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Volume 32, Issue 3
Numerical Identification of Nonlocal Potentials in Aggregation

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

Commun. Comput. Phys., 32 (2022), pp. 638-670.

Published online: 2022-09

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  • 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.

  • AMS Subject Headings

93C15, 35R30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-32-638, author = {He , YuchenKang , Sung HaLiao , WenjingLiu , Hao and Liu , Yingjie}, title = {Numerical Identification of Nonlocal Potentials in Aggregation}, journal = {Communications in Computational Physics}, year = {2022}, volume = {32}, number = {3}, pages = {638--670}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0177}, url = {http://global-sci.org/intro/article_detail/cicp/21041.html} }
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 -

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.

Yuchen He, Sung Ha Kang, Wenjing Liao, Hao Liu & Yingjie Liu. (2022). Numerical Identification of Nonlocal Potentials in Aggregation. Communications in Computational Physics. 32 (3). 638-670. doi:10.4208/cicp.OA-2021-0177
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