14 December 2025, Volume 46 Issue 4

    

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  PROGRESS IN THE LARGE-SCALE EIGENSOLVER GCGE

  Wang Zijing, Guo Zhaotong, Liu Haochen, Xie Hehu

  Journal on Numerica Methods and Computer Applications. 2025, 46(4): 265-282. https://doi.org/10.12288/szjs.s2025-1037

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  Large-scale algebraic eigenvalue problems are widely used in fields such as materials science and engineering structure analysis. In multi-scale complex systems, with the problem size increasing and the accumulation of errors induced by scale disparities, traditional eigenvalue algorithms face significant challenges in accuracy, stability and computational efficiency, and in some cases even fail to achieve effective solutions. This paper presents recent progress in the development of the GCGE eigenvalue solver, focusing on the design and implementation of efficient parallel algorithms aimed at enhancing its stability and adaptability in complex systems. To address the performance and stability bottlenecks of conventional solvers when dealing with complex small-scale problems and large-scale ill-conditioned problems, we introduce several improvements to the GCG algorithm. These include a diagonal normalization preconditioning strategy to improve convergence and numerical robustness, and orthogonalization strategies based on L² inner products and stiffness matrix inner products to enhance numerical performance for cases involving indefinite mass matrices. Furthermore, GCGE has been integrated into the SLEPc software framework to support the solution of Hermitian eigenvalue problems and dominant eigenvalue problems, and has been modularly deployed on the Baltam Platform, demonstrating strong potential for engineering applications.
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  A NEXT-GENERATION ALGEBRAIC MULTIGRID LIBRARY FOR SEMI-STRUCTURED GRIDS

  Jia Zhaopeng, Zong Yi, Zhang Chensong, Sun Jian, Mu Longjiang, Wang Jianchun, Xu Xiaowen, Wang Xinliang, Yu Peinan, Xue Wei

  Journal on Numerica Methods and Computer Applications. 2025, 46(4): 283-295. https://doi.org/10.12288/szjs.s2025-1046

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  Algebraic multigrid (AMG) is an efficient method for solving linear equation systems as preconditioners. Semi-structured AMG utilizes structured information for efficient computation and supports the presence of unstructured information, thus achieving both high performance and high flexibility, making it widely used in various scenarios of scientific and engineering computing. However, the current mainstream semi-structured AMG solvers still have significant deficiencies in absolute speed and scalability. Therefore, we developed SemiStructMG. On the one hand, it utilizes multidimensional coarsening to reduce complexity, improving single step running speed and scalability; On the other hand, it considers interblock connections in the smoother and interpolation operators, improving convergence in various complex problems. We tested Semi-StructMG in benchmark tests and multiple realworld applications, and achieved speedup of 5.97x, 15.2x, and 3.85x compared to SSAMG, Split and BoomerAMG in hypre.
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  X-SOLVER: A LIBRARY OF ITERATIVE SOLUTION METHODS ON HETEROGENEOUS CLUSTERS

  Wang yunting, Yang shaofeng, He xin, Tan guangming

  Journal on Numerica Methods and Computer Applications. 2025, 46(4): 296-320. https://doi.org/10.12288/szjs.s2025-1047

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  The X-Solver library aims to implement and optimize Krylov subspace solution methods and preconditioners to solve large and sparse linear systems of equations on distributed memory clusters equipped with many-core GPUs as accelerators. Motivated by the requirements from real-world applications and the trend of hardware integrations, we manage three achievements, i.e., more effective support for distributed calculations, better exploitation of characteristics of modern many-core architectures, and portability across different heterogeneous platforms. Numerical experiments demonstrate advantages over other state-of-the-art libraries in a broad range of applications on the targeting hardware.
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  A BLOCK ALGEBRAIC MULTIGRID METHOD AND ITS PARALLEL OPTIMIZATION FOR FULLY IMPLICIT RESERVOIR SIMULATION

  Zhao Li, Li Yanyan, Wang Baohua, Zhang Chensong

  Journal on Numerica Methods and Computer Applications. 2025, 46(4): 321-345. https://doi.org/10.12288/szjs.s2025-1041

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  The block algebraic multigrid (BAMG) method is an efficient preconditioning technique for solving discretized systems of partial differential equations and can be applied to multiphase flow models in porous media. In this work, we investigate the impact of different norm selections in block-matrix coarsening and a diagonal-block-based classical interpolation strategy on the convergence behavior of the BAMG method for multiphase, multicomponent reservoir simulations. To address the parallel bottleneck during the setup phase, we introduce the “delayed update” and “structure-preserving update” strategies, and develop an adaptive-setup-based BAMG algorithm guided by a tolerable iteration growth threshold criterion. Furthermore, we perform in-depth optimization and vectorization of performancecritical components in the solve phase, including the smoother and residual computation. Numerical experiments demonstrate the proposed method’s advantages in terms of convergence, efficiency, and parallel scalability. For example, in tests involving grids with over one hundred million elements, the proposed method reduces computation time by 57.6% and improves parallel efficiency by 38.1% when using 8192 CPU cores compared to the traditional method.
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  ON THE INEXACT RESTRICTED ADDITIVE AND MULTIPLICATIVE SCHWARZ PRECONDITIONERS FOR THREE-TEMPERATURE RADIATION DIFFUSION PROBLEMS

