Grid Transfer Operators Research Articles

Smoothed aggregation for difficult stretched mesh and coefficient variation problems

AbstractFour adaptations of the smoothed aggregation algebraic multigrid (SA‐AMG) method are proposed with an eye toward improving the convergence and robustness of the solver in situations when the discretization matrix contains many weak connections. These weak connections can cause higher than expected levels of fill‐in within the coarse discretization matrices and can also give rise to suboptimal smoothing within the prolongator smoothing phase. These smoothing drawbacks are due to the relatively small size of some diagonal entries within the filtered matrix that one obtains after dropping the weak connections. The new algorithms consider modifications to the Jacobi‐like step that defines the prolongator smoother, modifications to the filtered matrix, and also direct modifications to the resulting grid transfer operators. Numerical results are given illustrating the potential benefits of the proposed adaptations.

A Symbol-Based Analysis for Multigrid Methods for Block-Circulant and Block-Toeplitz Systems

In the literature, there exist several studies on symbol-based multigrid methods for the solution of linear systems having structured coefficient matrices. In particular, the convergence analysis for such methods has been obtained in an elegant form in the case of Toeplitz matrices generated by a scalar-valued function. In the block-Toeplitz setting, that is, in the case where the matrix entries are small generic matrices instead of scalars, some algorithms have already been proposed regarding specific applications, and a first rigorous convergence analysis has been performed in [M. Donatelli et al., Numer. Linear Algebra Appl., 28 (2021), e2356]. However, with the existent symbol-based theoretical tools, it is still not possible to prove the convergence of many multigrid methods known in the literature. This paper aims to generalize the previous results, giving more general sufficient conditions on the symbol of the grid transfer operators. In particular, we treat matrix-valued trigonometric polynomials which can be nondiagonalizable and singular at all points, and we express the new conditions in terms of the eigenvectors associated with the ill-conditioned subspace. Moreover, we extend the analysis to the V-cycle method, proving a linear convergence rate under stronger conditions, which resemble those given in the scalar case. In order to validate our theoretical findings, we present a classical block structured problem stemming from an FEM approximation of a second order differential problem. We focus on two multigrid strategies that use the geometric and the standard bisection grid transfer operators and prove that both fall into the category of projectors satisfying the proposed conditions. In addition, using a tensor product argument, we provide a strategy to construct efficient V-cycle procedures in the block multilevel setting.

Multigrid methods for block‐Toeplitz linear systems: convergence analysis and applications

AbstractIn the past decades, multigrid methods for linear systems having multilevel Toeplitz coefficient matrices with scalar entries have been widely studied. On the other hand, only few papers have investigated the case of block entries, where the entries are small generic matrices of fixed size instead of scalars. In that case the efforts of the researchers have been mainly devoted to specific applications, focusing on algorithmic proposals, but with marginal theoretical results. In this paper, we propose a general two‐grid convergence analysis, proving an optimal convergence rate independent of the matrix size, in the case of positive definite block‐Toeplitz matrices with generic blocks. In particular, the proof of the approximation property is not a straightforward generalization of the scalar case and, in fact, we have to require a specific commutativity condition on the block symbol of the grid transfer operator. According to the analysis, we define a class of grid transfer operators satisfying the previous theoretical conditions and we propose a strategy to ensure fast multigrid convergence even for more than two grids. Among the numerous applications that lead to the block‐Toeplitz structure, high‐order Lagrangian finite element methods and staggered discontinuous Galerkin methods are considered in the numerical results, confirming the effectiveness of our proposal and the correctness of the proposed theoretical analysis.

Multigrid preconditioners for anisotropic space-fractional diffusion equations

We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space. The use of Crank-Nicolson in time and finite differences in space leads to dense Toeplitz-like linear systems. Multigrid strategies that exploit such structure are particularly effective when the fractional orders are both close to 2. We seek to investigate how structure-based multigrid approaches can be efficiently extended to the case where only one of the two fractional orders is close to 2, i.e., when the fractional equation shows an intrinsic anisotropy. Precisely, we design a multigrid (block-banded–banded-block) preconditioner whose grid transfer operator is obtained with a semi-coarsening technique and that has relaxed Jacobi as smoother. The Jacobi relaxation parameter is estimated by using an automatic symbol-based procedure. A further improvement in the robustness of the proposed multigrid method is attained using the V-cycle with semi-coarsening as smoother inside an outer full-coarsening. Several numerical results confirm that the resulting multigrid preconditioner is computationally effective and outperforms current state of the art techniques.

