Parallel Multigrid Method Research Articles

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.

Local and parallel multigrid method for semilinear Neumann problem with nonlinear boundary condition

Local and parallel multigrid method for semilinear Neumann problem with nonlinear boundary condition

Parallel multigrid method for solving inverse problems

We considered in this work the linear operator equation and used the Landweber iterative method as an iterative solver. After that, we used the multigrid method as an optimization method for obtaining an approximation solution with a highly accurate and fast process. A new parallel algorithm for the multigrid process has been developed. The proposed algorithm is based on a V-cycle mixed with the two-grid method. This modification of the V-cycle provides for parallel computing for each level. A coarse grid operator with a residual right-hand side vector for each coarse grid is provided. This parallel algorithm is used to accelerate and enhance computation for the solution of the iteration method in solving the inverse ill-posed problems. The necessary cost-time computation for all stages and processes for the parallel V-cycle algorithm has been done. A numerical experiment on solving the IVP (initial value problem) for the heat equation showed that the new parallel algorithm is much more efficient than the sequential method.•The study of iteration algorithms and mathematical experiments reveals a slow rate of convergence.•The Multigrid method is often used to speed up the rate of convergence of iterative methods, which is an effective method of solving large systems of linear algebra equations.•The approximation solution for the linear algebra equations was found by using the parallel method with the multigrid method.

A parallel space-time multigrid method for the eddy-current equation

We expand the applicabilities and capabilities of an already existing space-time parallel method based on a block Jacobi smoother. First we formulate a more detailed criterion for spatial coarsening, which enables the method to deal with unstructured meshes and varying material parameters. Further we investigate the application to the eddy-current equation, where the non-trivial kernel of the curl operator causes severe problems. This is remedied with a new nodal auxiliary space correction. We proceed to identify convergence rates by local Fourier analysis and numerical experiments. Finally, we present a numerical experiment which demonstrates its excellent scaling properties.

Local and parallel multigrid method for semilinear elliptic equations

This paper presents a new type of local and parallel multigrid method to solve semilinear elliptic equations. The proposed method does not directly solve the semilinear elliptic equations on each layer of the multigrid mesh sequence, but transforms the semilinear elliptic equations into several linear elliptic equations on the multigrid mesh sequence and some low-dimensional semilinear elliptic equations on the coarsest mesh. Furthermore, the local and parallel strategy is used to solve the involved linear elliptic equations. Since solving large-scale semilinear elliptic equations in fine space, which can be fairly time-consuming, is avoided, the proposed local and parallel multigrid scheme will significantly improve the solving efficiency for the semilinear elliptic equations. Besides, compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only requires the Lipschitz continuation property of the nonlinear term. We make a rigorous theoretical analysis of the presented local and parallel multigrid scheme, and propose some numerical experiments to support the theory.

Parallel solution of saddle point systems with nested iterative solvers based on the Golub‐Kahan Bidiagonalization

SummaryThe Golub‐Kahan bidiagonalization is widely used in the singular value decomposition of rectangular matrices and has been generalized to an iterative solver for symmetric indefinite linear systems with a two‐by‐two block structure. In this work, we present a scalability study of this generalized solver as implemented in a recent release of the parallel numerical library PETSc (Portable, Extensible Toolkit for Scientific Computation). We present an improved solver performance for the two‐dimensional (2D) Stokes equations as compared to previous work. Furthermore, we investigate the performance of different parallel inner solvers in the outer Golub‐Kahan iteration for a three‐dimensional Stokes problem. The study includes parallel sparse direct solvers and multigrid methods. When increasing the number of cores for a fixed total problem size, the solver exhibits good speedups of up to 50% at the 1024 core count. For the tests in which the total problem size grows while the workload in each core stays constant, the parallel performance of the solver scales almost linearly with the increase in the core counts. In particular, the computation time increases only by about 15% when the number of cores increases from 80 to 1024 for a 2D test case.

Mesh generation for thin layered domains and its application to parallel multigrid simulation of groundwater flow

The generation of detailed three dimensional meshes for the simulation of groundwater flow in thin layered domains is crucial to capture important properties of the underlying domains and to reach a satisfying accuracy. At the same time, this level of detail poses high demands both on suitable hardware and numerical solver efficiency. Parallel multigrid methods have been shown to exhibit near optimal weak scalability for massively parallel computations of density driven flow. A fully automated parameterized algorithm for prism based meshing of coarse grids from height data of individual layers is presented. Special structures like pinch outs of individual layers are preserved. The resulting grid is used as a starting point for parallel mesh and hierarchy creation through interweaved projected refinement and redistribution. Efficiency and applicability of the proposed approach are demonstrated for a parallel multigrid based simulation of a realistic sample problem.

