Coarse Grid Operators Research Articles

Coarse-grid operator optimization in multigrid reduction in time for time-dependent Stokes and Oseen problems

Coarse-grid operator optimization in multigrid reduction in time for time-dependent Stokes and Oseen problems

Efficient Multigrid Reduction-in-Time for Method-of-Lines Discretizations of Linear Advection

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes in space-time known as characteristic components. We propose an alternative coarse-grid operator that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the proposed coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate speed-up over sequential time-stepping.

Scalable multi-level deflation preconditioning for highly indefinite time-harmonic waves

Recent research efforts aimed at iteratively solving time-harmonic waves have focused on a broad range of techniques to accelerate convergence. In particular, for the famous Helmholtz equation, deflation techniques have been studied to accelerate the convergence of Krylov subspace methods. In this work, we extend the two-level deflation method to a multilevel deflation method for (heterogeneous) Helmholtz and elastic wave problems. By using higher-order deflation vectors, we show that up to the level where the coarse-grid linear systems remain indefinite, the near-zero eigenvalues of the these coarse-grid operators remain aligned with the near-zero eigenvalues of the fine-grid operator, keeping the spectrum of the preconditioned system away from the origin. Combining this with the well-known CSLP-preconditioner, we obtain a scalable solver for the highly indefinite linear systems. This can be attributed to a close to wave number independent convergence and an optimal use of the CSLP-preconditioner on the indefinite levels. There, we approximate the CSLP-preconditioner, while allowing the complex shift to be small, by using inner Bi-CGSTAB iterations instead of a multigrid F-cycle. The proposed method shows very promising results for the more challenging two- and three-dimensional heterogeneous time-harmonic wave problems.

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.

On divergence- and gradient-preserving coarse-graining for finite volume primitive equation ocean models

We consider the problem of coarse-graining in the context of finite-volume fluid models. If a variable is defined on a high-resolution grid it may be coarse-grained so that it is defined on a grid of lower resolution. In general this will cause some information about the variable to be lost. In particular, horizontal divergences, gradients or other operators calculated on the coarse grid after projecting may differ from those calculated on the fine grid. In some cases we are able to choose averaging weights for coarse-graining such that the coarse-grid operators will give a result approximating that of the corresponding fine-grid operators applied on the fine grid. In this work we derive general conditions on the averaging weights that allow the divergence and gradient to be preserved. These conditions are applied to a regular triangular mesh with B-grid variable placement in which the fine-grid resolution is some integer multiple N of the coarse-grid resolution. For this case we find particular values for the averaging weights that preserve the divergence, and a set of different averaging weights that preserve the gradient. We observe that the vertical component of the curl is also preserved by the same coarse-graining that preserves divergence. These coarse-grainings are applied to data from FESOM2 simulations and we demonstrate that using the particular coarse-grainings derived herein gives an overall reduction in L1 error when compared with other methods.

A multigrid method with reduced phase error for 2D damped Helmholtz equations

This paper deals with a new multigrid method with reduced phase error for solving 2D damped Helmholtz equations. The method is obtained by taking the high‐effective, reduced phase error 5‐point finite difference (FD) scheme as a coarse grid operator and the regular 5‐point FD scheme as a fine grid operator. It is found that the proposed method gives a faster convergent rate than the regular multigrid method. A local mode Fourier analysis confirms the validity of our proposed method. Finally, some numerical results demonstrate the efficiency of the method.

