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Strategies with algebraic multigrid method for Coupled Systems

LOBACHEVSKII JOURNAL OF MATHEMATICS 2024年第1期45卷 251-261页

作者： Konshin, I. Terekhov, K. Vassilevski, Yu. Russian Acad Sci Fed Res Ctr Comp Sci & Control Moscow 119333 Russia Russian Acad Sci Marchuk Inst Numer Math Moscow 119333 Russia Sechenov First Moscow State Med Univ Moscow 119991 Russia Sirius Univ Sci & Technol Soci 354340 Russia

We consider strategies for algebraic multigrid-based solution of coupled systems arising from the anisotropic diffusion problem, the two-phase filtration problem, the Stokes and Navier-Stokes equations for incompressible fluid. We consider specific strategies for each problem: the direct application of the algebraic multigrid (AMG), the constrained pressure residual method with AMG preconditioner for the pressure block, the Bramble-Pasciak conjugate-gradient and biconjugate gradient stabilized methods with AMG preconditioner for the elliptic part of the (linearized) discrete operator. The strategies reveal an elliptic part of each system, which is efficiently addressed by the AMG method. As a result, the presented methods demonstrate linear complexity with respect to the size of the coupled problems.

关键词： algebraic multigrid method CPR method Bramble-Pasciak method mimetic finite difference Stokes equations Navier-Stokes equations

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Stationary Stokes solver for single-phase flow in porous media: A blastingly fast solution based on algebraic multigrid method using GPU

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ADVANCES IN WATER RESOURCES 2023年 171卷

作者： Evstigneev, Nickolay M. Ryabkov, Oleg I. Gerke, Kirill M. Russian Acad Sci Fed Res Ctr Comp Sci & Control Moscow Russia Russian Acad Sci Schmidt Inst Phys Earth Moscow Russia

The paper is focused on high efficiency Stokes solver that is applied to the incompressible flow in porous media. Computational domains are represented by binarized 3D computed tomography voxel models. The solution procedure is constructed around a fast algebraic multigrid solver that utilizes the power of graphics processing units (GPUs). In order to minimize memory footprint and accelerate the solver, a simple MAC-type staggered finite difference discretization is used and a coupled Stokes saddle-point type system is solved directly on a GPU. The MAC discretization is discussed, taking particular care of internal solid boundaries around pores. The method includes topological domain analysis on a GPU, which removes isolated volumes with no flow in the domain (based on the connected component labeling), depending on the boundary conditions. We consider various types of boundary conditions and efficient parallel strategies for the GPUs, including fast matrix assembly and residual regularization. Our method is extensively benchmarked both against analytical solutions and applied problems from digital rock physics in terms of computational wall time, precision, approximation order, and convergence. We demonstrate that it takes up to 5-23 s on a modern GPU card to obtain a solution with 1.10-6 error residuals for 3D geometries with 300-4503 voxels and a porosity range of 5-37%.

关键词： Stokes flow Porous media algebraic multigrid method General purpose GPU Krylov subspace solvers Permeability Permeability tensor

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Learning-based local weighted least squares for algebraic multigrid method

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JOURNAL OF COMPUTATIONAL PHYSICS 2023年第1期493卷

作者： Wang, Fan Gu, Xiang Sun, Jian Xu, Zongben Xi An Jiao Tong Univ Sch Math & Stat 28 Xianning West Rd Xian 710049 Shaanxi Peoples R China

algebraic multigrid (AMG) is an effective iterative algorithm for solving large-scale linear systems. One challenge of constructing the AMG algorithm is the determination of the prolongation operator, which affects the convergence rate of AMG and is problemdependent. In this paper, we propose a new Learning-based Local Weighted Least Squares (L-LWLS) method to construct the prolongation operator of AMG. Specifically, we construct the prolongation operator by solving the LWLS model with learned spatially-varying weights. We use the gradient descent algorithm to optimize the model with a learned initialization of the solution. Then the constructed prolongation operator is further corrected by a learned correction function to improve the convergence rate of AMG. We conduct experiments on solving graph Laplacian linear systems, diffusion partial differential equations, and Helmholtz equations. Experiments show that the proposed method can construct a better prolongation operator leading to a faster convergence rate than the compared methods, including the classical AMG, the smoothed aggregation AMG, the bootstrap AMG, and the learning-based AMG method. The results show that the proposed method can generalize well to different parameter distributions and problem sizes, i.e., the number of variables in the linear system.

关键词： algebraic multigrid method Prolongation operator Local weighted least squares model Deep learning

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An efficient algebraic multigrid method for second-order elliptic equation on polygonal domains

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MATHEMATICAL methodS IN THE APPLIED SCIENCES 2021年第0期

作者： Li, Ming Honghe Univ Dept Math & Stat Mengzi 661100 Yunnan Peoples R China

Based on a coarsening strategy of adjacency matrix, a new algebraic prolongation operator is developed for standard V-cycle multigrid method to accelerate the whole process. An efficient algebraic multigrid (EAMG) method is proposed for solving large-scale linear system, which arise from finite element (FE) discretization of second-order elliptic boundary value problem. Numerical experiments on polygonal domains are conducted to demonstrate that the EAMG computation is more efficient than standard method.