An Operator Smoothing with ILU(0) for Aggregation-based Algebraic Multigrid

Due to the potential optimal convergence, algebraic multigrid is widely used in solving large sparse linear systems. The aggregation-based version is one of the widely used methods for its cheap cost and easy implementation. But its convergence is often slow compared to other versions. In this paper, a smoothing technique based on incomplete LU factorization without fill-in is presented to the operators on each level. Each operator is approximately factorized, and the derived lower and upper triangular factors are approximately inverted, which are applied to the operator from both sides to improve the diagonal dominance, and then the effectiveness of the smoothing process and the accuracy of the grid transfer operators. The numerical results show that when incorporated into the preconditioned conjugate gradient iteration, the convergence rate is greatly improved, and though the time used for setup is larger, the time used for iteration and the overall time can also be reduced for large-scale systems.

Anisotropic bivariate subdivision with applications to multigrid

In this paper, motivated by applications to multigrid, we present families of anisotropic, bivariate, interpolating and approximating subdivision schemes. We study the minimality and polynomial generation/reproduction properties of both families and Hölder regularity of their prominent representatives. From the symbols of the proposed subdivision schemes, we define bivariate grid transfer operators for anisotropic multigrid methods. We link the generation/reproduction properties of subdivision to the convergence and optimality of the corresponding multigrid methods. We illustrate the performance of our subdivision based grid transfer operators on examples of anisotropic Laplacian and biharmonic problems.

三维薄结构热传导问题分层二次元方程的多水平方法

When the finite element method is applied to analyze the 3D thin-walled structures, some thin hexahedral elements are usually used in order to reduce the number of elements, and the corresponding higher-order elements are preferred since they have some obvious advantages in the calculation accuracy, the anti-distortion degree and so on. However, they have much higher computational complexity than the lower-order (e.g., linear) elements and the coefficient matrix of the linear algebraic equation system is severely ill-conditioned. The convergence of the commonly used solvers will deteriorate with the increasing size of the problem. An efficient and robust multi-level method was presented for the hierarchical quadratic discretizations of 3D thin-walled structures through combination of two special local block Gauss-Seidel smoothers and the DAMG algorithm based on the distance matrix. Since a hierarchical basis is used, those algebraic criteria are not needed to judge the relationships between the unknown variables and the geometric node types, and the grid transfer operators are also trivial. This makes it easy to find the coarse level (linear element) matrix derived directly from the fine level matrix, and thus the overall efficiency is greatly improved. The numerical results verify the efficiency and robustness of the proposed method.

Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations

Fractional diffusion equations (FDEs) are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance. In this paper, we focus on a two-dimensional space-FDE problem discretized by means of a second order finite difference scheme obtained as combination of the Crank–Nicolson scheme and the so-called weighted and shifted Grünwald formula.By fully exploiting the Toeplitz-like structure of the resulting linear system, we provide a detailed spectral analysis of the coefficient matrix at each time step, both in the case of constant and variable diffusion coefficients. Such a spectral analysis has a very crucial role, since it can be used for designing fast and robust iterative solvers. In particular, we employ the obtained spectral information to define a Galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped-Jacobi as smoother, and to prove the linear convergence rate of the corresponding two-grid method. The theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the V-cycle method and the geometric multigrid. On this basis, we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for Krylov methods. Several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate.

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

We introduce a family of implementations of low-order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers, and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based on the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free. Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realize full approximation storage (FAS) within the additive environment where, amortized, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realize a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as an enabler for full multigrid (FMG) cycling—the grid unfolds throughout the computation—allows us to reduce the cost per unknown. The present work primary contributes toward software realization and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR, and vectorization-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.

