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A novel direct Helmholtz solver in inhomogeneous media based on the operator Fourier transform functional calculus

Max Cubillos, Edwin Jimenez

Abstract

This article presents novel numerical algorithms based on pseudodifferential operators for fast, direct, solution of the Helmholtz equation in 1D, 2D, and 3D inhomogeneous unbounded media. The proposed approach relies on an Operator Fourier Transform (OFT) representation of pseudodifferential operators (ΨDO) which frame the problem of computing the inverse Helmholtz operator with a spatially-dependent wave speed in terms of two sequential applications of an inverse square root ΨDO. The OFT representation of the action of the inverse square root ΨDO, in turn, can be effected as a superposition of solutions of a pseudo-temporal initial-boundary-value problem for a paraxial equation. The OFT framework offers several advantages over traditional direct and iterative approaches for the solution of the Helmholtz equation. The operator integral transform is amenable to standard quadrature methods and the required pseudo-temporal paraxial equation solutions can be obtained using any suitable numerical method. A specialized quadrature is derived to evaluate the OFT efficiently and an alternating direction implicit method, used in conjunction with standard finite differences, is used to solve the requisite component paraxial equation problems. Numerical studies in 1D, 2D, and 3D are presented to confirm the expected OFT-based Helmholtz solver convergence rate. In addition, the efficiency and versatility of our proposed approach is demonstrated by tackling nontrivial wave propagation problems, including two-dimensional plane wave scattering from a geometrically complex inhomogeneity, three-dimensional scattering from turbulent channel flow and plane wave transmission through a spherically-symmetric gradient-index Luneburg lens. All computations, 3D problems which involve solving the Helmholtz equation with more than one billion complex unknowns, are performed in a single workstation.

A novel direct Helmholtz solver in inhomogeneous media based on the operator Fourier transform functional calculus

Abstract

This article presents novel numerical algorithms based on pseudodifferential operators for fast, direct, solution of the Helmholtz equation in 1D, 2D, and 3D inhomogeneous unbounded media. The proposed approach relies on an Operator Fourier Transform (OFT) representation of pseudodifferential operators (ΨDO) which frame the problem of computing the inverse Helmholtz operator with a spatially-dependent wave speed in terms of two sequential applications of an inverse square root ΨDO. The OFT representation of the action of the inverse square root ΨDO, in turn, can be effected as a superposition of solutions of a pseudo-temporal initial-boundary-value problem for a paraxial equation. The OFT framework offers several advantages over traditional direct and iterative approaches for the solution of the Helmholtz equation. The operator integral transform is amenable to standard quadrature methods and the required pseudo-temporal paraxial equation solutions can be obtained using any suitable numerical method. A specialized quadrature is derived to evaluate the OFT efficiently and an alternating direction implicit method, used in conjunction with standard finite differences, is used to solve the requisite component paraxial equation problems. Numerical studies in 1D, 2D, and 3D are presented to confirm the expected OFT-based Helmholtz solver convergence rate. In addition, the efficiency and versatility of our proposed approach is demonstrated by tackling nontrivial wave propagation problems, including two-dimensional plane wave scattering from a geometrically complex inhomogeneity, three-dimensional scattering from turbulent channel flow and plane wave transmission through a spherically-symmetric gradient-index Luneburg lens. All computations, 3D problems which involve solving the Helmholtz equation with more than one billion complex unknowns, are performed in a single workstation.
Paper Structure (22 sections, 3 theorems, 113 equations, 4 figures, 2 tables)

This paper contains 22 sections, 3 theorems, 113 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem description
  3. Electromagnetic waves
  4. Acoustic waves
  5. Pseudodifferential operators and the Operator Fourier Transform
  6. Using the OFT to solve differential equations
  7. The OFT applied to the inverse square-root Helmholtz operator
  8. Numerical computation of the OFT
  9. Temporal and spatial discretization of the paraxial equation
  10. Exponential time-stepping
  11. OFT quadrature
  12. Error analysis
  13. Numerical results
  14. Convergence studies for the $\left[ I + \Delta / \kappa^2 \right]^{-1/2}$ and $\left[ I + \Delta / \kappa^2 \right]^{-1}$ operators
  15. Two dimensional plane wave scattering from an inhomogeneous obstacle
  16. ...and 7 more sections

Key Result

Theorem 1

Let $f(A)$ be given by equation eq:SqRtHelmOFT and let where the weights are given by equation eq:OFTWeights and the sequence of time steps is given by equations eq:ExpTimeSteps and eq:ExpTimeStepsPrms. Then the error is bounded by where, fixing $\sigma$ and $R$ and letting $T \rightarrow \infty$ and $\Delta t_0 \rightarrow 0$, the error terms are given asymptotically by $E_1$ is the error incu

Figures (4)

  • Figure 1: Total field magnitude $|v|=|v^{i}+v^{s}|$ for plane wave scattering through a variable wave speed inhomogeneity in the shape of the U.S. Air Force logo. Total field magnitude with highlighted logo and surrounding partially reflected/transmitted waves for (a) 25 wavelengths and (b) 100 wavelengths, respectively, along each dimension. Elevated plots of the total field magnitude for (c) 25 wavelengths and (d) 100 wavelengths, which highlight the interaction of the incoming wave with the inhomogeneity.
  • Figure 2: Plane wave scattering from turbulent channel flow. (a) Problem schematic of a plane wave with wavenumber $\kappa = 78.5$ traveling in the direction $(1, -0.5, 0) / \sqrt{1.25}$ impinging on the channel wall located at $x_2 = 0$. (b) Variable refraction coefficient $m(\boldsymbol x)$ derived from turbulent flow data. (c) Horizontal cross-section pseudocolor plot at $x_2=0.25$ of the real part of the scattered field $v^{s}$. (d)-(e) Vertical pseudocolor planes with and without the turbulent refraction inhomogeneity for the real part of the scattered field $v^{s}$ which show the multiple wave scattering that results from interaction with the channel wall and the complex surrounding inhomogeneous medium.
  • Figure 3: Plane wave transmission through a Luneburg lens. (a) A plane wave travels from left to right and the total field magnitude $|v|$ peak reveals the focusing of the transmitted wave just after it exits the lens. (b) Total field magnitude cross-sections highlighting the plane wave focusing through the spherical gradient index lens. (c) Real part of the scattered field with the lens removed reveals the internal wave interaction inside the lens. (d) Close-up of a cross-section of the total field magnitude $|v|$ which confirms the correct location of the focusing region centered on the lens boundary.
  • Figure 4: Integration contour for Proposition \ref{['prop:Winding']}.

Theorems & Definitions (9)

  • Example 1
  • Example 2
  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • proof : Proof of Theorem \ref{['thm:Error']}