Advances in High-Dimensional Scientific Computing

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: 10 September 2026 | Viewed by 854

Special Issue Editor


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Guest Editor
Geotechnical and Structural Engineering Center, Shandong University, Shandong 250061, China
Interests: cross-application research of geotechnical engineering based on scientific calculations; high-precision numerical calculation and numerical analysis; partial differential equation and its inversion calculation; data modeling and calculation; risk estimation and quantification

Special Issue Information

Dear Colleagues,

With the continuous advancement of computational capabilities, solving high-dimensional data and complex problems has become one of the key challenges in modern engineering and scientific research. This Special Issue aims to gather research on high-dimensional numerical computing, efficient solutions for partial differential equations, and large-scale data modeling and computation, exploring their innovative applications in geotechnical engineering. In particular, we will focus on how high-precision numerical computing and numerical analysis methods can address complex problems in geotechnical and structural engineering, such as simulating the mechanical behavior of geotechnical media, risk assessment, and uncertainty quantification.

Additionally, the Special Issue will highlight how high-dimensional computational techniques facilitate the effective solution of partial differential equations and their inverse calculations, providing more precise and efficient tools for engineering design and decision-making. By showcasing the latest research developments, this Special Issue aims to promote the broad application of high-dimensional scientific computing in geotechnical engineering, offering new perspectives for future research directions.

Prof. Dr. Xu Guo
Guest Editor

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Keywords

  • non-local
  • peridynamics
  • fractional order
  • high-dimensional scientific computing
  • numerical computation
  • partial differential equations
  • inverse computation
  • geotechnical engineering
  • risk assessment
  • uncertainty quantification
  • large-scale data modeling
  • high-precision numerical analysis
  • structural engineering

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Published Papers (1 paper)

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Review

23 pages, 1934 KB  
Review
High-Dimensional Numerical Methods for Nonlocal Models
by Yujing Jia, Dongbo Wang and Xu Guo
Mathematics 2025, 13(21), 3512; https://doi.org/10.3390/math13213512 - 2 Nov 2025
Viewed by 624
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, [...] Read more.
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. Full article
(This article belongs to the Special Issue Advances in High-Dimensional Scientific Computing)
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