On the uniqueness of the discrete Calderon problem on multi-dimensional lattices

Abstract

In this work, we investigate the discrete Calderón problem on grid graphs of dimension three or higher, formed by hypercubic structures. The discrete Calderón problem is concerned with determining whether the discrete Dirichlet-to-Neumann (DtN) operator, which links boundary potentials to boundary current responses, can uniquely identify the conductivity values on the graph edges. We provide an affirmative answer to the question, thereby extending the classical uniqueness result of Curtis and Morrow for two-dimensional square lattices. The proof employs a novel slicing technique that decomposes the problem into lower-dimensional components. Additionally, we support the theoretical finding with numerical experiments that illustrate the effectiveness of the approach.

Figures (8)

• Figure 1: The interface partition within the domain $D\subset \mathbb{Z}^3$. The blue region is in $L_{t+1}^\mathcal{S}$.
• Figure 2: $J_t^\mathcal{S}$ (blue), $L_{t+1}$ (green) and $\partial D\setminus J_t^\mathcal{S}$ (yellow) in the three dimensional case and its projected view.
• Figure 3: Node deletion in dimension $d=3$. $L_t$$($blue$)$, $L_{t+1}$$($black$)$ and $K_t^-$$($orange$)$ for $t \le n+d-1$.
• Figure 4: $L_t$$($blue$)$, $L_{t+1}$$($black$)$, $K_t^-$$($orange$)$ and $K_{t+1}^+$$($green$)$ for $t \le n+d-1$$($left$)$ and $t> n+d-1$$($right$)$ in the 3D case.
• Figure 5: A schematic illustration of $\mathbf J_p^\mathbf u$ with different $p$ in the 3D case with $L_t$ (blue), $L_{t+1}$$($black$)$, $K_t^-$$($orange$)$ and $K_{t+1}^+$$($green$)$.

Theorems & Definitions (31)

• Theorem 1.1: Uniqueness
• Proposition 2.1
• proof
• Remark 2.1
• Proposition 2.2
• proof
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 3.1