Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Removing small wavenumber constraints in Side B of the Probe Method

Masaru Ikehata

Abstract

The Probe Method is an analytical reconstruction scheme for inverse obstacle problems utilizing the Dirichlet-to-Neumann map associated with the governing partial differential equation. It consists of two distinct parts: Side A and Side B. Both are based on the indicator sequence which is calculated from the Dirichlet-to-Neumann map acting on "needle-like" specialized solution of the governing equation for the background medium, whose energy is concentrated on an arbitrary given needle inside. In Side A, the limit of the indicator sequence-referred to as the indicator function-is computed before the needles touch the obstacle, and the boundary is identified as the point where this function first blows up. In contrast, Side B states the blow-up of the indicator sequence after the needles have come into contact with the obstacle. For the Helmholtz equation, the validity of Side B has long required a small wavenumber constraint. This paper finally removes this long-standing restriction, establishing the method's applicability for broader cases.

Removing small wavenumber constraints in Side B of the Probe Method

Abstract

The Probe Method is an analytical reconstruction scheme for inverse obstacle problems utilizing the Dirichlet-to-Neumann map associated with the governing partial differential equation. It consists of two distinct parts: Side A and Side B. Both are based on the indicator sequence which is calculated from the Dirichlet-to-Neumann map acting on "needle-like" specialized solution of the governing equation for the background medium, whose energy is concentrated on an arbitrary given needle inside. In Side A, the limit of the indicator sequence-referred to as the indicator function-is computed before the needles touch the obstacle, and the boundary is identified as the point where this function first blows up. In contrast, Side B states the blow-up of the indicator sequence after the needles have come into contact with the obstacle. For the Helmholtz equation, the validity of Side B has long required a small wavenumber constraint. This paper finally removes this long-standing restriction, establishing the method's applicability for broader cases.
Paper Structure (8 sections, 8 theorems, 66 equations)

This paper contains 8 sections, 8 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1.3
  3. Implication to Side B of the Probe Method
  4. A review on two sides of the Probe Method
  5. Blow-up of the indicator sequence
  6. Obstacle characterization
  7. Sound-soft obstacle
  8. A remaining problem

Key Result

Proposition 1.1

Given $x\in\Omega$ and $\sigma\in N_x$ let $\{v_n\}\in {\mathcal{N}}(x,\sigma)$. (a) Let $V$ be an arbitrary finite and open cone with vertex at $x$. We have (b) Given a point $z\in\sigma\setminus\{\sigma(0)\}$ let $B$ be an arbitrary open ball centered at $z$. We have

Theorems & Definitions (8)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Theorem 4.1