Aggregation equations are broadly used tomodel population dynamicswith nonlocal interactions,characterized by a potential in the equation.This paper considers the inverse problem of identifying the potential from a si...Aggregation equations are broadly used tomodel population dynamicswith nonlocal interactions,characterized by a potential in the equation.This paper considers the inverse problem of identifying the potential from a single noisy spatialtemporal 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 thismethod 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.展开更多
The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation...The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.展开更多
基金supported in part by Simons Foundation grant 282311 and 584960supported in part by NSF grant NSF-DMS 1818751 and NSF-DMS 2012652+1 种基金supported in part by HKBU 162784 and 179356supported in part by NSF grants DMS-1522585 and DMS-CDS&E-MSS-1622453.
文摘Aggregation equations are broadly used tomodel population dynamicswith nonlocal interactions,characterized by a potential in the equation.This paper considers the inverse problem of identifying the potential from a single noisy spatialtemporal 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 thismethod 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.
基金Supported by the National Nature Science Foundation of China(12101356,12101357,12071254,11771253)the National Science Foundation of Shandong Province(ZR2021QA065,ZR2020QA009,ZR2021MA047)the China Postdoctoral Science Foundation(2019M662313)。
文摘The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.
