Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

Symmetric collocation methods with RBFs allow approximation of the solution of a partial differential equation, even if the right‐hand side is only known at scattered data points, without needing to generate a grid. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. This cost can be alleviated somewhat by considering compactly supported RBFs and a multiscale technique. But the condition number and sparsity will still deteriorate with the number of data points. Therefore, we study certain block diagonal and triangular preconditioners. We investigate ideal preconditioners and determine the spectra of the preconditioned matrices before proposing more practical preconditioners based on a restricted additive Schwarz method with coarse grid correction. Numerical results verify the effectiveness of the preconditioners. Copyright © 2015 John Wiley & Sons, Ltd.     

Algebraic multilevel preconditioning of finite element matrices using local Schur complements

We consider an algebraic multilevel preconditioning technique for SPD matrices arising from finite element discretization of elliptic PDEs. In particular, we address the case of non‐M matrices. The method is based on element agglomeration and assumes access to the individual element matrices. The left upper block of the considered multiplicative two‐level preconditioner is approximated using incomplete factorization techniques. The coarse‐grid element matrices are simply Schur complements computed from local neighbourhood matrices, i.e. small collections of element matrices. Assembling these local Schur complements results in a global Schur complement approximation that can be analysed by regarding (local) macro elements. These components, when combined in the framework of an algebraic multilevel iteration, yield a robust and efficient linear solver. The presented numerical experiments include also the Lamé differential equation for the displacements in the two‐dimensional plane‐stress elasticity problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Multilevel approaches for FSAI preconditioning

Factorized sparse approximate inverse (FSAI) preconditioners are robust algorithms for symmetric positive matrices, which are particularly attractive in a parallel computational environment because of their inherent and almost perfect scalability. Their parallel degree is even redundant with respect to the actual capabilities of the current computational architectures. In this work, we present two new approaches for FSAI preconditioners with the aim of improving the algorithm effectiveness by adding some sequentiality to the native formulation. The first one, denoted as block tridiagonal FSAI, is based on a block tridiagonal factorization strategy, whereas the second one, domain decomposition FSAI, is built by reordering the matrix graph according to a multilevel k‐way partitioning method followed by a bandwidth minimization algorithm. We test these preconditioners by solving a set of symmetric positive definite problems arising from different engineering applications. The results are evaluated in terms of performance, scalability, and robustness, showing that both strategies lead to faster convergent schemes regarding the number of iterations and total computational time in comparison with the native FSAI with no significant loss in the algorithmic parallel degree.     

A general approach to analyse preconditioners for two‐by‐two block matrices

Two‐by‐two block matrices arise in various applications, such as in domain decomposition methods or when solving boundary value problems discretised by finite elements from the separation of the node set of the mesh into ‘fine’ and ‘coarse’ nodes. Matrices with such a structure, in saddle point form arise also in mixed variable finite element methods and in constrained optimisation problems. A general algebraic approach to construct, analyse and control the accuracy of preconditioners for matrices in two‐by‐two block form is presented. This includes both symmetric and nonsymmetric matrices, as well as indefinite matrices. The action of the preconditioners can involve element‐by‐element approximations and/or geometric or algebraic multigrid/multilevel methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Local preconditioners for two‐level non‐overlapping domain decomposition methods

We consider additive two‐level preconditioners, with a local and a global component, for the Schur complement system arising in non‐overlapping domain decomposition methods. We propose two new parallelizable local preconditioners. The first one is a computationally cheap but numerically relevant alternative to the classical block Jacobi preconditioner. The second one exploits all the information from the local Schur complement matrices and demonstrates an attractive numerical behaviour on heterogeneous and anisotropic problems. We also propose two implementations based on approximate Schur complement matrices that are cheaper alternatives to construct the given preconditioners but that preserve their good numerical behaviour. Through extensive computational experiments we study the numerical scalability and the robustness of the proposed preconditioners and compare their numerical performance with well‐known robust preconditioners such as BPS and the balancing Neumann–Neumann method. Finally, we describe a parallel implementation on distributed memory computers of some of the proposed techniques and report parallel performances. Copyright © 2001 John Wiley & Sons, Ltd.     

