Removing small wavenumber constraints in Side B of the Probe Method

TL;DR

This paper extends the Probe Method for inverse obstacle problems by removing the small wavenumber constraint in Side B, enabling broader applicability for the Helmholtz equation.

Contribution

It proves the validity of Side B of the Probe Method without the previously necessary small wavenumber restriction.

Findings

The method now applies to larger wavenumbers for Helmholtz equations.

The removal of the constraint broadens the method's practical use.

Theoretical validation of Side B's applicability is established.

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.