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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E1: Mathematics and Computer Science".

Deadline for manuscript submissions: 31 October 2026 | Viewed by 9053

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Special Issue Editors

Dr. Hao Jiang

E-Mail Website
Guest Editor

College of Computer, National University of Defense Technology, Changsha, China
Interests: numerical analysis; high-performance computing

Prof. Dr. Zhe Quan

E-Mail Website
Guest Editor

College of Computer Science and Electronic Engineering, Hunan University, Changsha, China
Interests: high-performance computing; compiler; machine learning

Special Issue Information

Dear Colleagues,

High-performance computing, optimization and simulation have led to advances in several areas of science, engineering and technology. Recent developments permit the successful completion of computationally intensive problems such as those in chemistry, physics, aerospace, energy, material, healthcare, automobile development, etc. However, further research is required for developing new algorithms, methods and models for new hardware and software architecture, since many heterogeneous many-core processors with accelerators or coprocessors are now an integral part of modern computing systems, especially supercomputers. To fulfill the increasing computing demands, strategies based on heterogeneous computing, mixed precision computing, communication-avoiding computing, etc., have been adopted.

The suggested topics include the following:

High-performance computing;
Numerical analysis;
Accurate computing;
Mixed-precision algorithm;
Large-scale heterogeneous computing;
Performance tuning;
Optimization;
Parallelization;
Modeling and simulation.

We also welcome papers that explore new programming models and languages appropriate for the new and emerging domains and applications of HPC.

Dr. Hao Jiang
Dr. Zhe Quan
Guest Editors

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.

Keywords

high-performance computing
parallel algorithms
quantum computations
error analysis
heterogenous computing
mixed precision
communication-avoiding computing
performance tuning
optimization
parallelization
simulation

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Published Papers (6 papers)

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Research

33 pages, 647 KB

Open AccessArticle

New Mathematics for Computer Performance: Array Algebra and Cost Functions

by Gaétan Hains and Lenore Mullin

Mathematics 2026, 14(9), 1479; https://doi.org/10.3390/math14091479 - 28 Apr 2026

Viewed by 326

Abstract

MoA (mathematics of arrays) is a theory of parallel operations on arrays that can describe all known algorithms in linear algebra, signal processing, and HPC because they are based on primitive recursion and array shapes. Mapping parallel algorithms to computer architectures remains more of an art than a science, and specific mathematical techniques are needed to provide a basis for performance evaluation at a level abstract enough to constitute an experimental science. In this paper we present a methodology for parallel code generation from MoA expressions. Then, we relate the MoA operators to the linear space of memory elements in computer architecture. Finally, we define a theory of execution costs that is based on classical operations research and is formally related to MoA-based parallel code generation. This constitutes a formalized and mechanizable approach to performance prediction, portability and optimization. Full article

(This article belongs to the Special Issue Advances in High-Performance Computing, Optimization and Simulation)

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38 pages, 3715 KB

Open AccessArticle

Stable and Efficient Gaussian-Based Kolmogorov–Arnold Networks

by Pasquale De Luca, Emanuel Di Nardo, Livia Marcellino and Angelo Ciaramella

Mathematics 2026, 14(3), 513; https://doi.org/10.3390/math14030513 - 31 Jan 2026

Viewed by 740

Abstract

Kolmogorov–Arnold Networks employ learnable univariate activation functions on edges rather than fixed node nonlinearities. Standard B-spline implementations require

O (3 K W)

parameters per layer (K basis functions, W connections). We introduce shared Gaussian radial basis functions with learnable centers [...] Read more.

Kolmogorov–Arnold Networks employ learnable univariate activation functions on edges rather than fixed node nonlinearities. Standard B-spline implementations require

O (3 K W)

parameters per layer (K basis functions, W connections). We introduce shared Gaussian radial basis functions with learnable centers

μ_{k}^{(l)}

and widths

σ_{k}^{(l)}

maintained globally per layer, reducing parameter complexity to

O (K W + 2 L K)

for L layers—a threefold reduction, while preserving Sobolev convergence rates

O (h^{s - Ω})

. Width clamping at

σ_{min} = 10^{- 6}

and tripartite regularization ensure numerical stability. On MNIST with architecture

[784, 128, 10]

and

K = 5

, RBF-KAN achieves

87.8 %

test accuracy versus

89.1 %

for B-spline KAN with

1.4 \times

speedup and 33% memory reduction, though generalization gap increases from

1.1 %

2.7 %

due to global Gaussian support. Physics-informed neural networks demonstrate substantial improvements on partial differential equations: elliptic problems exhibit a

45 \times

reduction in PDE residual and maximum pointwise error, decreasing from

1.32

0.18

; parabolic problems achieve a

2.1 \times

accuracy gain; hyperbolic wave equations show a

19.3 \times

improvement in maximum error and a

6.25 \times

reduction in

L^{2}

norm. Superior hyperbolic performance derives from infinite differentiability of Gaussian bases, enabling accurate high-order derivatives without polynomial dissipation. Ablation studies confirm that coefficient regularization reduces mean error by 40%, while center diversity prevents basis collapse. Optimal basis count

K \in [3, 5]

balances expressiveness and overfitting. The architecture establishes Gaussian RBFs as efficient alternatives to B-splines for learnable activation networks with advantages in scientific computing. Full article

(This article belongs to the Special Issue Advances in High-Performance Computing, Optimization and Simulation)

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21 pages, 538 KB

Open AccessArticle

Evaluation of GPU-Accelerated Edge Platforms for Stochastic Simulations: Performance and Energy Efficiency Analysis

by Pilsung Kang

Mathematics 2025, 13(20), 3305; https://doi.org/10.3390/math13203305 - 16 Oct 2025

