On the convergence of the no-response test for the heat equation

TL;DR

This paper proves the convergence of the no-response test (NRT) for the heat equation, establishing conditions under which the indicator function correctly identifies cavities inside a heat conductor without relying on duality.

Contribution

It provides a direct proof of NRT convergence for the heat equation, expanding the theoretical understanding of domain sampling methods in inverse problems.

Findings

Proves NRT convergence without duality assumptions.

Shows indicator function finiteness characterizes cavity inclusion.

Utilizes analytical extension properties of heat solutions.

Abstract

Domain sampling methods called the range test (RT) and no-response test (NRT), and their duality are known for several inverse scattering problems and an inverse boundary value problem for the Laplace operator (see Section 1 for more details). In our previous work [21], we established the duality between the NRT and RT, and demonstrated the convergence of the RT for the heat equation. We also provided numerical studies for both methods. However, we did not address the convergence for the NRT. As a continuation of this work, we prove the convergence of the NRT without using the duality. Specifically, assuming there exists a cavity $D$ inside a heat conductor $\Omega$, we define an indicator function $I_{NRT}(G)$ for a prescribed test domain $G$, where $\overline G\subset\Omega$ (i.e., $G\Subset\Omega$). By using the analytical extension property of solutions to the heat equation with respect to the spatial variables, we prove the convergence result given as $I_{NRT}(G)<\infty$ if and only if $\overline{D}\subset \overline{G}$, provided that the solution to the heat equation cannot be analytically extended across the boundary of the cavity. Thus, we complete the theoretical study of both methods. Here the analytic extension of solutions does not require the property that the solutions are real analytic with respect to the space variables. However, for the proof of the mentioned convergence result, we fully use this property.