论文标题
聚集非本地电位的数值鉴定
Numerical Identification of Nonlocal Potential in Aggregation
论文作者
论文摘要
聚集方程广泛用于建模种群动力学与非局部相互作用的人群动力学，其特征是方程中的潜力。本文考虑了从单个嘈杂的时空过程中识别潜力的反问题。由于数值差异的不稳定性，在存在噪声的情况下，该识别具有挑战性。我们提出了一种强大的基于模型的技术来通过最大程度地降低正规数据保真度项来识别潜力，并将正则化作为总变化和平方拉普拉斯式。 Bregman拆分方法用于解决正则优化问题。我们的方法通过使用连续的分化技术来对噪声具有鲁棒性。我们考虑其他约束，例如紧凑的支持和对称约束，以进一步提高性能。我们还应用这种方法来识别时变电势并确定基于代理的系统中的相互作用内核。包括一个和二维中的各种数值示例，以验证所提出方法的有效性和鲁棒性。
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.
