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A Local Model Reduction Method Based on $k$-Nearest-Neighbors for Parametrized Nonlocal Problems

A Local Model Reduction Method Based on $k$-Nearest-Neighbors for Parametrized Nonlocal Problems

Year:    2025

Author:    Caixia Nan, Qiuqi Li, Huailing Song

Communications in Computational Physics, Vol. 37 (2025), Iss. 1 : pp. 220–249

Abstract

In this paper, the model reduction method based on $k$-nearest-neighbors is provided for the parametrized nonlocal partial differential equations (PDEs). In comparison to standard local PDEs, the stiffness matrix of the corresponding nonlocal model 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-neighbors with the model reduction method to construct the efficient surrogate models of the parametrized nonlocal problems. This method is an offline-online mechanism. In the offline phase, we develop the full-order model by using the quadratic finite element method (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 the nonlocal problems. Subsequently, for the linear and nonlinear nonlocal problems with random inputs, the numerical results illustrate the efficiency and accuracy of the surrogate models.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2024-0028

Communications in Computational Physics, Vol. 37 (2025), Iss. 1 : pp. 220–249

Published online:    2025-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    30

Keywords:    Parametrized nonlocal PDEs surrogate model quadratic finite element method proper orthogonal decomposition dynamic mode decomposition $k$-nearest-neighbors.

Author Details

Caixia Nan

Qiuqi Li

Huailing Song