  Yue Xiaoqiang, Wang Yiyang, Pan Xianyun

  Journal on Numerica Methods and Computer Applications. 2025, 46(4): 346-370. https://doi.org/10.12288/szjs.s2025-1029

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  The three-temperature radiation diffusion equations provide an accurate mathematical framework for describing the physical phenomenon where radiation energy propagates through a medium, undergoing scattering, absorption, and emission processes. Based on an overlapping decomposition strategy on physical quantities, we proposed restricted additive and multiplicative Schwarz preconditioning and iterative algorithms in [Yue X, He J, Xu X, Shu S, Wang L. Commun. Comput. Phys., 2022, 32: 829-849], but did not provide any convergence analysis or numerical verification. In this work, we propose reasonable approximation assumptions for the sub-matrix inverses within the involved Schur complements. Accordingly, we construct the inexact restricted additive and multiplicative Schwarz preconditioning algorithms. Motivated by the concepts of norm- and field-of-values-equivalences, we conduct the optimal convergence analyses for their preconditioned generalized minimal residual iterative methods under the stability conditions necessary for ensuring the symmetrypreserving finite volume element discretization (or its associated coefficient matrix) to yield a unique solution. Furthermore, we validate the effectiveness of the theoretical results through experimental test cases derived from realistic simulations of hydrodynamic instability during the deceleration phase of a laser-driven spherical implosion.
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  A FEATURE-DRIVEN ADAPTIVE UA-AMG PRECONDITIONING ALGORITHM FOR RADIATION DIFFUSION PROBLEMS

  He Jianmeng, Shu Shi, Wei Jie, Yue Xiaoqiang

  Journal on Numerica Methods and Computer Applications. 2025, 46(4): 371-385. https://doi.org/10.12288/szjs.s2025-1043

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  Radiation diffusion problems are widespread in multi-physics coupling fields such as astrophysics and inertial confinement fusion. Algebraic Multigrid (AMG) methods based on the physical and algebraic features of a problem have become a hot topic in the field of multigrid research. This paper proposes T2T2-ILU(0)-V-FGMRES solver for efficiently solving linearized discrete systems derived from three-temperature radiation diffusion equations. This solver is a PGMRES solver employing a common unsmoothed aggregation AMG (UA-AMG) preconditioner. Furthermore, several physical and algebraic features are extracted from different discrete systems to enhance the computational performance. Based on these features, we develop an adaptive UA-AMG preconditioned FGMRES solver, termed Adapt-UA-AMG-FGMRES. Numerical experiments show that the new solver exhibits better robustness and computational efficiency. Compared to the T2T2-ILU(0)-V-FGMRES and HMIS-V-FGMRES solvers (best common non-aggregation AMG preconditioners), the CPU time of the new solver is reduced by approximately 49.1% and 25.3%, respectively. It should be noted that the algorithmic design principles can be easily extended to more general model problems, such as multi-group radiative diffusion equations.
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  MIXED-PRECISION OPTIMIZATION FOR THE PANGULU SOLVER ON THE MT-3000 HETEROGENEOUS MULTI-ZONE PROCESSOR

  Zhang Siwei, Li Junxian, Li Yida, Liu Weifeng

  Journal on Numerica Methods and Computer Applications. 2025, 46(4): 386-397. https://doi.org/10.12288/szjs.s2025-1048

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  In large-scale linear system solving, traditional sparse direct solvers typically rely on a single precision computation, making it difficult to flexibly balance computational efficiency and numerical accuracy. To address this issue, a mixed-precision optimization algorithm based on the distributed sparse direct solver PanguLU is proposed, tailored for the heterogeneous multi-zone processor MT-3000. The algorithm dynamically selects storage precision for matrix blocks based on their spatial location and numerical sensitivity, enabling mixedprecision computation during numerical factorization. Meanwhile, a pipelining mechanism that decouples computation precision from storage precision for GEMM subtasks is designed. Experimental results show that the proposed method achieves a speedup between 1.04 and 1.19 times in the numerical factorization phase, while reducing the relative residual by a factor between 1.97 and 4.15 compared to baselines of a single precision, thereby effectively controlling accuracy loss while improving computational performance.
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  SURVEY ON RANDOMIZED NUMERICAL ALGORITHMS FOR SPARSE LINEAR ALGEBRAIC EQUATIONS

  Du Hao, Xu Xiaowen

  Journal on Numerica Methods and Computer Applications. 2025, 46(4): 398-410. https://doi.org/10.12288/szjs.s2025-1067

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  Randomized numerical algorithms have undergone rapid development in recent years, exhibiting significant potential for practical applications and offering novel methodologies for solving large-scale linear systems. This paper reviewed the state of the art of three primary randomized algorithms for sparse linear systems, examined their features, computational complexity, and key challenges. Based on this analysis, we evaluates the current research progress, identifies gaps between current capabilities and practical application requirements, and proposes future research directions for randomized algorithms targeting large-scale practical problems.