Multigrid methods: Grid transfer operators and subdivision schemes

The convergence rate of a multigrid method depends on the properties of the smoother and the so-called grid transfer operator. In this paper we define and analyze new grid transfer operators with a generic cutting size which are applicable for high order problems. We enlarge the class of available geometric grid transfer operators by relating the symbol analysis of the coarse grid correction with the approximation properties of univariate subdivision schemes. We show that the polynomial generation property and stability of a subdivision scheme are crucial for convergence and optimality of the corresponding multigrid method. We construct a new class of grid transfer operators from univariate primal binary and ternary pseudo-spline symbols. Our numerical results illustrate the behavior of the new grid transfer operators and provide promising preliminary results for the bivariate case.

Parallel Multiscale Linear Solver for Highly Detailed Reservoir Models

Summary To realize the potential of the latest high-performance computing (HPC) architectures for reservoir simulation, scalable linear solvers are necessary. We describe a parallel algebraic multiscale solver (AMS) for the pressure equation of heterogeneous reservoir models. AMS is a two-level algorithm that uses domain decomposition with a localization assumption. In AMS, basis functions, which are local (subdomain) solutions computed during the setup phase, are used to construct the coarse-scale system and grid-transfer operators between the fine and coarse levels. The solution phase is composed of two stages: global and local. The global stage involves solving the coarse-scale system and interpolating the solution to the fine grid. The local stage involves application of a smoother on the fine-scale approximation. The design and implementation of a scalable AMS on multicore and many-core architectures, including the decomposition, memory allocation, data flow, and compute kernels, are described in detail. These adaptations are necessary to obtain good scalability on state-of-the-art HPC systems. The specific methods and parameters, such as the coarsening ratio (Cr), basis-function solver, and relaxation scheme, have significant effects on the asymptotic convergence rate and parallel computational efficiency. The balance between convergence rate and parallel efficiency as a function of Cr and the local stage parameters is analyzed in detail. The performance of AMS is demonstrated using heterogeneous 3D reservoir models, including geostatistically generated fields and models derived from SPE10 (Christie and Blunt 2001). The problems range in size from several million to 128 million cells. AMS shows excellent behavior for handling fixed-size problems as a function of the number of cores (so-called strong scaling). Specifically, for a 128-million-cell problem, a ninefold speedup is obtained on a single-node 12-core shared-memory architecture (dual-socket multicore Intel Xeon E5-2620-v2), and more than 12-fold on a single-node 20-core shared-memory architecture (dual-socket multicore Intel Xeon E5-2690-v2). These are encouraging results given the limited memory bandwidth that cores can share within a single node, which tends to be the major bottleneck for truly scalable solvers. We also compare the robustness and performance of our method with the parallel system algebraic mutligrid (SAMG) solver (Stüben 2012) from Fraunhofer SCAI.

Energy-efficient multigrid smoothers and grid transfer operators on multi-core and GPU clusters

We investigate time and energy to solution for the CPU- and GPU-based execution of the compute intensive smoother and grid transfer operators in a geometric multigrid linear solver. We use a hybrid parallel implementation for both shared and distributed memory multi-core host systems comprising CUDA-capable devices. Our numerical experiments are designed to assess the effect of combining an MPI-parallel multigrid framework with OpenMP host threads or CUDA accelerators instead of MPI-only CPU computations for various parallel setups. We present runtime and energy measurements from a quad-CPU test system equipped with two GPUs. We find that using an accelerated asynchronous smoother can yield substantial savings of time and energy to solution over using a host-only Jacobi smoother in small and medium sized host systems with one or two multi-core CPUs. The acceleration of the grid transfer operators also yields a benefit, yet smaller than the benefit from the smoother. For large host systems a hybrid MPI-OpenMP parallelization turns out to be most beneficial with respect to energy consumption, although it is not the fastest option.

