Sensitivity analysis for solutions to heterogeneous nonlocal systems

TL;DR

This paper investigates how solutions to complex nonlocal systems respond to data and parameter changes, providing explicit stability bounds and validating them through numerical simulations of heterogeneous kernels and discontinuous solutions.

Contribution

It offers new stability results for nonlocal systems with heterogeneous kernels, including explicit bounds and analysis of nonlinear sensitivities, supported by numerical validation.

Findings

Explicit bounds on solution stability under data perturbations

Analysis of solutions with heterogeneous and discontinuous kernels

Numerical simulations confirming theoretical results

Abstract

The paper presents a collection of results on continuous dependence for solutions to nonlocal problems under perturbations of data and system parameters. The integral operators appearing in the systems capture interactions via heterogeneous kernels that exhibit different types of weak singularities, space dependence, even regions of zero-interaction. The stability results showcase explicit bounds involving the measure of the domain and of the interaction collar size, nonlocal Poincar\'e constant, and other parameters. In the nonlinear setting the bounds quantify in different $L^p$ norms the sensitivity of solutions under different nonlinearity profiles. The results are validated by numerical simulations showcasing discontinuous solutions, varying horizons of interactions, and symmetric and heterogeneous kernels.