Numerical Identification of Nonlocal Potentials in Aggregation

Aggregation equations are broadly used tomodel population dynamicswith nonlocal interactions,characterized by a potential in the equation.This paper considers the inverse problem of identifying the potential from a single noisy spatialtemporal 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 thismethod 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.