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3D Controlled-source electromagnetic modelling in anisotropic media using secondary potentials and a cascadic multigrid solver

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GEOPHYSICAL PROSPECTING 2025年第1期73卷 284-302页

作者： Pan, Kejia Wang, Jinxuan Han, Xu Ren, Zhengyong Ling, Weiwei Guo, Rongwen Cent South Univ Sch Math & Stat Changsha 410083 Peoples R China China Univ Geosci Sch Geophys & Geomat Wuhan Peoples R China Cent South Univ Sch Geosci & Infophys Changsha Peoples R China Cent South Univ Minist Educ Key Lab Metallogen Predict Nonferrous Met & Geol E Changsha Peoples R China Jiangxi Coll Appl Technol Ganzhou Peoples R China

Quantitative interpretation of the data from controlled-source electromagnetic methods, whether via forward modelling or inversion, requires solving a considerable number of forward problems, and multigrid methods are often employed to accelerate the solving process. In this study, a new extrapolation cascadic multigrid method is employed to solve the large sparse complex linear system arising from the finite element approximation of Maxwell's equations using secondary potentials. The equations using secondary potentials are discretized by the classic nodal finite element method on nonuniform rectilinear grids. The resulting linear systems are solved by the extrapolation cascadic multigrid method with a new prolongation operator and preconditioned Stabilized bi-conjugate gradient method smoother. High-order interpolation and global extrapolation formulas are utilized to construct the multigrid prolongation operator. The extrapolation cascadic multigrid method with the new prolongation operator is easier to implement and more flexible in application than the original one. Finally, several synthetic examples including layered models, models with anisotropic anomalous bodies or layers, are used to validate the accuracy and efficiency of the proposed method. Numerical results show that the extrapolation cascadic multigrid method improves the efficiency of 3D controlled-source electromagnetic forward modelling a lot, compared with traditional iterative solvers and some state-of-the-art methods or software (e.g., preconditioned flexible generalized minimal residual method, emg3d) in the considered models and grid settings. The efficiency benefit is more evident as the number of unknowns increases, and the proposed method is more efficient at low frequencies. The extrapolation cascadic multigrid method can also be used to solve systems of equations arising from related applications, such as induction logging, airborne electromagnetic, etc.

关键词： anisotropy controlled-source electromagnetic modelling electrical properties multigrid solver electromagnetics moldeling numerical study

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Stress-aware large-scale mesh editing using a domain-decomposed multigrid solver

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COMPUTER AIDED GEOMETRIC DESIGN 2018年 63卷 17-30页

作者： Xu, Weiwei Yang, Haifeng Yang, Yin Wang, Yiduo Zhou, Kun State Key Lab of CAD&CG Zhejiang University China Electrical and Computer Engineering Department University of New Mexico USA

In this paper, we develop a domain-decomposed subspace and multigrid solver to analyze the stress distribution for large-scale finite element meshes with millions of degrees of freedom. Through the domain decomposition technique, the shape editing directly updates the data structure of local finite element matrices. Doing so avoids the expensive factorization step in a direct solver and provides users with a progressive feedback of the stress distribution corresponding to the mesh operations: a fast preview is achieved through the subspace solver, and the multigrid solver refines the preview result if the user needs to examine the stress distribution carefully at certain design stages. Our system constructs the subspace for stress analysis using reduced constrained modes and builds a three-level multigrid solver through the algebraic multigrid method. We remove mid-edge nodes and lump unknowns with the Schur complement method. The updating and solving of the large global stiffness matrix are implemented in parallel after the domain decomposition. Experimental results show that our solver outperforms the parallel Intel MKL solver. Speedups of 50%-100% can be achieved for large-scale meshes with reasonable pre-computation costs when setting the stopping criterion of the multigrid solver to be 1e-3 relative error. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Domain Decomposition multigrid solver Stress analysis

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Stable Electromagnetic Modeling Using a multigrid solver on Stretching Grids: The Magnetotelluric Example

