Numerical Identification of Nonlocal Potential in Aggregation

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.

References in corpus (9)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• A combined first and second order variational approach for image reconstruction
• The derivation of Swarming models: Mean-Field Limit and Wasserstein distances
• Nonparametric inference of interaction laws in systems of agents from trajectory data
• Formation of clumps and patches in self-aggregation of finite size particles
• Multiplication in Sobolev Spaces, Revisited
• An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications to Surface Smoothing and Imaging
• Learning interacting particle systems: diffusion parameter estimation for aggregation equations
• Robust Identification of Differential Equations by Numerical Techniques from a Single Set of Noisy Observation

Cited by in corpus (1)

• WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming