A Local Model Reduction Method Based on $k$-Nearest-Neighbors for Parametrized Nonlocal Problems (Communications in Computational Physics, 2025)
In this paper, the model reduction method based on $k$-nearest-neighbors isprovided for the parametrized nonlocal partial differential equations (PDEs). In comparison to standard local PDEs, the stiffness matrix of the corresponding nonlocalmodel loses sparsity due to the nonlocal interaction parameter $δ.$ Specially the nonlocal model contains uncertain parameters, enhancing the complexity of computation.In order to improve the computation efficiency, we combine the $k$-nearest-neighborswith the model reduction method to construct the efficient surrogate models of theparametrized nonlocal problems. This method is an offline-online mechanism. In theoffline phase, we develop the full-order model by using the quadratic finite elementmethod (FEM) to generate snapshots and employ the model reduction method to process the snapshots and extract their key characters. In the online phase, we utilize $k$-nearest-neighbors regression to construct the surrogate model. In the numerical experiments, we first verify the convergence rate when applying quadratic FEM to thenonlocal problems. Subsequently, for the linear and nonlinear nonlocal problems withrandom inputs, the numerical results illustrate the efficiency and accuracy of the surrogate models.