Local and Parallel Multigrid Method for Nonlinear Eigenvalue Problems

In this paper, a new type of local and parallel algorithm is proposed to solve nonlinear eigenvalue problem based on multigrid discretization. Instead of solving the nonlinear eigenvalue problem directly in each level mesh, our method converts the nonlinear eigenvalue problem in the finest mesh to a linear boundary value problem on each level mesh and some nonlinear eigenvalue problems on the coarsest mesh. Further, the involved linear boundary value problems are solved using the local and parallel strategy. As no nonlinear eigenvalue problem is being solved directly on the fine spaces, which is time-consuming, this new type of local and parallel multigrid method evidently improves the efficiency of nonlinear eigenvalue problem solving. We provide a rigorous theoretical analysis for our algorithm and present details on numerical simulations to support our theory.

A full multigrid method for the Steklov eigenvalue problem

ABSTRACTThis paper introduces a kind of parallel multigrid method for solving Steklov eigenvalue problem based on the multilevel correction method. Instead of the common costly way of directly solving the Steklov eigenvalue problem on some fine space, the new method contains some boundary value problems on a series of multilevel finite element spaces and some steps of solving Steklov eigenvalue problems on a very low dimensional space. The linear boundary value problems are solved by some multigrid iteration steps. We will prove that the computational work of this new scheme is truly optimal, the same as solving the corresponding linear boundary value problem. Besides, this multigrid scheme has a good scalability by using parallel computing technique. Some numerical experiments are presented to validate our theoretical analysis.

Correction to: 3D Parallel Multigrid Methods for Real-Time Fluid Simulation

Unfortunately, the co-author’s given name has been published incorrectly in the original online publication. The correct given name should be: Suiyun.

Hybrid parallel multigrid preconditioner based on automatic mesh coarsening for 3D metal forming simulations

SummaryA parallel multigrid (MG) method is developed to reduce the large computational costs involved by the finite element simulation of highly viscous fluid flows, especially those resulting from metal forming applications, which are characterized by using a mixed velocity/pressure implicit formulation, unstructured meshes of tetrahedra, and frequent remeshings. The developed MG method follows a hybrid approach where the different levels of nonnested meshes are geometrically constructed by mesh coarsening, while the linear systems of the intermediate levels result from the Galerkin algebraic approach. A linear O(N) convergence rate is expected (with N being the number of unknowns), while keeping software parallel efficiency. These objectives lead to selecting unusual MG smoothers (iterative solvers) for the upper grid levels and to developing parallel mesh coarsening algorithms along with parallel transfer operators between the different levels of partitioned meshes. Within the utilized PETSc library, the developed MG method is employed as a preconditioner for the usual conjugate residual algorithm because of the symmetric undefinite matrix of the system to solve. It shows a convergence rate close to optimal, an excellent parallel efficiency, and the ability to handle the complex forming problems encountered in 3‐dimensional hot forging, which involve large material deformations and frequent remeshings.

Analysis of parallel multigrid methods in real-time fluid simulation

The multigrid method has been widely used in computational fluid dynamics (CFD) numerical calculations because of its strong convergence. To achieve real-time simulation of a fluid in computer graphics (CG), the operation efficiency is also a significant factor to consider except for operational accuracy. For this problem, we introduced two multigrid cycling schemes, V-Cycle and full multigrid (FMG). Moreover, we have proposed a simple geometric multigrid method (GMG), and compared with the existing wide application of algebraic multigrid (AMG). All the calculations are the solution of parallel computing of GPU in this paper. The results showed that our approaches have improved the algorithm’s computational speed and convergence time, which prominently enhanced the efficiency of the fluid simulation.

Algorithm-based fault recovery of adaptively refined parallel multilevel grids

On future extreme scale computers, it is expected that faults will become an increasingly serious problem as the number of individual components grows and failures become more frequent. This is driving the interest in designing algorithms with built-in fault tolerance that can continue to operate and that can replace data even if part of the computation is lost in a failure. For fault-free computations, the use of adaptive refinement techniques in combination with finite element methods is well established. Furthermore, iterative solution techniques that incorporate information about the grid structure, such as the parallel geometric multigrid method, have been shown to be an efficient approach to solving various types of partial different equations. In this article, we present an advanced parallel adaptive multigrid method that uses dynamic data structures to store a nested sequence of meshes and the iteratively evolving solution. After a fail-stop fault, the data residing on the faulty processor will be lost. However, with suitably designed data structures, the neighbouring processors contain enough information so that a consistent mesh can be reconstructed in the faulty domain with the goal of resuming the computation without having to restart from scratch. This recovery is based on a set of carefully designed distributed algorithms that build on the existing parallel adaptive refinement routines, but which must be carefully augmented and extended.