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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS 2016年第3期13卷 334-338页

作者： Li, Gang Zhang, Lili Han, Bo Ocean Univ China Coll Marine Geosci Qingdao 266100 Peoples R China Chinese Acad Sci Key Lab Petr Resources Res Inst Geol & Geophys Beijing 100029 Peoples R China

Electromagnetic (EM) surveys are widely used in geophysical study. Reliable 3-D forward modeling is required for the inversion and interpretation of geophysical data. We have introduced a novel method for 3-D EM modeling using stretching grids, which uses a multigrid iterative solver for solving the formed linear system of equations based on the staggered finite-difference method. The developed algorithm is applied for the simulation of the magnetotelluric fields. We have tested this new algorithm using synthetic geoelectric models. The proposed multigrid method using stretching grids is also compared with the preconditioned Krylov-subspace solvers, the biconjugate gradients stabilized method, and the generalized minimum residual method. The developed multigrid method is proved to be more stable and requires less iterations to converge.

关键词： Electromagnetic (EM) Krylov solver multigrid solver staggered finite-difference (SFD) stretching grids three-dimensional (3-D)

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An Efficient Geometric multigrid solver for Viscous Liquids

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PROCEEDINGS OF THE ACM ON COMPUTER GRAPHICS AND INTERACTIVE TECHNIQUES 2019年第2期2卷 1–21页

作者： Aanjaneya, Mridul Han, Chengguizi Goldade, Ryan Batty, Christopher Rutgers State Univ New Brunswick NJ 08901 USA Univ Waterloo Waterloo ON Canada

We present an efficient geometric multigrid solver for simulating viscous liquids based on the variational approach of Batty and Bridson [2008]. Although the governing equations for viscosity are elliptic, the strong coupling between different velocity components in the discrete stencils mandates the use of more exotic smoothing techniques to achieve textbook multigrid efficiency. Our key contribution is the design of a novel box smoother involving small and sparse systems (at most 9 x 9 in 2D and 15 x 15 in 3D), which yields excellent convergence rates and performance improvements of 3.5x - 13.8x over a naive multigrid approach. We employ a hybrid approach by using a direct solver only inside the box smoother and keeping the remaining pipeline assembly-free, allowing our solver to efficiently accommodate more than 194 million degrees of freedom, while occupying a memory footprint of less than 16 GB. To reduce the computational overhead of using the box smoother, we precompute the Cholesky factorization of the subdomain system matrix for all interior degrees of freedom. We describe how the variational formulation, which requires volume weights computed at the centers of cells, edges, and faces, can be naturally accommodated in the multigrid hierarchy to properly enforce boundary conditions. Our proposed multigrid solver serves as an excellent preconditioner for Conjugate Gradients, outperforming existing state-of-the-art alternatives. We demonstrate the efficacy of our method on several high resolution examples of viscous liquid motion including two-way coupled interactions with rigid bodies.

关键词： viscosity liquid simulation variational approach multigrid solver

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An efficient geometric multigrid solver for viscous liquids

An efficient geometric multigrid solver for viscous liquids

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Proceedings of the ACM on Computer Graphics and Interactive Techniques

作者： Aanjaneya, Mridul Han, Chengguizi Goldade, Ryan Batty, Christopher Rutgers University United States University of Waterloo Canada

We present an efficient geometric multigrid solver for simulating viscous liquids based on the variational approach of Batty and Bridson [2008]. Although the governing equations for viscosity are elliptic, the strong coupling between different velocity components in the discrete stencils mandates the use of more exotic smoothing techniques to achieve textbook multigrid efficiency. Our key contribution is the design of a novel box smoother involving small and sparse systems (at most 9 × 9 in 2D and 15 × 15 in 3D), which yields excellent convergence rates and performance improvements of 3.5× - 13.8× over a naïve multigrid approach. We employ a hybrid approach by using a direct solver only inside the box smoother and keeping the remaining pipeline assembly-free, allowing our solver to efficiently accommodate more than 194 million degrees of freedom, while occupying a memory footprint of less than 16 GB. To reduce the computational overhead of using the box smoother, we precompute the Cholesky factorization of the subdomain system matrix for all interior degrees of freedom. We describe how the variational formulation, which requires volume weights computed at the centers of cells, edges, and faces, can be naturally accommodated in the multigrid hierarchy to properly enforce boundary conditions. Our proposed multigrid solver serves as an excellent preconditioner for Conjugate Gradients, outperforming existing state-of-the-art alternatives. We demonstrate the efficacy of our method on several high resolution examples of viscous liquid motion including two-way coupled interactions with rigid bodies. © 2019 Association for Computing Machinery.