关键词： algebraic multigrid method finite element method polygonal domain prolongation operator

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algebraic multigrid methodS FOR METRIC-PERTURBED COUPLED PROBLEMS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2024年第3期46卷 A1461-A1486页

作者： Budis, Ana Hu, Xiaozhe Kuchta, Miroslav Mardal, Kent-andre Zikatanov, Ludmil Simula Res Lab N-0164 Oslo Norway Tufts Univ Dept Math Medford MA 02155 USA Univ Oslo Dept Math N-0316 Blindern Oslo Norway Penn State Univ Dept Math University Pk PA 16802 USA

We develop multilevel methods for interface-driven multiphysics problems that can be coupled across dimensions and where complexity and strength of the interface coupling deteriorates the performance of standard methods. We focus on aggregation-based algebraic multigrid methods with custom smoothers that preserve the coupling information on each coarse level. We prove that, with the proper choice of subspace splitting, we obtain uniform convergence in discretization and physical parameters in the two-level setting. Additionally, we show parameter robustness and scalability with regard to the number of the degrees of freedom of the system on several numerical examples related to the biophysical processes in the brain, namely, the electric signaling in excitable tissue modeled by bidomain, the extracellular-membrane-intracellular (EMI) model, and reduced EMI equations.

关键词： algebraic multigrid method preconditioning iterative method coupled problems graph Laplacian

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Fast linear solvers for incompressible CFD simulations with compatible discrete operator schemes

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COMPUTATIONAL & APPLIED MATHEMATICS 2025年第1期44卷 1-36页

作者： Jang, Yongseok Bonelle, Jerome Kruse, Carola Hulsemann, Frank Ruede, Ulrich Univ Paris Saclay ONERA Chatillon France EDF Lab Chatou Chatou France CERFACS Toulouse France EDF LAB Paris Saclay Palaiseau France FAU Erlangen Nurnberg Dept Comp Sci Erlangen Germany

Finding a robust and efficient solver for (non-)symmetric systems that arise in incompressible Computational Fluid Dynamics (CFD) is of great interest to both academia and industry. We consider the Compatible Discrete Operator (CDO) discretization that has recently been devised for CFD simulations in the context of incompressible Stokes and Navier-Stokes flows. The discrete problems resulting from CDO schemes yield large saddle-point systems that require relevant numerical methods suitable to deal with large indefinite and poorly conditioned linear systems. In this paper, we focus on two segregated methods: the augmented Lagrangian Uzawa method and the generalized Golub-Kahan bidiagonalization, as well as a monolithic method based on an algebraic transformation by change of variables. We also employ algebraic multigrid (AMG) preconditioned Krylov solvers such as the Flexible Conjugate Gradient (FCG) method, and the Flexible Generalized Minimal Residual (FGMRES) method, to solve the linear systems. Using the CFD software code_saturne, we compare the numerical performance with respect to the choice of linear solvers and numerical strategies for the saddle-point problem. In the numerical experiments, the AMG preconditioned Krylov methods show robustness in test cases of Stokes and Navier-Stokes problems.

关键词： algebraic multigrid method Compatible discrete operator Incompressible Navier-Stokes Saddle-point problems

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On a multilevel Levenberg-Marquardt method for the training of artificial neural networks and its application to the solution of partial differential equations

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OPTIMIZATION methodS & SOFTWARE 2022年第1期37卷 361-386页

作者： Calandra, H. Gratton, S. Riccietti, E. Vasseur, X. Ctr Sci & Tech Jean Feger TOTAL Pau France Univ Toulouse INPT IRIT 2 Rue CamichelBP 7122 F-31071 Toulouse 7 France ENSEEIHT 2 Rue CamichelBP 7122 F-31071 Toulouse 7 France Univ Toulouse ISAE SUPAERO Toulouse 4 France

In this paper, we propose a new multilevel Levenberg-Marquardt optimizer for the training of artificial neural networks with quadratic loss function. This setting allows us to get further insight into the potential of multilevel optimization methods. Indeed, when the least squares problem arises from the training of artificial neural networks, the variables subject to optimization are not related by any geometrical constraints and the standard interpolation and restriction operators cannot be employed any longer. A heuristic, inspired by algebraic multigrid methods, is then proposed to construct the multilevel transfer operators. We test the new optimizer on an important application: the approximate solution of partial differential equations by means of artificial neural networks. The learning problem is formulated as a least squares problem, choosing the nonlinear residual of the equation as a loss function, whereas the multilevel method is employed as a training method. Numerical experiments show encouraging results related to the efficiency of the new multilevel optimization method compared to the corresponding one-level procedure in this context.

