# Sensor placement via large deviations in the Eikonal equation

## TL;DR

This paper presents a novel method for optimal sensor placement in a region by approximating distance functions through elliptic PDEs, combining geometric analysis with classical PDE results, and demonstrating effectiveness via simulations.

## Contribution

It introduces a new PDE-based approach for sensor placement optimization using large deviations in the Eikonal equation, integrating geometric analysis with classical PDE results.

## Key findings

- Effective sensor placement achieved in simulations
- Approximation of distance functions via elliptic PDEs
- Method outperforms traditional geometric approaches

## 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.