关键词： Liquids Degrees of freedom (mechanics) Viscosity Cholesky factorizations Computational overheads Geometric multigrids Liquid simulations multigrid solver Smoothing techniques Variational approaches Variational formulation

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AN AUTOADAPTIVE solver ASSOCIATED WITH A multigrid INPUT DATA ON A MACINTOSH MICROCOMPUTER

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ADVANCES IN ENGINEERING SOFTWARE 1993年第1期18卷 31-40页

作者： DEVESLUD, CL MAURICE, G ECOLE MINES NANCY SCI & GENIE SURFACES LABF-54042 NANCYFRANCE

We present an autoadaptive multigrid finite element solver for a Macintosh microcomputer. The multigrid input data allows dynamic, problem-oriented optimization of the required computer memory and program execution time, if and only if grid refinement is pursued locally in order to enhance precision. This interactive optimization takes account of all of the pertinent local variables. Numerical experiments applied to examples of classical elasticity illustrate the power of this strategy.

关键词： multigrid solver AUTOADAPTIVE INPUT DATA MICROCOMPUTER

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Implementing the discontinuous-Galerkin finite element method using graph neural networks with application to diffusion equations

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NEURAL NETWORKS 2025年 185卷 107061页

作者： Li, Linfeng Xiang, Jiansheng Chen, Boyang Heaney, Claire E. Dargaville, Steven Pain, Christopher C. Imperial Coll London Dept Earth Sci & Engn Prince Consort Rd London SW7 2BP England Imperial Coll London Ctr AI Phys Modelling Imperial X White City Campus London W12 7SL England

Machine learning (ML) has benefited from both software and hardware advancements, leading to increasing interest in capitalising on ML throughout academia and industry. There have been efforts in the scientific computing community to leverage this development via implementing conventional partial differential equation (PDE) solvers with machine learning packages, most of which rely on structured spatial discretisation and fast convolution algorithms. However, unstructured meshes are favoured in problems with complex geometries. To bridge this gap, we propose to implement the unstructured Finite Element Method (FEM) on simplicial meshes with graph neural networks. This paper is the first to implement an unstructured mesh FEM solver using graph neural networks. All compute-intensive algorithms in the solver are represented with either convolutional or graph neural networks. Specifically, the FEM solver uses a discontinuous Galerkin formulation with an interior penalty method for spatial discretisation and a multigrid preconditioned Krylov solver as the linear solver. The multigrid method has been designed to suit the data structure within the ML package and adopts the commonly used U-Net architecture for this. A hierarchy of coarsened meshes is generated from p-multigrid and algebraic node agglomeration guided by either a space-filling curve or a smoothed aggregation algorithm. The solver is verified and assessed for solving the diffusion problems. The solver shows the theoretical convergence rate of ( p + 1) order. Compared with a highly optimised implementation, the solver running on GPU can reach promising throughput in terms of matrix operator evaluation at 6.8 MDOF/s. The method can easily extend to other PDEs and computing platforms beyond CPU and GPU.

