High-Dimensional Numerical Methods for Nonlocal Models

Yujing Jia, Dongbo Wang and Xu Guo ()
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Yujing Jia: School of Civil Engineering, Shandong University, Jinan 250061, China
Dongbo Wang: School of Civil Engineering, Shandong University, Jinan 250061, China
Xu Guo: School of Civil Engineering, Shandong University, Jinan 250061, China

Mathematics, 2025, vol. 13, issue 21, 1-23

Abstract: Nonlocal models offer a unified framework for describing long-range spatial interactions and temporal memory effects. The review briefly outlines several representative physical problems, including anomalous diffusion, material fracture, viscoelastic wave propagation, and electromagnetic scattering, to illustrate the broad applicability of nonlocal systems. However, the intrinsic global coupling and historical dependence of these models introduce significant computational challenges, particularly in high-dimensional settings. From the perspective of algorithmic strategies, the review systematically summarizes high-dimensional numerical methods applicable to nonlocal equations, emphasizing core approaches for overcoming the curse of dimensionality, such as structured solution frameworks based on FFT, spectral methods, probabilistic sampling, physics-informed neural networks, and asymptotically compatible schemes. By integrating recent advances and common computational principles, the review establishes a dual “problem review + method review” structure that provides a systematic perspective and valuable reference for the modeling and high-dimensional numerical simulation of nonlocal systems.

Keywords: nonlocal models; high-dimensional computation; curse of dimensionality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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