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A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: the Neumann case

Fenglong Qu, Bo Zhang, Haiwen Zhang

TL;DR

This work studies direct and inverse scattering by a locally perturbed infinite plane under a Neumann boundary condition in 2D. It introduces a novel integral equation defined on a bounded curve augmented by an auxiliary boundary, solved efficiently via the RCIP method even at large wavenumbers, with solvability relying on $k^2$ not being a Dirichlet eigenvalue in the auxiliary domain $D_{tilde}$. For the inverse problem, uniqueness is established from the far-field pattern $u^ romto^ abla( heta;d)$ for all observation directions and incident directions at fixed $k$, and a Newton-type reconstruction using multi-frequency data is developed based on the Fréchet differentiability of the far-field map $F_{d,k}[h]$ and its linearization via a linearized NP problem. Numerical experiments demonstrate stable, accurate recovery of multi-scale surface profiles, validating the method for high frequencies and suggesting extensions to 3D Helmholtz and Maxwell equations.

Abstract

This paper is concerned with direct and inverse scattering by a locally perturbed infinite plane (called a locally rough surface in this paper) on which a Neumann boundary condition is imposed. A novel integral equation formulation is proposed for the direct scattering problem which is defined on a bounded curve (consisting of a bounded part of the infinite plane containing the local perturbation and the lower part of a circle) with two corners and some closed smooth artificial curve. It is a nontrivial extension of our previous work on direct and inverse scattering by a locally rough surface from the Dirichlet boundary condition to the Neumann boundary condition [{\em SIAM J. Appl. Math.}, 73 (2013), pp. 1811-1829]. In this paper, we make us of the recursively compressed inverse preconditioning (RCIP) method developed by Helsing to solve the integral equation which is efficient and capable of dealing with large wave numbers. For the inverse problem, it is proved that the locally rough surface is uniquely determined from a knowledge of the far-field pattern corresponding to incident plane waves. Further, based on the novel integral equation formulation, a Newton iteration method is developed to reconstruct the locally rough surface from a knowledge of multiple frequency far-field data. Numerical examples are also provided to illustrate that the reconstruction algorithm is stable and accurate even for the case of multiple-scale profiles.

A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: the Neumann case

TL;DR

This work studies direct and inverse scattering by a locally perturbed infinite plane under a Neumann boundary condition in 2D. It introduces a novel integral equation defined on a bounded curve augmented by an auxiliary boundary, solved efficiently via the RCIP method even at large wavenumbers, with solvability relying on not being a Dirichlet eigenvalue in the auxiliary domain . For the inverse problem, uniqueness is established from the far-field pattern for all observation directions and incident directions at fixed , and a Newton-type reconstruction using multi-frequency data is developed based on the Fréchet differentiability of the far-field map and its linearization via a linearized NP problem. Numerical experiments demonstrate stable, accurate recovery of multi-scale surface profiles, validating the method for high frequencies and suggesting extensions to 3D Helmholtz and Maxwell equations.

Abstract

This paper is concerned with direct and inverse scattering by a locally perturbed infinite plane (called a locally rough surface in this paper) on which a Neumann boundary condition is imposed. A novel integral equation formulation is proposed for the direct scattering problem which is defined on a bounded curve (consisting of a bounded part of the infinite plane containing the local perturbation and the lower part of a circle) with two corners and some closed smooth artificial curve. It is a nontrivial extension of our previous work on direct and inverse scattering by a locally rough surface from the Dirichlet boundary condition to the Neumann boundary condition [{\em SIAM J. Appl. Math.}, 73 (2013), pp. 1811-1829]. In this paper, we make us of the recursively compressed inverse preconditioning (RCIP) method developed by Helsing to solve the integral equation which is efficient and capable of dealing with large wave numbers. For the inverse problem, it is proved that the locally rough surface is uniquely determined from a knowledge of the far-field pattern corresponding to incident plane waves. Further, based on the novel integral equation formulation, a Newton iteration method is developed to reconstruct the locally rough surface from a knowledge of multiple frequency far-field data. Numerical examples are also provided to illustrate that the reconstruction algorithm is stable and accurate even for the case of multiple-scale profiles.

Paper Structure

This paper contains 10 sections, 8 theorems, 89 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Solvability of the direct problem via the integral equation method
  3. Numerical solution of the direct scattering problem
  4. The numerical algorithm
  5. Numerical experiments
  6. Uniqueness of the inverse problem
  7. Numerical solution of the inverse problem
  8. Reconstruction algorithm with multi-frequency data
  9. Numerical examples of the inverse problem
  10. Conclusion

Key Result

Theorem 2.1

\newlabelthm1-uni The Neumann problem (NP) has at most one solution $u^s\in H^1_{loc}(D_+)$.

Figures (7)

  • Figure 2.1: The scattering problem from a locally rough surface
  • Figure 3.1: Geometry of the coarse and fine meshes
  • Figure 3.2: Geometry of the problem
  • Figure 3.3: The relative error ${|u^\infty(\hat{x};y)-u^\infty_{app}(\hat{x};y)|}/{|u^\infty(\hat{x};y)|}$ of the far-field pattern with $n_{pan}=5N$, $N=1,2,3,\ldots, 24$, respectively.
  • Figure 5.1: The reconstructed curve (dashed line) at $k=1,5,9,13,$ respectively, from $5\%$ noisy far-field data for two incident fields $u^i(x;d_l),l=1,2$, with $d_1=(\cos(-\pi/3),\sin(-\pi/3))$ and $d_2=(\cos(-2\pi/3),\sin(-2\pi/3))$, where the real curve is denoted by the solid line.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • ...and 12 more