Geometric Multigrid Method Research Articles

Kernel multigrid on manifolds

Kernel methods for solving partial differential equations work coordinate-free on the surface and yield high approximation rates for smooth solutions. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods for data fitting problems, but the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs is a challenging problem which has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a τ≥2-cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting solver can be accelerated by using the Lagrange function decay and obtain satisfying convergence rates by a rigorous analysis. In particular, we show that the computational cost of the linear solver scales log-linear in the degrees of freedom.

A semi-automatic method for block-structured hexahedral meshing of aortic dissections.

The article presents a semi-automatic approach to generating structured hexahedral meshes of patient-specific aortas ailed by aortic dissection. The condition manifests itself as a formation of two blood flow channels in the aorta, as a result of a tear in the inner layers of the aortic wall. Subsequently, the morphology of the aorta is greatly impacted, making the task of domain discretization highly challenging. The meshing algorithm presented herein is automatic for the individual lumina, whereas the tears require user interaction. Starting from an input (triangle) surface mesh, we construct an implicit surface representation as well as a topological skeleton, which provides a basis for the generation of a block-structure. Thereafter, the mesh generation is performed via transfinite maps. The meshes are structured and fully hexahedral, exhibit good quality and reliably match the original surface. As they are generated with computational fluid dynamics in mind, a fluid flow simulation is performed to verify their usefulness. Moreover, since the approach is based on valid block-structures, the meshes can be made very coarse (around 1000 elements for an entire aortic dissection domain), and thus promote using solvers based on the geometric multigrid method, which is typically reliant on the presence of a hierarchy of coarser meshes.

Hp-Multigrid Preconditioner for a Divergence-Conforming HDG Scheme for the Incompressible Flow Problems

In this study, we present an hp-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier–Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The hp-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we develop a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix–Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput Methods Appl Mech Engrg 311:304–326, 2022), which allows for the direct application of geometric multigrid theory on the Crouzeix–Raviart discretization. The numerical experiments demonstrate the robustness of the proposed hp-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi and Olshanskii (SIAM J Sci Comput 28:2095–2113, 2006) and Farrell et al. (SIAM J Sci Comput 41:A3073–A3096, 2019), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier–Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to 103\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^3$$\\end{document}.

Hp-Robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p ≥ 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator which comes with the solver is hp-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

A correction function-based kernel-free boundary integral method for elliptic PDEs with implicitly defined interfaces

This work addresses a novel version of the kernel-free boundary integral (KFBI) method for solving elliptic PDEs with implicitly defined irregular boundaries and interfaces. We focus on boundary value problems and interface problems, which are reformulated into boundary integral equations and solved with the matrix-free GMRES method. In the KFBI method, evaluating boundary and volume integrals only requires solving equivalent but much simpler interface problems in a bounding box, for which fast solvers such as FFTs and geometric multigrid methods are applicable. For the simple interface problem, a correction function is introduced for both the evaluation of right-hand side correction terms and the interpolation of a non-smooth potential function. A mesh-free collocation method is proposed to compute the correction function near the interface. The new method avoids complicated derivation for derivative jumps of the solution and is easy to implement, especially for the fourth-order method in three space dimensions. Various numerical examples are presented, including challenging cases such as high-contrast coefficients, arbitrarily close interfaces and heterogeneous interface problems. The reported numerical results verify that the proposed method is both accurate and efficient.

Meshless Geometric Multigrid Method for Complex Geometries with Improved Cell Coarsening Algorithm

In this paper, we present an improved multicloud methodology to extend the multicloud convergence accelerator, which depends on meshless discretization on coarse level computation of the geometric multigrid method, to cell finite volume (CFV) on three-dimensional space. To achieve this, we first identify the primary challenge in the meshless coarsening application to CFV. Specifically, the challenge lies in the poorly coarsened grid, which results in a degradation of the convergence rate. The reason for this is that the coarsening stencils were coupled with the discretization stencil. To address the issue, we proposed a new coarsening strategy on CFV that adequately utilizes node information on the fine-level mesh. The proposed methodology shows superior coarsening rates compared to the basic meshless cell coarsening, regardless of the dimension and type of mesh elements, resulting in significant speedup in convergence. Various computational fluid dynamics (CFD) simulations with three-dimensional complex geometries are presented, and their results demonstrate approximately seven times faster convergence compared to the single-grid method. The overall results verify that the multicloud method with the proposed coarsening procedure provides a powerful tool for practical CFD simulations by reducing the computation time, and it enables researchers and engineers to simulate and optimize real-world systems more efficiently.

