Numerical Identification of Nonlocal Potentials in Aggregation (Communications in Computational Physics, 2022)
Aggregation equations are broadly used to model population dynamics withnonlocal 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 theinstability of numerical differentiation. We propose a robust model-based techniqueto identify the potential by minimizing a regularized data fidelity term, and regularization is taken as the total variation and the squared Laplacian. A split Bregmanmethod is used to solve the regularized optimization problem. Our method is robustto noise by utilizing a Successively Denoised Differentiation technique. We consideradditional constraints such as compact support and symmetry constraints to enhancethe performance further. We also apply this method to identify time-varying potentialsand identify the interaction kernel in an agent-based system. Various numerical examples in one and two dimensions are included to verify the effectiveness and robustnessof the proposed method.