Numerical Identification of Nonlocal Potentials in Aggregation

Yuchen He, Sung Ha Kang, Wenjing Liao, Hao Liu*, Yingjie Liu

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

2 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)638-670
Number of pages33
JournalCommunications in Computational Physics
Issue number3
Publication statusPublished - Sept 2022

Scopus Subject Areas

  • Physics and Astronomy (miscellaneous)

User-Defined Keywords

  • Aggregation equation
  • nonlocal potential
  • PDE identification
  • Bregman iteration
  • operator splitting.


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