Fundamental data structures for matrix-free finite elements on hybrid tetrahedral grids

This paper presents efficient data structures for the implementation of matrix-free finite element methods on block-structured, hybrid tetrahedral grids. It provides a complete categorization of all geometric sub-objects that emerge from the regular refinement of the unstructured, tetrahedral coarse grid and describes efficient iteration patterns and analytical linearization functions for the mapping of coefficients to memory addresses. This foundation enables the implementation of fast, extreme-scalable, matrix-free, iterative solvers, and in particular geometric multigrid methods by design. Their application to the variable-coefficient Stokes system subject to an enriched Galerkin discretization and to the curl-curl problem discretized with Nédélec edge elements showcases the flexibility of the implementation. Finally, the solution of a curl-curl problem with 1.6 ⋅ 10 11 (more than one hundred billion) unknowns on more than 32,000 processes with a matrix-free full multigrid solver demonstrates its extreme-scalability.

A parallel geometric multigrid method for adaptive topology optimization

This work presents an efficient parallel geometric multigrid (GMG) implementation for preconditioning Krylov subspace methods solving differential equations using non-conforming meshes for discretization. The approach does not constrain such meshes to the typical multiscale grids used by Cartesian hierarchical grid methods, such as octree-based approaches. It calculates the restriction and interpolation operators for grid transferring between the non-conforming hierarchical meshes of the cycle scheme. Using non-Cartesian grids in topology optimization, we reduce the mesh size discretizing only the design domain and keeping the geometry of boundaries in the final design. We validate the GMG method operating on non-conforming meshes using an adaptive density-based topology optimization method, which coarsens the finite elements dynamically following a weak material estimation criterion. The GMG method requires the generation of the hierarchical non-conforming meshes dynamically from the one used by the adaptive topology optimization to analyze to the one coarsening all the mesh elements until the coarsest level of the mesh hierarchy. We evaluate the performance of the adaptive topology optimization using the GMG preconditioner operating on non-conforming meshes using topology optimization on a fine-conforming mesh as the reference. We also test the strong and weak scaling of the parallel GMG preconditioner with two three-dimensional topology optimization problems using adaptivity, showing the computational advantages of the proposed method.

Fast Multigrid Simulations of Pinning in REBCO With Highly Resistive Nanorods

A geometric multigrid method is applied to the frozen-field time-dependent Ginzburg–Landau equations for anisotropic superconductors with spatially varying material properties. This reduces the simulation time for critical current determination in large 3D systems by up to two orders of magnitude compared with a single grid. Critical current density calculations are presented for a model of REBCO tape with nanorod artificial pinning centers. The model consists of an anisotropic superconducting matrix with embedded nanorods which are treated as isotropic highly resistive cylinders. The effects of varying the density, splay, and conductivity of the nanorods are considered. Increasing the resistivity of the nanorods profoundly affects the critical current density in high fields, where current paths are percolative in the vicinity of matrix-pin interfaces and persist well above the upper critical field of the superconducting matrix.

A Geometric Multigrid Method for Space-Time Finite Element Discretizations of the Navier–Stokes Equations and its Application to 3D Flow Simulation

We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier–Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for generalized minimal residual iterations. Its performance properties are demonstrated for 2D and 3D benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the deal.II finite element library. The concepts are flexible and can be transferred to similar software platforms.

Multigrid accelerated projection method on GPU cluster for the simulation of turbulent flows

ABSTRACT A graphics processing unit (GPU)-enabled numerical procedure based on the projection method is developed for simulating incompressible turbulent flows. The pressure Poisson equation is efficiently solved using the V-cycle geometric multigrid method. Additionally, the coarse grid aggregation (CGA) technique enhances the multigrid level of multi-GPU simulations, resulting in significant performance improvements. The validity of the proposed method is confirmed through direct numerical simulations of the turbulent lid-driven cavity flows at a Reynolds number of 3200. The computed mean, and turbulence quantities closely match the available measured data, validating the accuracy of the approach. For the cubic cavity under consideration, the optimized minimum grid sizes for multigrid and CGA are determined to be 83 and 323, respectively. An additional speedup of approximately ≈2.3 to ≈2.6 is achieved by employing CGA. In terms of performance, the current implementation demonstrates compatibility with the lattice Boltzmann method while also being three times faster than the explicit weakly compressible scheme. The superior performance of the GPU implementation over CPU is further highlighted, with a remarkable one thousandfold speedup observed between the Nvidia Tesla V100 and a single core of the Intel I7-6900K (8 cores). Specifically, the performance of one Tesla V100 is found to be equivalent to 125 I7-6900K central processing units.

