Numerical Identification of Nonlocal Potential in Aggregation

TL;DR

This paper introduces a robust numerical method for identifying nonlocal potentials in aggregation models from noisy data, using regularization and denoising techniques, applicable to static and time-varying potentials.

Contribution

It develops a novel regularized optimization approach with denoising for stable potential identification from noisy data, including constraints and time-varying cases.

Findings

Method effectively recovers potentials in noisy environments.

Regularization and denoising improve stability and accuracy.

Applicable to both static and dynamic potentials in multiple dimensions.

Abstract

Aggregation equations are broadly used to model population dynamics with nonlocal interactions, characterized by a potential in the equation. This paper considers the inverse problem of identifying the potential from a single noisy spatial-temporal process. The identification is challenging in the presence of noise due to the instability of numerical differentiation. We propose a robust model-based technique to identify the potential by minimizing a regularized data fidelity term, and regularization is taken as the total variation and the squared Laplacian. A split Bregman method is used to solve the regularized optimization problem. Our method is robust to noise by utilizing a Successively Denoised Differentiation technique. We consider additional constraints such as compact support and symmetry constraints to enhance the performance further. We also apply this method to identify time-varying potentials and identify the interaction kernel in an agent-based system. Various numerical examples in one and two dimensions are included to verify the effectiveness and robustness of the proposed method.