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Efficient multi-GPU implementations of preconditioned conjugate gradient method

Advances in engineering Software 2025年 207卷

作者： Ma, Tailai Gou, Zhihong Xu, Ningyi Sun, Shuli LTCS Department of Mechanics and Engineering Science College of Engineering Peking University Beijing100091 China School of Electronic Information and Electrical Engineering Shanghai Jiao Tong University Shanghai200030 China

Many numerical methods rely on solving sparse linear systems, such as finite element analysis and computational fluid dynamics. In many simulations, solving sparse linear systems has become a bottleneck, often consuming a significant portion of the total simulation time. The Preconditioned Conjugate Gradient (PCG) method is an efficient iterative method for solving large sparse linear systems, and it is well-suited for GPU parallelization. As the scale of the problems to be solved increases, higher efficiency in PCG solvers is critical. This paper proposes two efficient implementations of the PCG method, aimed at further accelerating its computational performance on GPUs. First, we optimize one of the most time-consuming operation SpMV (Sparse Matrix-Vector Multiplication) in PCG using a block storage format for efficient global memory access. To this end, we introduce two novel SpMV algorithms: warp-fixed-mma and warp-balanced-mma. Next, we present two PCG implementations: PCG-cuda-graphs and PCG-persistent-kernel. Both implementations use the proposed SpMV algorithms. The experimental results on the 2904 matrices from the SuiteSparse Matrix Collection show that our best-performing SpMV algorithm in FP64 precision outperforms four latest SpMV methods cuSPARSE, DASP (Specific Dense Matrix Multiply-Accumulate Units Accelerated General Sparse Matrix-Vector Multiplication), CSR5 and TileSpMV by a factor of on average 2.2x, 3.1x, 2.48x and 3.20x on A100, respectively. Compared to the PCG implementation in PETSc (the Portable, Extensible Toolkit for Scientific Computation), the two proposed PCG implementations achieve average speedups of 1.41x and 1.63x on the NVIDIA A100 GPU, and 1.43x and 1.66x on the NVIDIA H100 GPU on 21 representative matrices. Additionally, for large matrices, our proposed multi-GPU parallel PCG-persistent-kernel implementation delivers impressive speedup compared to the single-GPU implementation. © 2025 Elsevier Ltd

关键词： Conjugate gradient method

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Mesh Deformation Method Based on Mean Value Coordinates Interpolation

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Acta Mechanica Solida Sinica 2016年第1期29卷 1-12页

作者： Shuli Sun Shuming Lv Yuan Yuan Mingwu Yuan LTCS Department of Mechanics and Engineering ScienceCollege of EngineeringPeking University

Mesh deformation technique is widely used in many application fields, and has re- ceived a lot of attentions in recent years. This paper focuses on the methodology and algorithm of algebraic type mesh deformation for unstructured mesh in numerical discretization. To preserve mesh quality effectively, an algebraic approach for two and three dimensional unstructured mesh is developed based on mean value coordinates interpolation combined with node visibility analysis. The proposed approach firstly performs node visibility analysis to find out the visible boundary for each grid point to be moved, then evaluates the mean value coordinates of each grid point with respect to all vertices on its visible boundary. Thus the displacements of grid points can be calculated by interpolating the boundary movement by the mean value coordinates. Compared with other methods, the proposed method has good deformation capability and predictable com- putational cost, with no need to select parameters or functions. Applications of mesh deformation in different fields are presented to demonstrate the effectiveness of the proposed approach. The results of numerical experiments exhibit not only superior deformation capability of the method in traditional applications of fluid dynamic grid, but also great potential in modeling for large deformation analysis and inverse design problems.

