Sensor placement via large deviations in the Eikonal equation

TL;DR

The paper studies optimal geometric sensor placement inside a region to minimize the maximal or mean distance to points in the domain, formalized as the Hausdorff distance $d^H(\omega, \Omega)$ with $|\omega|=c$. It replaces the non-differentiable objective with a PDE-based proxy using Varadhan's theorem: solve $w_\varepsilon-\varepsilon\Delta w_\varepsilon=0$ in a containing box and define $v_\varepsilon= -\sqrt{\varepsilon}\log w_\varepsilon$, which converges to the distance to the sensor set as $\varepsilon\to 0$; the viscous Eikonal equation $1-| abla v_\varepsilon|^2+\sqrt{\varepsilon}\Delta v_\varepsilon=0$ with $v_\varepsilon=0$ on the boundary then yields a computable proxy $f_{p,\varepsilon}$. The authors derive the gradient of this proxy with respect to sensor centers via shape derivatives and an adjoint problem, enabling gradient-based optimization of the sensor configuration. Numerical experiments demonstrate sensor placements for $N∈{1,2,3}$ in various domains, revealing symmetry-breaking phenomena and the importance of multi-start strategies to approach better local minima. Overall, the work provides a PDE-based, geometrically grounded initialization for PDE-informed sensor design with potential extensions to broader settings.

Abstract

In this work, we address the problem of optimally placing a finite number of sensors within a given region so as to minimize the mean or maximal distance to the points of the domain. To tackle this natural geometric performance criterion, formulated in terms of distance functions, we combine tools from geometric analysis with a classical result of Varadhan, which provides an efficient approximation of the distance function via the solution of a simple elliptic PDE. The effectiveness of the proposed approach is demonstrated through illustrative numerical simulations.

Figures (5)

• Figure 1: The box $D$ containing the domain $\Omega$ and the sensors $(B_i)$. As it can be seen in the figure, the box $D$ is chosen such that $d(B_4,\partial \Omega)=d(B_4,M_4)\leq d(M_4,\partial D)$
• Figure 2: Optimal placement of $N\in \{1,2,3\}$ sensors inside the country of Saudi Arabia for different values of $p$.
• Figure 3: Optimal placement of $N\in \{1,2,3\}$ sensors in rhombus for different values of $p$.
• Figure 4: Optimal placement of $N\in \{1,2,3\}$ sensors in rhombus for different values of $p$.
• Figure 5: Two critical configurations of the sensors obtained for $p=1$ by considering different initial positions: $(0.3,0)$ and $(-0.3,0)$ for the case on the left and $(0.1,0.1)$ and $(-0.3,-0.1)$ for the case on the right, which seems to be a global maximum.

Theorems & Definitions (8)

• Theorem 3.1
• Lemma 3.2
• proof
• Theorem 4.1
• proof
• Remark 4.2
• Remark 4.3
• Remark 4.4