TY - JOUR
T1 - Numerical Identification of Nonlocal Potentials in Aggregation
AU - He , Yuchen
AU - Kang , Sung Ha
AU - Liao , Wenjing
AU - Liu , Hao
AU - Liu , Yingjie
JO - Communications in Computational Physics
VL - 3
SP - 638
EP - 670
PY - 2022
DA - 2022/09
SN - 32
DO - http://doi.org/10.4208/cicp.OA-2021-0177
UR - https://global-sci.org/intro/article_detail/cicp/21041.html
KW - Aggregation equation, nonlocal potential, PDE identification, Bregman iteration, operator splitting.
AB - <p style="text-align: justify;">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.</p>