 3 \}$. There is a scatterer in the black region. Top left: Conv. Top right: fast-U. Bottom left: fast-UV. Bottom right: fast-UV-Vtailored.