Viewed by 1687

Abstract

With the increasing emphasis on energy-efficient computing, edge devices accelerated by graphics processing units (GPUs) are gaining attention for their potential in scientific workloads. These platforms support compute-intensive simulations under strict energy and resource constraints, yet their computational efficiency across architectures remains an open question. This study evaluates the performance of GPU-based edge platforms for executing the stochastic simulation algorithm (SSA), a widely used and inherently compute-intensive method for modeling biochemical and physical systems. Execution time, floating point throughput, and the trade-offs between cost and power consumption are analyzed, with a focus on how variations in core count, clock speed, and architectural features impact SSA scalability. Experimental results show that the Jetson Orin NX consistently outperforms Xavier NX and Orin Nano in both speed and efficiency, reaching up to 4.86 million iterations per second while operating under a 20 W power envelope. At the largest workload scale, it achieves 2102.7 ms/W in energy efficiency and 105.3 ms/USD in cost-performance—substantially better than the other Jetson devices. These findings highlight the architectural considerations necessary for selecting edge GPUs for scientific computing and offer practical guidance for deploying compute-intensive workloads beyond artificial intelligence (AI) applications. Full article

(This article belongs to the Special Issue Advances in High-Performance Computing, Optimization and Simulation)

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18 pages, 960 KB

Open AccessArticle

Hybrid Algorithm via Reciprocal-Argument Transformation for Efficient Gauss Hypergeometric Evaluation in Wireless Networks

by Jianping Cai and Zuobin Ying

Mathematics 2025, 13(15), 2354; https://doi.org/10.3390/math13152354 - 23 Jul 2025

Viewed by 653

Abstract

The rapid densification of wireless networks demands efficient evaluation of special functions underpinning system-level performance metrics. To facilitate research, we introduce a computational framework tailored for the zero-balanced Gauss hypergeometric function [...] Read more.

Ψ (x, y) ≜ {}_{2}F_{1} (1, x; 1 + x; - y)

, a fundamental mathematical kernel emerging in Signal-to-Interference-plus-Noise Ratio (SINR) coverage analysis of non-uniform cellular deployments. Specifically, we propose a novel Reciprocal-Argument Transformation Algorithm (RTA), derived rigorously from a Mellin–Barnes reciprocal-argument identity, achieving geometric convergence with

O (1 / y)

. By integrating RTA with a Pfaff-series solver into a hybrid algorithm guided by a golden-ratio switching criterion, our approach ensures optimal efficiency and numerical stability. Comprehensive validation demonstrates that the hybrid algorithm reliably attains machine-precision accuracy (

\sim 10^{- 16}

) within 1 μs per evaluation, dramatically accelerating calculations in realistic scenarios from hours to fractions of a second. Consequently, our method significantly enhances the feasibility of tractable optimization in ultra-dense non-uniform cellular networks, bridging the computational gap in large-scale wireless performance modeling. Full article

(This article belongs to the Special Issue Advances in High-Performance Computing, Optimization and Simulation)

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19 pages, 560 KB

Open AccessArticle

Accurate Sum and Dot Product with New Instruction for High-Precision Computing on ARMv8 Processor

by Kaisen Xie, Qingfeng Lu, Hao Jiang and Hongxia Wang

Mathematics 2025, 13(2), 270; https://doi.org/10.3390/math13020270 - 15 Jan 2025

Cited by 2 | Viewed by 2292

Abstract

The accumulation of rounding errors can lead to unreliable results. Therefore, accurate and efficient algorithms are required. A processor from the ARMv8 architecture has introduced new instructions for high-precision computation. We have redesigned and implemented accurate summation and the accurate dot product. The [...] Read more.

7 n - 5

and

10 n - 5

4 n - 2

and

7 n - 2

, compared with the classic compensated precision algorithms. It has been proven that our accurate summation and dot algorithms’ error bounds are

γ_{n - 1} γ_{n} cond + u

and

γ_{n} γ_{n + 1} cond + u

, where ‘cond’ denotes the condition number,

γ_{n} = n \cdot u / (1 - n \cdot u)

, and u denotes the relative rounding error unit. Our accurate summation and dot product achieved a 1.69× speedup and a 1.14× speedup, respectively, on a simulation platform. Numerical experiments also illustrate that, under round-towards-zero mode, our algorithms are as accurate as the classic compensated precision algorithms. Full article

(This article belongs to the Special Issue Advances in High-Performance Computing, Optimization and Simulation)

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22 pages, 689 KB

Open AccessArticle

GPU Accelerating Algorithms for Three-Layered Heat Conduction Simulations

by Nicolás Murúa, Aníbal Coronel, Alex Tello, Stefan Berres and Fernando Huancas

Mathematics 2024, 12(22), 3503; https://doi.org/10.3390/math12223503 - 9 Nov 2024

Cited by 3 | Viewed by 2096

Abstract

In this paper, we consider the finite difference approximation for a one-dimensional mathematical model of heat conduction in a three-layered solid with interfacial conditions for temperature and heat flux between the layers. The finite difference scheme is unconditionally stable, convergent, and equivalent to the solution of two linear algebraic systems. We evaluate various methods for solving the involved linear systems by analyzing direct and iterative solvers, including GPU-accelerated approaches using CuPy and PyCUDA. We evaluate performance and scalability and contribute to advancing computational techniques for modeling complex physical processes accurately and efficiently. Full article

(This article belongs to the Special Issue Advances in High-Performance Computing, Optimization and Simulation)

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