A parallel scalable multigrid method and HOC scheme for anisotropy elliptic problems

ABSTRACTAnisotropy problems are widely encountered in practical applications. In the present paper, a parallel and scalable multigrid (MG) method combined with the high-order compact difference scheme is employed to solve an anisotropy model equation on a rectangular domain. The present method is compared with the MG combined with the standard central difference scheme. Numerical results show that the MG method combined with the high-order compact difference scheme is more accurate and efficient than the MG with the standard central difference scheme for solving anisotropy elliptic problems. MG components (restriction, prolongation, relaxation, and cycling) and the corresponding parallelism characteristics are also discussed.

A parallel multigrid method for constrained minimization problems and its application to friction, contact, and obstacle problems

The parallel solution of constrained minimization problems requires special care to be taken with respect to the information transfer between the different subproblems. Here, we present a nonlinear decomposition approach which employs an additional nonlinear correction step along the processor interfaces. Our approach is generic in the sense that it can be applied to a wide class of minimization problems with strongly local nonlinearities, including even nonsmooth minimization problems. We also describe the implementation of our nonlinear decomposition method in the object oriented library ObsLib $$++$$++. The flexibility of our approach and its implementation is presented along different problem classes as obstacle problems, frictional contact problems and biomechanical applications. For the same examples, number of iterations, computation time, and parallelization speedup are measured, and the results demonstrate that the implementation scales reasonably well up to 4096 processors.

Resilience for Massively Parallel Multigrid Solvers

Fault tolerant massively parallel multigrid methods for elliptic partial differential equations are a step towards resilient solvers. Here, we combine domain partitioning with geometric multigrid methods to obtain fast and fault-robust solvers for three-dimensional problems. The recovery strategy is based on the redundant storage of ghost values, as they are commonly used in distributed memory parallel programs. In the case of a fault, the redundant interface values can be easily recovered, while the lost inner unknowns are recomputed approximately with recovery algorithms using multigrid cycles for solving a local Dirichlet problem. Different strategies are compared and evaluated with respect to performance, computational cost, and speedup. Especially effective are asynchronous strategies combining global solves with accelerated local recovery. By this, multiple faults can be fully compensated with respect to both the number of iterations and run-time. For illustration, we use a state-of-the-art petascale supercomputer to study failure scenarios when solving systems with up to $6 \cdot 10^{11}$ (0.6 trillion) unknowns.

Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems

We present and analyze a new space-time parallel multigrid method for parabolic equations. The method is based on arbitrarily high order discontinuous Galerkin discretizations in time and a finite element discretization in space. The key ingredient of the new algorithm is a block Jacobi smoother. We present a detailed convergence analysis when the algorithm is applied to the heat equation and determine asymptotically optimal smoothing parameters, a precise criterion for semi-coarsening in time or full coarsening, and give an asymptotic two grid contraction factor estimate. We then explain how to implement the new multigrid algorithm in parallel and show with numerical experiments its excellent strong and weak scalability properties.

Scalable implementation of the parallel multigrid method on massively parallel computers

We consider a scalable implementation of multigrid methods for elliptic problems for fusion simulations on current and future high performance computer (HPC) architectures. Multigrid methods are the most efficient available solvers for elliptic problems. But, these methods require to handle the operations on several coarser levels where the communication costs are higher than the computation costs. We use only one core from a certain coarser level and get performance improvements on a large number of cores. Also, we consider the OpenMP/MPI hybridization implementation which is fitted on multi-core CPU architectures. The hybridization can use a small number of MPI tasks on the same number of cores and achieve performance improvements on a large number of cores. We investigate the scaling properties of the parallel multigrid solver with structured discretization on a regular hexagonal domain on a massively parallel computer (IFERC-CSC).

Parallel multigrid method for solving elliptic equations

The proposed algorithm represents an efficient parallel implementation of the Fedorenko multigrid method and is intended for solving three-dimensional elliptic equations. Scalability is provided by the use of the Chebyshev iterations for solution of the coarsest grid equations and for construction of the smoothing procedures. The calculation results are given: they confirm the efficiency of the algorithm and scalability of the parallel code.

A new parallel cascadic multigrid method

In this paper, a new parallel cascadic multigrid method is studied with domain decomposition technique and extrapolation technique. Convergence analysis is given based on the superconvergence of the finite element. The numerical results show that the method is efficient.