A multigrid accelerated eigensolver for the Hermitian Wilson–Dirac operator in lattice QCD

Eigenvalues of the Hermitian Wilson–Dirac operator are of special interest in several lattice QCD simulations, e.g., for noise reduction when evaluating all-to-all propagators. In this paper we present a Davidson-type eigensolver that utilizes the structural properties of the Hermitian Wilson–Dirac operator Q to compute eigenpairs of this operator corresponding to small eigenvalues. The main idea is to exploit a synergy between the (outer) eigensolver and its (inner) iterative scheme which solves shifted linear systems. This is achieved by adapting the multigrid DD-αAMG algorithm to a solver for shifted systems involving the Hermitian Wilson–Dirac operator. We demonstrate that updating the coarse grid operator using eigenvector information obtained in the generalized Davidson method is crucial to achieve good performance when calculating many eigenpairs, as our study of the local coherence shows. We compare our method with the commonly used software-packages PARPACK and PRIMME in numerical tests, where we are able to achieve significant improvements, with speed-ups of up to one order of magnitude and a near-linear scaling with respect to the number of eigenvalues. For illustration we compare the distribution of the small eigenvalues of Q on a 64×323 lattice with what is predicted by the Banks–Casher relation in the infinite volume limit.

An efficient strategy for large scale 3D simulation of heterogeneous materials to predict effective thermal conductivity

The behavior of heterogeneous materials (e.g. composite, polycrystalline) is one of the most challenging subjects for researchers. Imaging techniques based on X-ray tomography permit one to obtain the inner structure of these materials. To account for such detailed information and understand the material behavior, the use of tomographic images as an input for numerical simulations is becoming more and more common. The main difficulty is the computational cost, mesh generation in the context of finite element simulations and the large discontinuities of the material properties, which can lead to divergence. The subject of this paper is to resolve these problems and understand the behavior of a material with high heterogeneity. The application of a MultiGrid method coupled with homogenization technique is proposed. The MultiGrid method is a well known method to increase convergence speed. The homogenization technique is applied to compute the coarse grid operators of the MultiGrid process. The combination of the two methods can efficiently deal with large property variations. Hybrid MPI/OpenMP parallel computing is used to save computational time. The influence of material heterogeneity is analyzed as well as the ratio of material properties for the thermal conduction. The effective thermal conductivity of material is illustrated.

Quasi-matrix-free Hybrid Multigrid on Dynamically Adaptive Cartesian Grids

We present a family of spacetree-based multigrid realizations using the tree’s multiscale nature to derive coarse grids. They align with matrix-free geometric multigrid solvers as they never assemble the system matrices, which is cumbersome for dynamically adaptive grids and full multigrid. The most sophisticated realizations use BoxMG to construct operator-dependent prolongation and restriction in combination with Galerkin/Petrov-Galerkin coarse-grid operators. This yields robust solvers for nontrivial elliptic problems. We embed the algebraic, problem-dependent, and grid-dependent multigrid operators as stencils into the grid and evaluate all matrix-vector products in situ throughout the grid traversals. Such an approach is not literally matrix-free as the grid carries the matrix. We propose to switch to a hierarchical representation of all operators. Only differences of algebraic operators to their geometric counterparts are held. These hierarchical differences can be stored and exchanged with small memory footprint. Our realizations support arbitrary dynamically adaptive grids while they vertically integrate the multilevel operations through spacetree linearization. This yields good memory access characteristics, while standard colouring of mesh entities with domain decomposition allows us to use parallel many-core clusters. All realization ingredients are detailed such that they can be used by other codes.

H-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

In this work we exploit agglomeration based h-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier–Stokes equations. As a distinctive feature h-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element L2 projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures. Significant execution time gains are documented.

Multigrid methods for convection–diffusion problems discretized by a monotone scheme

We study multigrid (MG) methods for the solution of systems of linear algebraic equations obtained from a stable discretization of convection–diffusion problems by an exponential fitting scheme. The latter ensures the stability of the simplest possible coarse grid operators obtained from Galerkin projections based on graph matching. Linear and nonlinear MG preconditioners are defined in the framework of algebraic multilevel iteration. The option of using polynomial smoothers is investigated in context of nonsymmetric problems and a systematic performance comparison is presented for various algorithms on a representative set of two- and three-dimensional test problems.

Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation

Many important physical phenomena can be described by the Helmholtz equation. We investigate to what extent the convergence of the shifted Laplacian preconditioner for the Helmholtz equation can be accelerated using deflation with multigrid vectors. We therefore present a unified framework for two published algorithms. The first deflates the preconditioned operator and requires no further preconditioning. The second deflates the original operator and combines deflation and preconditioning in a multiplicative fashion. We pursue two scientific contributions. First we show, using a model problem analysis, that both algorithms cluster the eigenvalues. The new and key insight here is that the near-kernel of the coarse grid operator causes a limited set of eigenvalues to shift away from the center of the cluster with a distance proportional to the wave number. This effect is less pronounced in the first algorithmic variant at the expense of a higher computational cost. In the second contribution we quantify for the first time the large amount of reduction in CPU-time that results from the clustering of eigenvalues and the reduction in iteration count. We report to this end on the findings of an implementation in PETSc on two and three-dimensional problems with constant and variable wave number.

Multigrid Methods for Tensor Structured Markov Chains with Low Rank Approximation

Tensor structured Markov chains are part of stochastic models of many practical applications, e.g., in the description of complex production or telephone networks. The most interesting question in Markov chain models is the determination of the stationary distribution as a description of the long-term behavior of the system. This involves the computation of the eigenvector corresponding to the dominant eigenvalue or equivalently the solution of a singular linear system of equations. Due to the tensor structure of the models the dimension of the operators grows rapidly and a direct solution without exploiting the tensor structure becomes infeasible. Algebraic multigrid methods have proven to be efficient when dealing with Markov chains without using tensor structure. In this work we present an approach to adapt the algebraic multigrid framework to the tensor frame, using not only the tensor structure in matrix-vector multiplications but also tensor structured coarse-grid operators and tensor representation...

Multigrid Methods with Newton-Gauss-Seidel Smoothing and Constraint Preserving Interpolation for Obstacle Problems

AbstractIn this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.

Non-Galerkin Multigrid Based on Sparsified Smoothed Aggregation

Algebraic multigrid (AMG) methods are known to be efficient in solving linear systems arising from the discretization of partial differential equations and other related problems. These methods employ a hierarchy of representations of the problem on successively coarser meshes. The coarse-grid operators are usually defined by (Petrov--)Galerkin coarsening, which is a projection of the original operator using the restriction and prolongation transfer operators. Therefore, these transfer operators determine the sparsity pattern and operator complexity of the multigrid hierarchy. In many scenarios the multigrid operators tend to become much denser as the coarsening progresses. Such behavior is especially problematic in parallel AMG computations, where it imposes an expensive communication overhead. In this work we present a new algebraic technique for controlling the sparsity pattern of the operators in the AMG hierarchy, independently of the choice of the restriction and prolongation. Our method is based on the aggregation multigrid framework, and it “sparsifies” smoothed aggregation operators while preserving their right and left near null-spaces. Numerical experiments for problems of convection-diffusion and diffusion with discontinuous coefficients demonstrate the efficiency and potential of this approach.

Closing the performance gap between an iterative frequency-domain solver and an explicit time-domain scheme for 3D migration on parallel architectures

Three-dimensional reverse-time migration with the constant-density acoustic wave equation requires an efficient numerical scheme for the computation of wavefields. An explicit finite-difference scheme in the time domain is a common choice. However, it requires a significant amount of disk space for the imaging condition. The frequency-domain approach simplifies the correlation of the source and receiver wavefields, but requires the solution of a large sparse linear system of equations. For the latter, we use an iterative Krylov solver based on a shifted Laplace multigrid preconditioner with matrix-dependent prolongation. The question is whether migration in the frequency domain can compete with a time-domain implementation when both are performed on a parallel architecture. Both methods are naturally parallel over shots, but the frequency-domain method is also parallel over frequencies. If we have a sufficiently large number of compute nodes, we can compute the result for each frequency in parallel and the required time is dominated by the number of iterations for the highest frequency. As a parallel architecture, we consider a commodity hardware cluster that consists of multicore central processing units (CPUs), each of them connected to two graphics processing units (GPUs). Here, GPUs are used as accelerators and not as an independent compute node. The parallel implementation of the 3D migration in frequency domain is compared to a time-domain implementation. We optimize the throughput of the latter with dynamic load balancing, asynchronous I/O, and compression of snapshots. Because the frequency-domain solver uses matrix-dependent prolongation, the coarse-grid operators require more storage than available on GPUs for problems of realistic size. Due to data transfer, there is no significant speedup using GPU-accelerators. Therefore, we consider an implementation on CPUs only. Nevertheless, with the parallelization over shots and frequencies, this approach could compete with the time-domain implementation on multiple GPUs.