Stabilization of algebraic multilevel iteration methods; additive methods

There exist two main versions of preconditioners of algebraic multilevel type, the additive and the multiplicative methods. They correspond to preconditioners in block diagonal and block matrix factorized form, respectively. Both can be defined and analysed as recursive two-by-two block methods. Although the analytical framework for such methods is simple, for many finite element approximations it still permits the derivation of the strongest results, such as optimal, or nearly optimal, rate of convergence and optimal, or nearly optimal order of computational complexity, when proper recursive global orderings of node points have been used or when they are applied for hierarchical basis function finite element methods for elliptic self-adjoint equations and stabilized in a certain way. This holds for general elliptic problems of second order, independent of the regularity of the problem, including independence of discontinuities of coefficients between elements and of anisotropy. Important ingredients in the methods are a proper balance of the size of the coarse mesh to the finest mesh and a proper solver on the coarse mesh. This paper presents in a survey form the basic results of such methods and considers in particular additive methods. This method has excellent parallelization properties. This revised version was published online in June 2006 with corrections to the Cover Date.     

A comparison of overlapping Schwarz methods and block preconditioners for saddle point problems

Three domain decomposition methods for saddle point problems are introduced and compared. The first two are block‐diagonal and block‐triangular preconditioners with diagonal blocks approximated by an overlapping Schwarz technique with positive definite local and coarse problems. The third is an overlapping Schwarz preconditioner based on indefinite local and coarse problems. Numerical experiments show that while all three methods are numerically scalable, the last method is almost always the most efficient. Copyright © 2000 John Wiley & Sons, Ltd.     

A multilevel block incomplete cholesky preconditioner for solving normal equations in linear least squares problems

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.     

Algebraic multilevel iteration method for Stieltjes matrices

The numerical solution of elliptic selfadjoint second-order boundary value problems leads to a class of linear systems of equations with symmetric, positive definite, large and sparse matrices which can be solved iteratively using a preconditioned version of some algorithm. Such differential equations originate from various applications such as heat conducting and electromagnetics. Systems of equations of similar type can also arise in the finite element analysis of structures. We discuss a recursive method constructing preconditioners to a symmetric, positive definite matrix. An algebraic multilevel technique based on partitioning of the matrix in two by two matrix block form, approximating some of these by other matrices with more simple sparsity structure and using the corresponding Schur complement as a matrix on the lower level, is considered. The quality of the preconditioners is improved by special matrix polynomials which recursively connect the preconditioners on every two adjoining levels. Upper and lower bounds for the degree of the polynomials are derived as conditions for a computational complexity of optimal order for each level and for an optimal rate of convergence, respectively. The method is an extended and more accurate algebraic formulation of a method for nine-point and mixed five- and nine-point difference matrices, presented in some previous papers.     

Local refinement techniques for elliptic problems on cell-centered grids

Summary Algebraic multilevel analogues of the BEPS preconditioner designed for solving discrete elliptic problems on grids with local refinement are formulated, and bounds on their relative condition numbers, with respect to the composite-grid matrix, are derived. TheV-cycle and, more generally,v-foldV-cycle multilevel BEPS preconditioners are presented and studied. It is proved that for 2-D problems theV-cycle multilevel BEPS is almost optimal, whereas thev-foldV-cycle algebraic multilevel BEPS is optimal under a mild restriction on the composite cell-centered grid. For thev-fold multilevel BEPS, the variational relation between the finite difference matrix and the corresponding matrix on the next-coarser level is not necessarily required. Since they are purely algebraically derived, thev-fold (v>1) multilevel BEPS preconditioners perform without any restrictionson the shape of subregions, unless the refinement is too fast. For theV-cycle BEPS preconditioner (2-D problem), a variational relation between the matrices on two consecutive grids is required, but there is no restriction on the method of refinement on the shape, or on the size of the subdomains.     

