numerical algorithm development

This work addresses the challenges of integrating sparse linear algebra libraries into scientific computing applications—such as computational fluid dynamics (CFD), power grid simulation, and cardiac electrophysiology—including poor maintainability, high cross-platform adaptation costs, and tight coupling between application code and low-level implementations. We propose a modular integration framework built upon Ginkgo, which achieves loose coupling via a unified abstract interface, explicit decoupling of algorithms from hardware backends, and runtime backend selection. From a software engineering perspective, the framework significantly reduces integration complexity while enhancing portability, testability, and long-term maintainability. Experimental evaluation demonstrates that the framework sustains high performance across heterogeneous platforms (CPU/GPU), shortens the hardware adaptation cycle, and enables efficient, sustainable multi-domain simulation.

This work addresses a key limitation of conventional LLM-based PDE solvers, which implicitly embed numerical strategies within generated code, making pre-execution validation and post-failure correction challenging. To overcome this, the authors propose AutoPDE, the first framework to explicitly model solution strategies as revisable, decoupled objects separate from implementation code. AutoPDE employs a three-stage pipeline—PDE type identification, numerical method selection, and adaptive parameter tuning—augmented by low-overhead trial solves and a reusable skill library to construct and refine strategies prior to code generation. Evaluated on the PDE Agent Bench, AutoPDE achieves a 54.5% pass rate, outperforming the strongest baseline by 14.2 percentage points, thereby substantially improving both the reliability and interpretability of AI-driven PDE solving.

This study systematically compares MATLAB, Mathematica, and Maple in solving ordinary differential equations (ODEs), partial differential equations (PDEs), and systems of differential equations. A unified benchmark suite—grounded in analytically tractable reference solutions—is employed to empirically evaluate the tools across five dimensions: syntactic usability, numerical accuracy, computational efficiency, visualization capability, and specialized solver functionality. Crucially, the work introduces a novel, problem-driven software selection framework that classifies tasks by intrinsic characteristics—including equation type, stiffness, and boundary condition complexity. Results indicate that Mathematica excels in symbolic solution derivation and medium-scale ODE accuracy; MATLAB demonstrates superior performance in large-scale numerical simulation and engineering-oriented PDE applications; and Maple offers distinctive advantages in special-function handling and analytic derivation. This is the first systematic, multidimensional comparative study of these major mathematical software platforms, thereby bridging a critical gap in computational tool evaluation and providing actionable, evidence-based guidance for scientific and engineering practice.

This work addresses the need for mathematically verifiable solutions to parameter identification, stability analysis, and optimization in control theory. We propose a unified framework integrating symbolic elimination with high-precision interval arithmetic. Implemented as the open-source Julia package PACE.jl, it is the first tool to deeply integrate discriminant variety construction, rational univariate representation (RUR), and arbitrary-precision interval arithmetic—enabling verified numerical solving for both parametric and non-parametric dynamical systems. Compared to conventional numerical methods, our approach significantly improves reliability and accuracy in stability certification and parameter estimation, delivering rigorously error-bounded solutions on benchmark control problems. The core contribution lies in the design and engineering implementation of a symbolic–numeric co-verification mechanism, establishing a new paradigm for trustworthy control algorithm development.

Scientific software selection frequently suffers from non-reproducible benchmarks due to multi-library, multi-metric evaluation and dynamic evolution—such as the introduction of new algorithms or modifications to test cases and evaluation criteria. This paper addresses numerical integration over arbitrary 2D/3D domains with implicit or parameterized boundaries (cut-cell quadrature), proposing the first automated benchmarking framework that systematically integrates CI/CD engineering practices into scientific computing workflows. The framework unifies GitHub Actions, Docker, Python-based scheduling, Jupyter-based report generation, and semantically versioned result archiving. It supports automated configuration, execution, visualization, and historical result comparison. It achieves >90% automation for benchmark tasks and regression detection; reduces integration time for new libraries or algorithms by 70%; and enables precise attribution of performance deviations to specific code commits. The framework significantly enhances reliability, reproducibility, and evolutionary adaptability in scientific software evaluation.

Modern computational fluid dynamics (CFD) urgently requires seamless integration of simulation into design, optimization, and data-driven workflows, confronting challenges in the co-design of physical models, numerical methods, heterogeneous hardware, and automatic differentiation. This work systematically evaluates the suitability of the Julia programming language for CFD, leveraging its unified language ecosystem, multiple dispatch, and type specialization to deeply integrate high performance, differentiability, and software composability. Empirical validation through distributed CPU/multi-GPU parallelism, performance-portable frameworks, and open-source CFD projects demonstrates the feasibility of native Julia-based CFD at scale and its advantages in differentiable workflows. Nevertheless, the maturity of Julia’s industrial toolchain still lags behind that of conventional languages.

This work proposes and implements a high-performance, unified numerical optimization framework in Rust, addressing the pervasive challenge of numerical optimization in scientific computing and engineering. The framework natively supports diverse constraint types, integrates multiple optimization algorithms, and offers a consistent, extensible interface design. As the first numerical optimization library in the Rust ecosystem to combine a rich suite of solvers, first-class constraint handling, and high computational efficiency, it demonstrates exceptional practicality and scalability across applications such as model fitting, simulation calibration, and machine learning training.

This work addresses the challenge of selecting the regularization parameter (nugget) in ill-posed linear systems arising in machine learning, where existing adaptive methods lack compatibility with automatic differentiation and suffer from computational inefficiency. To overcome these limitations, we introduce autonugget, a lightweight Python package fully compatible with JAX’s automatic differentiation framework. Our approach uniquely integrates Richardson extrapolation with Tikhonov regularized solutions computed across multiple nugget values, thereby preserving end-to-end differentiability while avoiding the information loss inherent in single-solution strategies. Experimental results demonstrate that autonugget significantly enhances solution accuracy and training stability without compromising rapid prototyping capabilities.

This work addresses the susceptibility of deep learning operators to numerical instability under finite-precision arithmetic, which can lead to error accumulation and result corruption. For the first time, the CESTAC stochastic rounding error analysis method is systematically introduced into the deep learning domain. The authors propose a unified software framework that enables automatic detection of operator-level numerical instability, precise localization of instability sources, and end-to-end stability monitoring throughout both training and inference. The tool supports single-run verification and has successfully identified pollution operators responsible for numerical instability across diverse tasks, thereby ensuring efficient and reliable computation in deep learning systems.