关键词： unstructured mesh mesh deformation mean value coordinates (MVC) visibilityanalysis domain partition

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INTERFACE EFFECT ON THE EFFECTIVE BULK MODULUS OF A PARTICLE-REINFORCED COMPOSITE

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Acta Mechanica Sinica 2004年第6期20卷 676-679页

作者：孙黎武义明黄筑平王建祥 LTCS and Department of Mechanics and Engineering Science Peking UniversityBeijing 100871China

Classical micromechanical methods for calculating the effective moduli of a heteroge- neous material are generalized to include the interface(surface)*** using Hashin's Composite Sphere Assemblage(CSA)model,a new expression of the bulk modulus for a particle-reinforced com- posite is *** is emphasized that the present study is within the finite-deformation framework such that the effective properties are not influenced by the interface stress itself solely,but influenced by the change of the interface stress due to changes of the shape and size of the *** some inadequacies in previous papers are pointed out.

关键词： effective bulk modulus interface effect Hashin's Composite Sphere Assemblage

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Matching Boundary Conditions for a Heterogeneous Atomic Chain

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Chinese Physics Letters 2016年第8期33卷 1-4页

作者：陈星如刘舒瑜唐少强 HEDPS CAPT and LTCSDepartment of Mechanics and Engineering ScienceCollege of EngineeringPeking University

We propose accurate boundary treatments for a heterogeneous atomic chain, in terms of matching boundary conditions （MBCs）. The main challenge lies in reproducing the physical reflection across the boundary to a correct amount. With reflection coefficients we demonstrate that the accuracy is improved when more atoms are used under the boundary condition. The inclusion of an atom in the embedded sublattice B may considerably enhance the performance. Numerical testing illustrates the effectiveness of the proposed MBCs.

关键词： for on it Matching Boundary Conditions for a Heterogeneous Atomic Chain of is MBC that

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A New Method for Separation of Waves in Improving the Conventional SHPB Technique

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Chinese Physics Letters 2006年第11期23卷 3045-3048页

作者：刘凯欣李旭东 LTCS and Department of Mechanics and Engineering Science College of Engineering Peking University Beijing 100871

We present a new wave separation method to enable the split Hopkinson pressure bar （SHPB） technique to break through the limitation of the length of the incident bar and greatly to increase its measurable maximum strain. At the same time the dispersion effect of the elastic wave is significantly reduced. The fundamental principle of the new method is proven rigorously. The feasibility and credibility of the new method are also verified by experiments.

关键词： PRESSURE BAR

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Effect of geometric parameters and grazing incidence on magnetic polaritons excited in 1D multi-groove gratings

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science China(Technological sciences) 2021年第8期64卷 1589-1598页

作者： Su ChengShuai Fu CeJi LTCS and Department of Mechanics and Engineering Science College of EngineeringPeking UniversityBeijing100871China

In this paper,effect of geometric parameters and grazing incidence on excitation of magnetic polaritons(MPs)in a 1D multigroove grating with different groove depths made of silver was *** results reveal that when the distance between grooves is sufficiently small,the resonance wavelengths of MPs excited in the grooves of unequal depths exhibit red shift with decrease of the distance,contrary to the case with equal groove *** shift of the MP resonance wavelengths was explained with the LC circuit ***,it was found that the resonance wavelengths of MPs depend linearly on the groove depths except when the difference between the groove depths is *** the grooves have equal depths,a large drop of the absorptance can occur due to the interaction and cancellation of the electric field vectors in the region between the ***,the results show that when a TM wave is at grazing incidence,MPs can be excited simultaneously in the grooves at a blue-shifted wavelength due to reduction of the effective capacitance,resulting in a dramatic enhancement of the ***,the results in this work may provide useful guidance in the design of wavelength-selective absorbers based on MPs.

关键词： multi-groove grating magnetic polaritons absorptance LC circuit model

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Atomic diffusion across Ni_(50)Ti_(50)Cu explosive welding interface:Diffusion layer thickness and atomic concentration distribution

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Chinese Physics B 2014年第6期23卷 446-451页

作者：陈仕洋武振伟刘凯欣 LTCS and Department of Mechanics and Engineering Science College of Engineering Peking University Center for Applied Physics and Technology Peking University

Molecular dynamics simulations are carried out to study atomic diffusion in the explosive welding process of NisoTis0-Cu （at.%）. By using a hybrid method which combines molecular dynamics simulation and classical diffusion the- ory, the thickness of the diffusion layer and the atomic concentration distribution across the welding interface are obtained. The results indicate that the concentration distribution curves at different times have a geometric similarity. According to the geometric similarity, the atomic concentration distribution at any time in explosive welding can be calculated. NisoTis0- Cu explosive welding and scanning electron microscope experiments are done to verify the results. The simulation results and the experimental results are in good agreement.