Domain decomposition for multiscale PDEs

We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises often in practice, for example in the computation of flows in heterogeneous porous media, in both the deterministic and (Monte–Carlo simulated) stochastic cases. We consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid. We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions, the preconditioner can still be robust even for large coefficient variation inside domains, when the classical method fails to be robust. In particular our estimates prove very precisely the previously made empirical observation that the use of low-energy coarse spaces can lead to robust preconditioners. We go on to consider coarse spaces constructed from multiscale finite elements and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary (i.e., two-scale) media model problems. Moreover numerical experiments show that the new preconditioner has greatly improved performance over standard preconditioners even in the random coefficient case. We show also how the analysis extends in a straightforward way to multiplicative versions of the Schwarz method. We would like to thank Bill McLean for very useful discussions concerning this work. We would also like to thank Maksymilian Dryja for helping us to improve the result in Theorem 4.3.     

Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non‐conforming FEM systems

Preconditioners based on various multilevel extensions of two‐level finite element methods (FEM) lead to iterative methods which have an optimal order computational complexity with respect to the size of the system. Such methods were first presented in Axelsson and Padiy (SIAM. J. Sci. Stat. Comp. 1990; 20 :1807) and Axelsson and Vassilevski (Numer. Math. 1989; 56 :157), and are based on (recursive) two‐level splittings of the finite element space. The key role in the derivation of optimal convergence rate estimates is played by the constant γ in the so‐called Cauchy–Bunyakowski–Schwarz (CBS) inequality, associated with the angle between the two subspaces of the splitting. It turns out that only existence of uniform estimates for this constant is not enough but accurate quantitative bounds for γ have to be found as well. More precisely, the value of the upper bound for γ∈(0,1) is part of the construction of various multilevel extensions of the related two‐level methods. In this paper, an algebraic two‐level preconditioning algorithm for second‐order elliptic boundary value problems is constructed, where the discretization is done using Crouzeix–Raviart non‐conforming linear finite elements on triangles. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinements are not nested. To handle this, a proper two‐level basis is considered, which enables us to fit the general framework for the construction of two‐level preconditioners for conforming finite elements and to generalize the method to the multilevel case. The major contribution of this paper is the derived estimates of the related constant γ in the strengthened CBS inequality. These estimates are uniform with respect to both coefficient and mesh anisotropy. To our knowledge, the results presented in the paper are the first such estimates for non‐conforming FEM systems. Copyright © 2004 John Wiley & Sons, Ltd.     

A Two‐Parameter Stabilized Finite Element Method for Incompressible Flows

Based on two‐grid discretizations, a two‐parameter stabilized finite element method for the steady incompressible Navier–Stokes equations at high Reynolds numbers is presented and studied. In this method, a stabilized Navier–Stokes problem is first solved on a coarse grid, and then a correction is calculated on a fine grid by solving a stabilized linear problem. The stabilization term for the nonlinear Navier–Stokes equations on the coarse grid is based on an elliptic projection, which projects higher‐order finite element interpolants of the velocity into a lower‐order finite element interpolation space. For the linear problem on the fine grid, either the same stabilization approach (with a different stabilization parameter) as that for the coarse grid problem or a completely different stabilization approach could be employed. Error bounds for the discrete solutions are estimated. Algorithmic parameter scalings of the method are also derived. The theoretical results show that, with suitable scalings of the algorithmic parameters, this method can yield an optimal convergence rate. Numerical results are provided to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 425–444, 2017     

A comparison of abstract versions of deflation,balancing and additive coarse grid correction preconditioners