关键词： algebraic multigrid method artificial neural network Levenberg-Marquardt method multilevel optimization method partial differential equation

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A PARALLEL AUXILIARY GRID algebraic multigrid method FOR GRAPHIC PROCESSING UNITS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2013年第3期35卷 C263-C283页

作者： Wang, Lu Hu, Xiaozhe Cohen, Jonathan Xu, Jinchao Penn State Univ Dept Math University Pk PA 16802 USA Penn State Univ Ctr Computat Math & Applicat University Pk PA 16802 USA NVIDIA Res Ann Arbor MI 48105 USA

In this paper, we develop a new parallel auxiliary grid algebraic multigrid (AMG) method to leverage the power of graphic processing units (GPUs). In the construction of the hierarchical coarse grid, we use a simple and fixed coarsening procedure based on a region quadtree generated from an auxiliary grid. This allows us to explicitly control the sparsity patterns and operator complexities of the AMG solver. This feature provides (nearly) optimal load balancing and predictable communication patterns on shape regular grids, which makes our new algorithm suitable for parallel computing, especially on GPUs. We also design a parallel smoother based on the special coloring of the quadtree to accelerate the convergence rate and improve the parallel performance of this solver. Based on the CUDA toolkit [NVIDIA CUDA Programming Guide, NVIDIA Corp., 2010], we implemented our new parallel auxiliary grid AMG method on GPUs and the numerical results of this implementation demonstrate the efficiency of our new method for (nearly) isotropic problems. The results achieve an average speedup of over 4 on quasi-uniform grids and 2 on shape regular grids when compared to the AMG implementation in CUSP [M. Garland and N. Bell, CUSP: Generic Parallel Algorithms for Sparse Matrix and Graph Computations, http://***/(2010)].

关键词： algebraic multigrid method aggregation nonlinear AMLI-cycle auxiliary grid GPU parallel algorithm

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AI-enhanced iterative solvers for accelerating the solution of large-scale parametrized systems

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 2024年第2期125卷 e7372-e7372页

作者： Nikolopoulos, Stefanos Kalogeris, Ioannis Stavroulakis, George Papadopoulos, Vissarion Natl Tech Univ Athens Inst Struct Anal & Antiseism Res Sch Civil Engn Athens Greece

Recent advances in the field of machine learning open a new era in high performance computing for challenging computational science and engineering applications. In this framework, the use of advanced machine learning algorithms for the development of accurate and cost-efficient surrogate models of complex physical processes has already attracted major attention from scientists. However, despite their powerful approximation capabilities, surrogate model predictions are still far from being near to the 'exact' solution of the problem. To address this issue, the present work proposes the use of up-to-date machine learning tools in order to equip a new generation of iterative solvers of linear equation systems, capable of very efficiently solving large-scale parametrized problems at any desired level of accuracy. The proposed approach consists of the following two steps. At first, a reduced set of model evaluations is performed using a standard finite element methodology and the corresponding solutions are used to establish an approximate mapping from the problem's parametric space to its solution space using a combination of deep feedforward neural networks and convolutional autoencoders. This mapping serves as a means of obtaining very accurate initial predictions of the system's response to new query points at negligible computational cost. Subsequently, an iterative solver inspired by the algebraic multigrid method in combination with Proper Orthogonal Decomposition, termed POD-2G, is developed that successively refines the initial predictions of the surrogate model towards the exact solution. The application of POD-2G as a standalone solver or as preconditioner in the context of preconditioned conjugate gradient methods is demonstrated on several numerical examples of large scale systems, with the results indicating its strong superiority over conventional iterative solution schemes.

关键词： algebraic multigrid method concolutional neural networks iterative solvers large-scale parametrized systems preconditioned conjugate gradient method

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GPU-Accelerated Field Simulation of HVAC Gas Insulated Lines 20

GPU-Accelerated Field Simulation of HVAC Gas Insulated Lines

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20th Biennial IEEE Conference on Electromagnetic Field Computation (IEEE CEFC)

作者： Hensel, Hendrik Henkel, Marvin-Lucas Haussmann, Norman Joergens, Christoph Stroka, Steven Clemens, Markus Univ Wuppertal Chair Elect Theory D-42119 Wuppertal Germany

ISBN: (数字)9781665468336

ISBN: (纸本)9798350346404

High voltage alternating current gas insulated systems are utilized in narrow spaces. In operation electric field peaks can occur and may impair the functioning of the system. The knowledge of the electric field distribution in gas insulated lines is essential to ensure a reliable operation of the system. This can be determined by numerical simulations and requires a fast linear solver. Utilizing the AmgX library on NVIDIA graphic processing units allows a fast solution of a Poisson system of linear equations with millions of degrees of freedom within fractions of a second.

关键词： algebraic multigrid method finite element method gas insulated transmission lines graphics processing unit (GPU)

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