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Numerical identification of nonlocal potentials in aggregation. (English) Zbl 1500.65054

Summary: 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.

MSC:

65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
49M41 PDE constrained optimization (numerical aspects)
49N60 Regularity of solutions in optimal control
60H50 Regularization by noise
92C17 Cell movement (chemotaxis, etc.)
92D25 Population dynamics (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
93A16 Multi-agent systems
35R30 Inverse problems for PDEs

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