Advances in Numerical Analysis and High-Performance Scientific Computing

Dear Colleagues,

This Special Issue aims to gather cutting-edge research at the critical intersection of numerical analysis and HPC, focusing on the development, analysis, and application of mathematical algorithms that are efficient, scalable, and robust on modern parallel computing platforms. We seek contributions that not only advance the theoretical underpinnings of numerical methods but also demonstrate their practical effectiveness in solving large-scale, complex problems across science and engineering.

We invite submissions covering, but not limited to, the following key areas:

1. Advanced Numerical Algorithms and Analysis

• High-order and adaptive methods: Finite element methods (FEM), finite difference methods (FDM), finite volume methods (FVM), and spectral methods that achieve high accuracy and adaptivity.
• Fast solvers and preconditioning: Development and parallel implementation of fast linear and nonlinear solvers, including multilevel methods (e.g., geometric and algebraic multigrid), domain decomposition, and highly scalable preconditioning techniques.
• Time integration: Implicit, explicit, and stiff solvers designed for stability and efficiency in parallel environments.
• Uncertainty quantification (UQ): Numerical methods for propagation and quantification of uncertainty in large-scale simulations (e.g., stochastic collocation, Monte Carlo methods).

2. High-Performance and Parallel Computing

• Scalable implementations: Studies on parallelization strategies (MPI, OpenMP, threading, hybrid models) and performance optimization across diverse HPC architectures (CPU clusters, GPU accelerators).
• Numerical linear algebra for big data: Techniques for handling massive, sparse, and dense linear systems, large-scale eigenvalue problems, and randomized numerical linear algebra.
• Data-driven scientific computing: Integration of machine learning and deep learning techniques (e.g., physics-informed neural networks—PINNs) to accelerate or enhance traditional numerical simulations.
• Software and tool development: Reports on new numerical libraries, frameworks, or tools designed specifically for parallel scientific computing.

3. Applications

• Validated numerical solutions to challenging inverse problems and optimization problems.
• Large-scale simulation of physical phenomena (e.g., computational fluid dynamics (CFD), electromagnetics, materials science, quantum physics, and weather/climate modeling) that require exascale or petascale computing resources.

We encourage submissions that detail both the rigorous mathematical analysis and the practical high-performance implementation and scaling results of the proposed numerical techniques.

Dr. Thái Anh Nhan
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.