In this paper we consider various preconditioners for the conjugate gradient (CG) method to solve large linear systems of equations with symmetric positive definite system matrix. We continue the comparison between abstract versions of the deflation, balancing and additive coarse grid correction preconditioning techniques started in (SIAM J. Numer. Anal. 2004; 42 :1631–1647; SIAM J. Sci. Comput. 2006; 27 :1742–1759). There the deflation method is compared with the abstract additive coarse grid correction preconditioner and the abstract balancing preconditioner. Here, we close the triangle between these three methods. First of all, we show that a theoretical comparison of the condition numbers of the abstract additive coarse grid correction and the condition number of the system preconditioned by the abstract balancing preconditioner is not possible. We present a counter example, for which the condition number of the abstract additive coarse grid correction preconditioned system is below the condition number of the system preconditioned with the abstract balancing preconditioner. However, if the CG method is preconditioned by the abstract balancing preconditioner and is started with a special starting vector, the asymptotic convergence behavior of the CG method can be described by the so‐called effective condition number with respect to the starting vector. We prove that this effective condition number of the system preconditioned by the abstract balancing preconditioner is less than or equal to the condition number of the system preconditioned by the abstract additive coarse grid correction method. We also provide a short proof of the relationship between the effective condition number and the convergence of CG. Moreover, we compare the A‐norm of the errors of the iterates given by the different preconditioners and establish the orthogonal invariants of all three types of preconditioners. Copyright © 2008 John Wiley & Sons, Ltd.     

Commuting projections on graphs

Motivated by the increasing importance of large‐scale networks typically modeled by graphs, this paper is concerned with the development of mathematical tools for solving problems associated with the popular graph Laplacian. We exploit its mixed formulation based on its natural factorization as product of two operators. The goal is to construct a coarse version of the mixed graph Laplacian operator with the purpose to construct two‐level, and by recursion, a multilevel hierarchy of graphs and associated operators. In many situations in practice, having a coarse (i.e., reduced dimension) model that maintains some inherent features of the original large‐scale graph and respective graph Laplacian offers potential to develop efficient algorithms to analyze the underlined network modeled by this large‐scale graph. One possible application of such a hierarchy is to develop multilevel methods that have the potential to be of optimal complexity. In this paper, we consider general (connected) graphs and function spaces defined on its edges and its vertices. These two spaces are related by a discrete gradient operator, ‘Grad’ and its adjoint, ‘ ? Div’, referred to as (negative) discrete divergence. We also consider a coarse graph obtained by aggregation of vertices of the original one. Then, a coarse vertex space is identified with the subspace of piecewise constant functions over the aggregates. We consider the ?₂‐projection Q_H onto the space of these piecewise constants. In the present paper, our main result is the construction of a projection π_H from the original edge‐space onto a properly constructed coarse edge‐space associated with the edges of the coarse graph. The projections π_H and Q_H commute with the discrete divergence operator, that is, we have Div π_H = Q_H div. The respective pair of coarse edge‐space and coarse vertex‐space offer the potential to construct two‐level, and by recursion, multilevel methods for the mixed formulation of the graph Laplacian, which utilizes the discrete divergence operator. The performance of one two‐level method with overlapping Schwarz smoothing and correction based on the constructed coarse spaces for solving such mixed graph Laplacian systems is illustrated on a number of graph examples. Copyright © 2013 John Wiley & Sons, Ltd.     

An efficient algebraic multigrid method for quadratic discretizations of linear elasticity problems on some typical anisotropic meshes in three dimensions

The quality of the mesh used in the finite element discretizations will affect the efficiency of solving the discreted linear systems. The usual algebraic solvers except multigrid method do not consider the effect of the grid geometry and the mesh quality on their convergence rates. In this paper, we consider the hierarchical quadratic discretizations of three‐dimensional linear elasticity problems on some anisotropic hexahedral meshes and present a new two‐level method, which is weakly independent of the size of the resulting problems by using a special local block Gauss–Seidel smoother, that is LBGS_v iteration when used for vertex nodes or LBGS_m iteration for midside nodes. Moreover, we obtain the efficient algebraic multigrid (AMG) methods by applying DAMG (AMG based on distance matrix) or DAMG‐PCG (PCG with DAMG as a preconditioner) to the solution of the coarse level equation. The resulting AMG methods are then applied to a practical example as a long beam. The numerical results verify the efficiency and robustness of the proposed AMG algorithms. Copyright © 2015 John Wiley & Sons, Ltd.     

