A finite element framework for simulating residential burglary in realistic urban geometries

TL;DR

This paper develops a finite element framework to simulate residential burglary using a PDE model derived from an agent-based approach, incorporating realistic urban geometries and boundary conditions, with efficient numerical schemes and open-source code.

Contribution

It introduces a novel finite element method with natural boundary conditions for a nonlinear PDE burglary model, enabling realistic urban simulations and efficient computations.

Findings

Robust and efficient numerical scheme for PDE burglary model

Successful simulation in realistic city geometry (Chicago)

Flexible framework for heterogeneous parameters

Abstract

We consider a partial differential equation (PDE) model to predict residential burglary derived from a probabilistic agent-based model through a mean-field limit operation. The PDE model is a nonlinear, coupled system of two equations in two variables (attractiveness of residential sites and density of criminals), similar to the Keller-Segel model for aggregation based on chemotaxis. Unlike previous works, which applied periodic boundary conditions, we enforce boundary conditions that arise naturally from the variational formulation of the PDE problem, i.e., the starting point for the application of a finite element method. These conditions specify the value of the normal derivatives of the system variables at the boundary. For the numerical solution of the PDE problem discretized in time and space, we propose a scheme that decouples the computation of the attractiveness from the computation of the criminal density at each time step, resulting in the solution of two linear algebraic systems per iteration. Through numerous numerical tests, we demonstrate the robustness and computational efficiency of this approach. Leveraging the flexibility allowed by the finite element method, we show results for spatially heterogeneous model parameters and a realistic geometry (city of Chicago). The paper includes a discussion of future perspectives to build multiscale, 'multi-physics' models that can become a tool for the community. The robust and efficient code developed for this paper, which is shared open-source, is intended as the solid base for this broader research program.