关键词： Graph neural networks Discontinuous-Galerkin finite element method multigrid solver U-Net Unstructured meshes

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Performance of preconditioned iterative and multigrid solvers in solving the three-dimensional magnetotelluric modeling problem using the staggered finite-difference method: a comparative study

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JOURNAL OF GEOPHYSICS AND ENGINEERING 2016年第1期13卷 1-10页

作者： Li, Gang Zhang, Lili Hao, Tianyao Ocean Univ China Minist Educ Key Lab Submarine Geosci & Prospecting Technol Qingdao 266100 Peoples R China Chinese Acad Sci Inst Geol & Geophys Key Lab Petr Resources Res Beijing 100029 Peoples R China

An effective solver for the large complex system of linear equations is critical for improving the accuracy of numerical solutions in three-dimensional (3D) magnetotelluric (MT) modeling using the staggered finite-difference (SFD) method. In electromagnetic modeling, the formed system of linear equations is commonly solved using preconditioned iterative relaxation methods. We present 3D MT modeling using the SFD method, based on former work. The multigrid solver and three solvers preconditioned by incomplete Cholesky decomposition the minimum residual method, the generalized product bi-conjugate gradient method and the bi-conjugate gradient stabilized method-are used to solve the formed system of linear equations. Divergence correction for the magnetic field is applied. We also present a comparison of the stability and convergence of these iterative solvers if divergence correction is used. Model tests show that divergence correction improves the convergence of iterative solvers and the accuracy of numerical results. Divergence correction can also decrease the number of iterations for fast convergence without changing the stability of linear solvers. For consideration of the computation time and memory requirements, the multigrid solver combined with divergence correction is preferred for 3D MT field simulation.

关键词： magnetotellurics staggered finite-difference method three-dimensional modeling preconditioned iterative solvers multigrid solver divergence correction

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Gradient-Domain Processing within a Texture Atlas

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ACM TRANSACTIONS ON GRAPHICS 2018年第4期37卷 1-14页

作者： Prada, Fabian Kazhdan, Misha Chuang, Ming Hoppe, Hugues Johns Hopkins Univ 3400 N Charles St Baltimore MD 21218 USA PerceptIn Inc 4633 Old Ironsides Dr Snata Clara CA 95054 USA Google Inc 601 N 34th St Seattle WA 98103 USA

Processing signals on surfaces often involves resampling the signal over the vertices of a dense mesh and applying mesh-based filtering operators. We present a framework to process a signal directly in a texture atlas domain. The benefits are twofold: avoiding resampling degradation and exploiting the regularity of the texture image grid. The main challenges are to preserve continuity across atlas chart boundaries and to adapt differential operators to the non-uniform parameterization. We introduce a novel function space and multigrid solver that jointly enable robust, interactive, and geometry-aware signal processing. We demonstrate our approach using several applications including smoothing and sharpening, multiview stitching, geodesic distance computation, and line integral convolution.

关键词： signal processing Laplacian filtering mesh parameterization multigrid solver domain decomposition

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A multigrid preconditioned numerical scheme for a reaction-diffusion system

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APPLIED MATHEMATICS AND COMPUTATION 2015年 254卷 266-276页

作者： Bhowmik, Samir Kumar Al Imam Mohammad Ibn Saud Islamic Univ IMSIU Coll Sci Dept Math & Stat Riyadh 11623 Saudi Arabia

Reaction diffusion operators have been used to model many engineering and biological systems. In this study we consider a reaction diffusion system modeling various engineering and life science problems. There are many algorithms to approximate such mathematical models. Most of the algorithms are conditionally stable and convergent. For a big time step size a Krylov subspace type solver for such models converges slowly or oscillates because of the presence of the diffusion term. Here we study a multigrid pre-conditioned generalized minimal residual method (GMRES) for such a model. We start with a five point scheme for the spatial integration and a method of lines for the temporal integration of the system of PDEs. Then we implement a multigrid iterative algorithm for the full discrete model, and show some numerical results to demonstrate the dominance of the solver. We analyze the convergence rate of such a multigrid iterative preconditioning algorithm. Reaction diffusion systems arise in many mathematical models and thus this study has many applicabilities. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Reaction-diffusion system Gray Scot model Predator prey model Finite difference scheme multigrid solver

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