关键词： diffusion interfaces explosive welding molecular dynamics

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Projection Improved SPAI Preconditioner for FGMRES

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Numerical Mathematics(Theory,Methods and Applications) 2023年第4期16卷 1035-1052页

作者： Mingguang Geng Shuli Sun LTCS Department of Mechanics and Engineering ScienceCollege of EngineeringPeking UniversityBeijing 100871China

Krylov subspace methods are widely used for solving sparse linear algebraic equations,but they rely heavily on preconditioners,and it is difficult to find an effective preconditioner that is efficient and stable for all *** this paper,a novel projection strategy including the orthogonal and the oblique projection is proposed to improve the preconditioner,which can enhance the efficiency and stability of *** proposed strategy can be considered as a minimization process,where the orthogonal projection minimizes the energy norm of error and the oblique projection minimizes the 2-norm of the residual,meanwhile they can be regarded as approaches to correct the approximation by solving low-rank inverse of the *** strategy is a wide-ranging approach and provides a way to transform the constant preconditioner into a variable *** paper discusses in detail the projection strategy for sparse approximate inverse(SPAI)preconditioner applied to flexible GMRES and conducts the numerical test for problems from different *** results show that the proposed projection strategy can significantly improve the solving process,especially for some non-converging and slowly convergence systems.

关键词： Generalized minimal residual preconditioner sparse approximate inverse oblique projection orthogonal projection

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A three-dimensional Eulerian method for the numerical simulation of high-velocity impact problems

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Chinese Physics B 2014年第3期23卷 338-347页

作者：吴士玉刘凯欣陈千一 LTCS and Department of Mechanics and Engineering Science College of Engineering Peking University Center for Applied Physics and Technology Peking University

In the present paper, a three-dimensional （3D） Eulerian technique for the 3D numerical simulation of high-velocity impact problems is proposed. In the Eulerian framework, a complete 3D conservation element and solution element scheme for conservative hyperbolic governing equations with source terms is given. A modified ghost fluid method is proposed for the treatment of the boundary conditions. Numerical simulations of the Taylor bar problem and the ricochet phenomenon of a sphere impacting a plate target at an angle of 60~ are carried out. The numerical results are in good agreement with the corresponding experimental observations. It is proved that our computational technique is feasible for analyzing 3D high-velocity impact problems.

关键词： three-dimensional numerical simulation conservation element and solution element (CE/SE)method ghost fluid method high-velocity impact

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ANALYTICAL SOLUTIONS FOR ELASTOSTATIC PROBLEMS OF PARTICLE-AND FIBER-REINFORCED COMPOSITES WITH INHOMOGENEOUS INTERPHASES

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Applied Mathematics and mechanics(English Edition) 2005年第3期26卷 336-344页

作者：段慧玲王建祥黄筑平黄红波 LTCS and Department of Mechanics and Engineering Science Peking UniversityBeijing 100871 P.R.China

By transforming the governing equations for displacement components into Riccati equations, analytical solutions for displacements, strains and stresses for Representive Volume Elements (RVEs) of particle_ and fiber_reinforced composites containing inhomo geneous interphases were obtained. The analytical solutions derived here are new and general for power_law variations of the elastic moduli of the inhomogeneous interphases. Given a power exponent, analytical expressions for the bulk moduli of the composites with inho mogeneous interphases can be obtained. By changing the power exponent and the coefficients of the power terms, the solutions derived here can be applied to inhomogeneous interphases with many different property profiles. The results show that the modulus variation and the thickness of the inhomogeneous interphase have great effect on the bulk moduli of the composites. The particle will exhibit a sort of “size effect”, if there is an interphase.

关键词： inhomogeneous interphase particle-reinforced composite fiber-reinforced composite analytical solution bulk modulus

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