An efficient multigrid algorithm for heterogeneous acoustic media sign‐indefinite high‐order FEM models

SummaryLarge‐scale scientific computing models are needed for the simulation of wave propagation especially for multiple frequency and high‐frequency models in complex heterogeneous media. Multigrid methods provide efficient iterative solvers for many large sign‐definite systems of equations resulting from physical models. Time‐harmonic wave propagation models lead to sign‐indefinite systems with eigenvalues in the left half of the complex plane. Thus standard multigrid approaches applied in conjunction with a low‐order finite difference or finite element method are not sufficient. In this work, we describe a high‐order finite element method model for multiple (low to high) frequency time‐harmonic acoustic wave propagation on general curved, non‐convex, and non‐smooth domains with heterogeneous media using a multigrid approximation of the shifted Laplacian operator as a preconditioner. We implement the model using an efficient geometric multigrid approach with parallel grid transfer operator calculations to simulate the model using the BiCGStab iterative solver. We demonstrate the efficiency and parallel performance of the computational model with multiple low (5 wavelength) to high‐frequency (100 wavelength) input incident waves. Copyright © 2016 John Wiley & Sons, Ltd.

An iterative multigrid regularization method for Toeplitz discrete ill-posed problems

Iterative regularization multigrid methods have been successful applied to signal/image deblurring problems. When zero-Dirichlet boundary conditions are im- posed the deblurring matrix has a Toeplitz structure and it is potentially full. A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations. The smoother has to be an iterative regularization method. The grid transfer operator should preserve the regularization property of the smoother. This paper improves the iterative multigrid method proposed in (11) introducing a wavelet soft-thresholding denoising post-smoother. Such post- smoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects. The resulting iterative multigrid regularization method stabilizes the iterations so that and imprecise (over) estimate of the stopping iteration does not have a deleterious effect on the computed solution. Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.

Generalized grid transfer operators for multigrid methods applied on Toeplitz matrices

In this paper we discuss classical sufficient conditions to be satisfied from the grid transfer operators in order to obtain optimal two-grid and V-cycle multigrid methods utilizing the theory for Toeplitz matrices. We derive relaxed conditions that allow the construction of special grid transfer operators that are computationally less expensive while preserving optimality. This is particularly useful when the generating symbol of the system matrix has a zero of higher order, like in the case of higher order PDEs. These newly derived conditions allow the use of rank deficient grid transfer operators. In this case the use of a pre-relaxation iteration that is lacking the smoothing property is proposed. Combining these pre-relaxations with the new rank deficient grid transfer operators yields a substantial reduction of the convergence rate and of the computational cost at each iteration compared with the classical choice for Toeplitz matrices. The proposed strategy, i.e. a rank deficient grid transfer operator plus a specific pre-relaxation, is applied to linear systems whose system matrix is a Toeplitz matrix where the generating symbol is a high-order polynomial. The necessity of using high-order polynomials as generating symbols for the grid transfer operators usually destroys the Toeplitz structure on the coarser levels. Therefore, we discuss some effective and computational cheap coarsening strategies found in the literature. In particular, we present numerical results showing near-optimal behavior while keeping the Toeplitz structure on the coarser levels.

A formalism for designing robust multigrid methods for Helmholtz‐like problems

This paper discusses templates for designing efficient and robust multigrid (MG) methods. All MG mechanisms are described by means of matrix functions (block symbols), which capture the properties of the full matrices. We propose a new strategy for deriving the block symbol by means of sparse matrices and of its representation in terms of the usual tensor product. Emphasis is laid on the indefinite Helmholtz equation, which is of great practical interest in real‐life applications. We take advantage of previously known patterns (the subblock smoother as preconditioner and the full projection as grid transfer operator) for designing MG methods that work as direct solvers. The corresponding numerical results are presented and the superiority of the combined use of a subblock smoother and a full projection is demonstrated. The condition number and positive definiteness of the subblock smoother especially for Helmholtz‐like problems are analyzed, and the corresponding requirements and bounds are given. Copyright © 2013 John Wiley & Sons, Ltd.