A Constrained Cauchy-Born Elasticity Accelerated Multigrid Method for Nanoindentation

AbstractWe introduce a new multigrid method to study the lattice statics model arising from nanoindentation. A constrained Cauchy-Born elasticity model is used as the coarse-grid operator. This method accelerates the relaxation process and considerably reduces the computational cost. In particular, it saves quite a bit when dislocations nucleate and move, as demonstrated by the simulation results.

A Multigrid Method for the Helmholtz Equation with Optimized Coarse Grid Corrections

We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let $G_{\rm c}$ denote the number of points per wavelength at the coarse level. If the coarse scale solutions are to approximate the true solutions, then the oscillatory nature of the solutions implies the requirement $G_{\rm c} > 2$. However, in examples the requirement is more like $G_{\rm c} \gtrsim 10$, in a trade-off involving also the amount of damping present and the number of multigrid iterations. We conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators. Standard 5-point finite differences in two dimensions are our first example. A new coarse scale 9-point operator is constructed to match the fine scale phase speeds. We then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator. The required $G_{\rm c}$ is reduced from about 10 to about 3.5, with less damping present so that waves propagate over $>$ 100 wavelengths in the new scheme. Next, we consider extensions of the method to more general cases. In three dimensions, comparable results are obtained with standard 7-point differences and optimized 27-point coarse grid operators, leading to an order of magnitude reduction in the number of unknowns for the coarsest scale linear system. Finally, we show how to include perfectly matched layers at the boundary, using a regular grid finite element method. Matching coarse scale operators can easily be constructed for other discretizations. The method is therefore potentially useful for a large class of discretized high-frequency Helmholtz equations.

Non-Galerkin Coarse Grids for Algebraic Multigrid

Algebraic multigrid (AMG) is a popular and effective solver for systems of linear equations that arise from discretized partial differential equations. While AMG has been effectively implemented on large scale parallel machines, challenges remain, especially when moving to exascale. In particular, stencil sizes (the number of nonzeros in a row) tend to increase further down in the coarse grid hierarchy, and this growth leads to more communication. Thus, as problem size increases and the number of levels in the hierarchy grows, the overall efficiency of the parallel AMG method decreases, sometimes dramatically. This growth in stencil size is due to the standard Galerkin coarse grid operator, $P^T A P$, where $P$ is the prolongation (i.e., interpolation) operator. For example, the coarse grid stencil size for a simple three-dimensional (3D) seven-point finite differencing approximation to diffusion can increase into the thousands on present day machines, causing an associated increase in communication costs...

Algebraic multilevel preconditioning in isogeometric analysis: Construction and numerical studies

We present algebraic multilevel iteration (AMLI) methods for isogeometric discretization of scalar second order elliptic problems. The construction of coarse grid operators and hierarchical complementary operators are given. Moreover, for a uniform mesh on a unit interval, the explicit representation of B-spline basis functions for a fixed mesh size h is given for p=2,3,4 and for C0- and Cp-1-continuity. The presented methods show h- and (almost) p-independent convergence rates. Supporting numerical results for convergence factor and iterations count for AMLI cycles (V-, linear W-, nonlinear W-) are provided. Numerical tests are performed, in two-dimensions on square domain and quarter annulus, and in three-dimensions on quarter thick ring.