Geometric multigrid method for solving Poisson's equation on octree grids with irregular boundaries

A method is presented to include irregular domain boundaries in a geometric multigrid solver. Dirichlet boundary conditions can be imposed on an irregular boundary defined by a level set function. Our implementation employs quadtree/octree grids with adaptive refinement, a cell-centered discretization and pointwise smoothing. Boundary locations are determined at a subgrid resolution by performing line searches. For grid blocks near the interface, custom operator stencils are stored that take the interface into account. For grid block away from boundaries, a standard second-order accurate discretization is used. The convergence properties, robustness and computational cost of the method are illustrated with several test cases. New version program summaryProgram Title: AfivoCPC Library link to program files:https://doi.org/10.17632/5y43rjdmxd.2Developer's repository link:https://github.com/MD-CWI/afivoLicensing provisions: GPLv3Programming language: FortranJournal reference of previous version: Comput. Phys. Commun. 233 (2018) 156–166. https://doi.org/10.1016/j.cpc.2018.06.018Does the new version supersede the previous version?: Yes.Reasons for the new version: Add support for internal boundaries in the geometric multigrid solver.Summary of revisions: The geometric multigrid solver was generalized in several ways: a coarse grid solver from the Hypre library is used, operator stencils are now stored per grid block, and methods for including boundaries via a level set function were added.Nature of problem: The goal is to solve Poisson's equation in the presence of irregular boundaries that are not aligned with the computational grid. It is assumed these irregular boundaries are defined by a level set function, and that a Dirichlet type boundary condition is applied. The main applications are 2D and 3D simulations with octree-based adaptive mesh refinement, in which the mesh frequently changes but the irregular boundaries do not.Solution method: A geometric multigrid method compatible with octree grids is developed, using a cell-centered discretization and point-wise smoothing. Near irregular boundaries, custom operator stencils are stored. Line searches are performed to locate interfaces with sub-grid resolution. To increase the methods robustness, this line search is modified on coarse grids if boundaries are otherwise not resolved. The multigrid solver uses OpenMP parallelization.

A Geometric Multigrid Method for 3D Magnetotelluric Forward Modeling Using Finite-Element Method

The traditional three-dimensional (3D) magnetotelluric (MT) forward modeling using Krylov subspace algorithms has the problem of low modeling efficiency. To improve the computational efficiency of 3D MT forward modeling, we present a novel geometric multigrid algorithm for the finite element method. We use the vector finite element to discretize Maxwell’s equations in the frequency domain and apply the Dirichlet boundary conditions to obtain large sparse complex linear equations for the solution of EM responses. To improve the convergence of the solution at low frequencies we use the divergence correction to correct the electric field. Then, we develop a V-cycle geometric multigrid algorithm to solve the linear equations system. To demonstrate the efficiency and effectiveness of our geometric multigrid method, we take three synthetic models (COMMEMI 3D-2 model, Dublin test model 1, modified SEG/EAEG salt dome model) and compare our results with the published ones. Numerical results show that the geometric multigrid algorithm proposed in this paper is much better than the commonly used Krylov subspace algorithms (such as SOR-GMRES, ILU-BICGSTAB, SOR-BICGSTAB) in terms of the iteration number, the solution time, and the stability, and thus is more suitable for large-scale 3D MT forward modeling.

Distributed Geometric Multigrid Method: Analysis of a V Cycle Truncation Level Criteria

This article presents the analysis of a V cycle truncation level criteria in a parallel implemention of a geometric multigrid method for solving partial differential equations, developed over a distributed memory system. The proposed system is implemented in C, using the Message Passing Interface library. A theoretical analysis of the proposed truncation level criteria is presented, and its evaluation is reported for the Poisson problem. The experimental analysis indicates that the proposed method achieves accurate speedup and computational efficiency, and shows a good scalability behavior to solve large problems by properly using more processing units.

