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Truncated kernel windowed Fourier projection: a fast algorithm for the 3D free-space wave equation

Nour G. Al Hassanieh, Alex H. Barnett, Leslie Greengard

TL;DR

This work introduces the Truncated-Kernel Windowed Fourier Projection (TK-WFP) to efficiently compute 3D free-space wave potentials from many point sources. By splitting the solution into a local part and a spectral history part, and by spatially truncating the kernel to control oscillations, the method achieves spectral accuracy with complexity largely linear in the number of sources and quasilinear in the number of Fourier modes. The authors provide rigorous decay results for the history coefficients and demonstrate strong numerical performance, including scenarios with up to a million sources, without requiring absorbing boundary conditions. TK-WFP offers a practical component for time-domain wave-scattering simulations and can complement integral-equation approaches in large-scale applications.

Abstract

We present a spectrally accurate fast algorithm for evaluating the solution to the scalar wave equation in free space driven by a large collection of point sources in a bounded domain. With $M$ sources temporally discretized by $N_t$ time steps of size $Δt$, a naive potential evaluation at $M$ targets on the same time grid requires $\mathcal O(M^2 N_t)$ work. Our scheme requires $\mathcal{O}\left((M + N^3\log N)N_t\right)$ work, where $N$ scales as $\mathcal O(1/Δt)$, i.e., the maximum signal frequency. This is achieved by using the recently-proposed windowed Fourier projection (WFP) method to split the potential into a local part, evaluated directly, plus a smooth history part approximated by an $N^3$-point equispaced discretization of the Fourier transform, where each Fourier coefficient obeys a simple recursion relation. The growing oscillations in the spectral representation (which would be present with a naive use of the Fourier transform) are controlled by spatially truncating the hyperbolic Green's function itself. Thus, the method avoids the need for absorbing boundary conditions. We demonstrate the performance of our algorithm with up to a million sources and targets at 6-digit accuracy. We believe it can serve as a key component in addressing time-domain wave equation scattering problems.

Truncated kernel windowed Fourier projection: a fast algorithm for the 3D free-space wave equation

TL;DR

This work introduces the Truncated-Kernel Windowed Fourier Projection (TK-WFP) to efficiently compute 3D free-space wave potentials from many point sources. By splitting the solution into a local part and a spectral history part, and by spatially truncating the kernel to control oscillations, the method achieves spectral accuracy with complexity largely linear in the number of sources and quasilinear in the number of Fourier modes. The authors provide rigorous decay results for the history coefficients and demonstrate strong numerical performance, including scenarios with up to a million sources, without requiring absorbing boundary conditions. TK-WFP offers a practical component for time-domain wave-scattering simulations and can complement integral-equation approaches in large-scale applications.

Abstract

We present a spectrally accurate fast algorithm for evaluating the solution to the scalar wave equation in free space driven by a large collection of point sources in a bounded domain. With sources temporally discretized by time steps of size , a naive potential evaluation at targets on the same time grid requires work. Our scheme requires work, where scales as , i.e., the maximum signal frequency. This is achieved by using the recently-proposed windowed Fourier projection (WFP) method to split the potential into a local part, evaluated directly, plus a smooth history part approximated by an -point equispaced discretization of the Fourier transform, where each Fourier coefficient obeys a simple recursion relation. The growing oscillations in the spectral representation (which would be present with a naive use of the Fourier transform) are controlled by spatially truncating the hyperbolic Green's function itself. Thus, the method avoids the need for absorbing boundary conditions. We demonstrate the performance of our algorithm with up to a million sources and targets at 6-digit accuracy. We believe it can serve as a key component in addressing time-domain wave equation scattering problems.

Paper Structure

This paper contains 19 sections, 5 theorems, 65 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Prior work
  3. Windowed Fourier projection and the decomposition of hyperbolic potentials
  4. The blending function
  5. The local and history part
  6. Approximation of the history part and choice of $\delta$
  7. Discretization of the Fourier integral
  8. Wavenumber truncation
  9. Choice of blending timescale
  10. Computation of the history part
  11. Computation and storage costs
  12. Computation of the local part
  13. Numerical results
  14. Convergence study
  15. Sources and targets on a smooth surface
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $\delta>0$ and $0<\epsilon<1$. Then eq:windowFT with $b = \ln(1/\epsilon)$ obeys, for any $\theta>1$, the bound

Figures (6)

  • Figure 1: Local and truncated history contributions to the solution $u(\bold x,t)$ at a target location $\bold x\in B = [-1,1]^3$ at the current time $t$. The left side depicts the space-time diagram (light cone) for the local $\ul$ part, while the central image is the same for the history $u_h$ part. The vertical axis shows the source time $\tau$ spanning $0$ to the current time $t$, and is common to all parts of the figure. (Only two of the three dimensions of space are shown; each temporal slice of the light "cone" is a sphere rather than a circle.) The color on the cones $|\bold x-\bold y|=t-\tau$ indicates the delay $t-\tau$. The right side graphs their respective temporal windowing functions; see \ref{['ul']} for the local part, and \ref{['eq:uexactFtrunc']} for the history part where the kernel $\mathcal{G}_A$ is smoothly truncated to the finite time horizon $t-\tau<A$. Note that $A-\delta \ge \mathop{\mathrm{diam}}\nolimits B$, so that this truncation has no effect in $B$.
  • Figure 2: Computed solution due to eight sources as in Section \ref{['sec:eightSources']}, at time $t = 6$, using $\Delta t=0.0102$. The same potential $\tilde{u}$ is shown as a cut-away of the volume (left), and on the plane $z = -0.01$ (right). At its center frequency (visible as the oscillations in the plot) the wave field has 30 wavelengths on each side of the cube $B$. The solution has 6 accurate digits.
  • Figure 3: Convergence of the maximum error at $t=6$, estimated on a grid, for the eight-source test of Section \ref{['sec:eightSources']}.
  • Figure 4: Results from the surface source and target distribution test of Section \ref{['sec:cruller']}. Left: computed solution at $t = 6$ at the $M=102400$ surface points, shown looking down the $z$-axis onto the $xy$-plane. Color indicates value of $u$, as per the colorscale shown. Right: CPU timing table for this experiment. Entries labeled with "est." are estimated by scaling up from the cost for a single time-step.
  • Figure 5: Computed solution at $t = 4$ (left) and $t = 6$ (right), for the test of Section \ref{['sec:randomSources']} with $10^6$ sources and targets. The solution is shown on a target grid of size $100^3$. There are $30$ wavelengths per side of cube at the maximum center frequency of the sources, and about 42 wavelengths per side at the $\epsilon$-bandlimit frequency. The relative maximum error at $t = 6$ is $\tilde{\mathcal{E}}_{\Delta t} = 1.4\times10^{-6}$.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Lemma 2.1
  • Proposition 3.1
  • Theorem 1
  • Remark 3.1: Bound on the truncation error
  • Remark 4.1
  • Lemma 4.1
  • Remark 4.2: History storage costs
  • Remark 5.1
  • Remark 5.2
  • Remark 7.1: Periodic case and FFT size ratio
  • ...and 3 more