Grid transfer operators for multigrid methods

AbstractLocal Fourier analysis (LFA) is a classical tool for proving convergence theorems for multigrid methods (MGMs). Analogously, the symbols of the involved matrices are studied to prove convergence results for MGMs for Toeplitz matrices. We show that in the case of elliptic partial differential equations (PDEs) with constant coefficients, the two different approaches lead to an equivalent optimality condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA which allows to deal not only with differential problems but also, e.g., with integral problems. A class of grid transfer operators related to the B‐spline's refinement equation is discussed as well. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

New robust nonconforming finite elements of higher order

We show that existing quadrilateral nonconforming finite elements of higher order exhibit a reduction in the order of approximation if the sequence of meshes is still shape-regular but consists no longer of asymptotically affine equivalent mesh cells. We study second order nonconforming finite elements as members of a new family of higher order approaches which prevent this order reduction. We present a new approach based on the enrichment of the original polynomial space on the reference element by means of nonconforming cell bubble functions which can be removed at the end by static condensation. Optimal estimates of the approximation and consistency error are shown in the case of a Poisson problem which imply an optimal order of the discretization error. Moreover, we discuss the known nonparametric approach to prevent the order reduction in the case of higher order elements, where the basis functions are defined as polynomials on the original mesh cell. Regarding the efficient treatment of the resulting linear discrete systems, we analyze numerically the convergence of the corresponding geometrical multigrid solvers which are based on the canonical full order grid transfer operators. Based on several benchmark configurations, for scalar Poisson problems as well as for the incompressible Navier–Stokes equations (representing the desired application field of these nonconforming finite elements), we demonstrate the high numerical accuracy, flexibility and efficiency of the discussed new approaches which have been successfully implemented in the FeatFlow software ( www.featflow.de). The presented results show that the proposed FEM-multigrid combinations (together with discontinuous pressure approximations) appear to be very advantageous candidates for efficient simulation tools, particularly for incompressible flow problems.

A General Interpolation Strategy for Algebraic Multigrid Using Energy Minimization

Algebraic multigrid methods solve sparse linear systems $Ax=b$ by automatic construction of a multilevel hierarchy. This hierarchy is defined by grid transfer operators that must accurately capture algebraically smooth error relative to the relaxation method. We propose a methodology to improve grid transfers through energy minimization. The proposed strategy is applicable to Hermitian, non-Hermitian, definite, and indefinite problems. Each column of the grid transfer operator P is minimized in an energy-based norm while enforcing two types of constraints: a defined sparsity pattern and preservation of specified modes in the range of P. A Krylov-based strategy is used to minimize energy, which is equivalent to solving $A P_j = \boldsymbol{0}$ for each column j of P, with the constraints ensuring a nontrivial solution. For the Hermitian positive definite case, a conjugate gradient (CG-)based method is utilized to construct grid transfers, while methods based on generalized minimum residual (GMRES) and CG on the normal equations (CGNR) are explored for the general case. The approach is flexible, allowing for arbitrary coarsenings, unrestricted sparsity patterns, straightforward long-distance interpolation, and general use of constraints, either user-defined or auto-generated. We conclude with numerical evidence in support of the proposed framework.

An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices

AbstractLocal Fourier analysis (LFA) is a classical tool for proving convergence theorems for multigrid methods (MGMs). In particular, we are interested in optimal convergence, i.e. convergence rates that are independent of the problem size. For elliptic partial differential equations (PDEs), a well‐known optimality result requires that the sum of the orders of the grid transfer operators is not lower than the order of the PDE approximated. Analogously, when dealing with MGMs for Toeplitz matrices, a well‐known optimality condition concerns the position and the order of the zeros of the symbols of the grid transfer operators. In this work we show that in the case of elliptic PDEs with constant coefficients, the two different approaches lead to an equivalent condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA, which allows to deal not only with differential problems but also for instance with integral problems. The equivalence of the two approaches gives the possibility of using grid transfer operators with different orders also for MGMs for Toeplitz matrices. We give also a class of grid transfer operators related to the B‐spline's refinement equation and study their geometric properties. Numerical experiments confirm the correctness of the proposed analysis. Copyright © 2010 John Wiley & Sons, Ltd.