Robust Optimal Well Control using an Adaptive Multigrid Reinforcement Learning Framework

Reinforcement learning (RL) is a promising tool for solving robust optimal well control problems where the model parameters are highly uncertain and the system is partially observable in practice. However, the RL of robust control policies often relies on performing a large number of simulations. This could easily become computationally intractable for cases with computationally intensive simulations. To address this bottleneck, an adaptive multigrid RL framework is introduced which is inspired by principles of geometric multigrid methods used in iterative numerical algorithms. RL control policies are initially learned using computationally efficient low-fidelity simulations with coarse grid discretization of the underlying partial differential equations (PDEs). Subsequently, the simulation fidelity is increased in an adaptive manner towards the highest fidelity simulation that corresponds to the finest discretization of the model domain. The proposed framework is demonstrated using a state-of-the-art, model-free policy-based RL algorithm, namely the proximal policy optimization algorithm. Results are shown for two case studies of robust optimal well control problems, which are inspired from SPE-10 model 2 benchmark case studies. Prominent gains in computational efficiency are observed using the proposed framework, saving around 60-70% of the computational cost of its single fine-grid counterpart.

A meshless geometric multigrid method for a grid with a high aspect ratio

A meshless geometric multigrid method for a grid with a high aspect ratio

Optimized sparse approximate inverse smoothers for solving Laplacian linear systems

In this paper we propose and analyze new efficient sparse approximate inverse (SAI) smoothers for solving the two-dimensional (2D) and three-dimensional (3D) Laplacian linear system with geometric multigrid methods. Local Fourier analysis shows that our proposed SAI smoother for 2D achieves a much smaller smoothing factor than the state-of-the-art SAI smoother studied in Bolten et al. (2016) [12]. The proposed SAI smoother for 3D cases provides smaller optimal smoothing factor than that of weighted Jacobi smoother. Numerical results validate our theoretical conclusions and illustrate the high-efficiency and high-effectiveness of our proposed SAI smoothers. Such SAI smoothers have the advantage of inherent parallelism.

Data-driven Multi-Grid solver for accelerated pressure projection

Pressure projection is the single most computationally expensive step in an unsteady incompressible fluid simulation. This work demonstrates the ability of data-driven methods to accelerate the approximate solution of the Poisson equation at the heart of pressure projection. Geometric Multi-Grid methods are identified as linear convolutional encoder–decoder networks and a data-driven smoother is developed using automatic differentiation to optimize the velocity-divergence projection. The new method is found to accelerate classic Multi-Grid methods by a factor of two to three with no loss of accuracy on eleven 2D and 3D flow cases including cases with dynamic immersed solid boundaries. The optimal parameters are found to transfer nearly 100% effectiveness as the resolution is increased, providing a robust approach for accelerated pressure projection of unsteady flows.

Two Methods to Improve the Efficiency of Supersonic Flow Simulation on Unstructured Grids

The paper presents two methods to improve the efficiency of supersonic flow simulation using arbitrarily shaped unstructured grids. The first method promotes increasing the numerical solution convergence rate and is based on the geometric multigrid method for initialization of the flow field. The method is used to obtain the initial field of distributed physical quantity values, which maximally corresponds to the converged solution. For this purpose, the problem simulation is performed on a series of coarse grids beginning from the coarsest one in this series. Upon completion of simulations, the solution obtained is interpolated to a finer grid and used for initialization of simulations on this grid. The second method allows increasing the numerical solution accuracy and is based on statically adapting the computational grid to the flow specifics. The static adaptation algorithm provides automatic refinement of the computational grid in the region of specific features of flow, such as shock waves typical for supersonic flows. This algorithm provides a better description of the shock-wave front owing to the local grid refinement, with the local refinement region being automatically selected. Results of using these methods are demonstrated for the two supersonic aerodynamics problems: the simulation of the bow shock strength at a given distance under axially symmetric body Seeb-ALR and a mock-up aircraft Lockheed Martin 1021. It is shown that in both cases, the numerical solution convergence rate is increased owing to the use of the geometric multigrid method for initialization and a higher quality and a higher accuracy of solution is gained owing to the local grid refinement (using static adaptation means) near the shock-wave front.

A restricted additive Vanka smoother for geometric multigrid

The solution of saddle-point problems, such as the Stokes equations, is a challenging task, especially in large-scale problems. Multigrid methods are one of the most efficient solvers for such systems of equations and can achieve convergence rates independent of the problem size. The smoother is a crucial component of multigrid methods and significantly affects its overall efficiency. We propose a Vanka-type smoother that we refer to as Restricted Additive Vanka and investigate its convergence in the context of adaptive geometric multigrid methods for the Stokes equations. The proposed smoother has the advantage of being an additive method and provides favorable properties in terms of algorithmic complexity, scalability and applicability to high-performance computing. We compare the performance of the smoother with two variants of the classical Vanka smoother using numerical benchmarks for the Stokes problem. We find that the restricted additive smoother achieves comparable convergence rates to the classical multiplicative Vanka smoother while being computationally less expensive per iteration, which results in faster solution runtimes.