CLC in AMG solvers for saddle‐point problems

An algebraic multigrid (AMG) solution method for saddle‐point problems is described. The indefinite nature of the saddle‐point matrix makes it unsuitable for the simple smoothing methods normally used in AMG. Moreover, even if presented in a stabilised form, as is done here, poorly conditioned matrices will be generated when constructing the coarse‐grid approximation. This is because, with each successive coarsening step, the off‐diagonal matrix blocks (of interfield coupling) reduce in size more slowly than the diagonal blocks (of intrafield coupling). Stabilised smoothing operators are therefore considered. The first is based on an incomplete decomposition of the complete system matrix into the product of lower‐triangular ( L ) and upper‐triangular ( U ) matrices, an ILU factorisation. The second is based on an exact block decomposition of an incomplete (simplified) system matrix into lower and upper block‐triangular matrices, a BILU factorisation. However, the degree of stabilisation thus established in the smoothing operators does not guarantee an efficient smoothing at all grid levels. There can still be inefficiency on the least‐stable coarser grids. The breakdown in efficiency begins at a grid level where the ratio of the inter‐ to intrafield coupling strengths exceeds a critical ratio. Provision is thus made for a further conditioning of coarse‐grid operators, a coarse‐level conditioning (CLC). This is another block‐LU factorisation that is only applied at and beyond the critical grid level. It is not applied directly to the operator for any chosen level but to its fine‐grid progenitor. Thus, while ILU and BILU use postcoarsening‐step factorisations, CLC uses precoarsening‐step factorisations. With CLC so deployed, ILU and BILU become efficient at all grid levels, resulting in an AMG convergence that is independent of the total number of grids.     

An order optimal solver for the discretized bidomain equations

The electrical activity in the heart is governed by the bidomain equations. In this paper, we analyse an order optimal method for the algebraic equations arising from the discretization of this model. Our scheme is defined in terms of block Jacobi or block symmetric Gauss–Seidel preconditioners. Furthermore, each block in these methods is based on standard preconditioners for scalar elliptic or parabolic partial differential equations (PDEs). Such preconditioners can be realized in terms of multigrid or domain decomposition schemes, and are thus readily available by applying ‘off‐the‐shelves’ software. Finally, our theoretical findings are illuminated by a series of numerical experiments. Copyright © 2006 John Wiley & Sons, Ltd.     

Two‐level preconditioners for serendipity finite element matrices

In this paper an approach to construct algebraic multilevel preconditioners for serendipity finite element matrices is presented. Two‐level preconditioners constructed in the paper allow to obtain multilevel preconditioners in serendipity case using multilevel preconditioners for linear finite element matrices. Copyright © 2006 John Wiley & Sons, Ltd.     

Robust iterative methods for elliptic problems with highly varying coefficients in thin substructures

Summary. In this paper we introduce a class of robust multilevel interface solvers for two-dimensional finite element discrete elliptic problems with highly varying coefficients corresponding to geometric decompositions by a tensor product of strongly non-uniform meshes. The global iterations convergence rate is shown to be of the order with respect to the number of degrees of freedom on the single subdomain boundaries, uniformly upon the coarse and fine mesh sizes, jumps in the coefficients and aspect ratios of substructures. As the first approach, we adapt the frequency filtering techniques [28] to construct robust smoothers on the highly non-uniform coarse grid. As an alternative, a multilevel averaging procedure for successive coarse grid correction is proposed and analyzed. The resultant multilevel coarse grid preconditioner is shown to have (in a two level case) the condition number independent of the coarse mesh grading and jumps in the coefficients related to the coarsest refinement level. The proposed technique exhibited high serial and parallel performance in the skin diffusion processes modelling [20] where the high dimensional coarse mesh problem inherits a strong geometrical and coefficients anisotropy. The approach may be also applied to magnetostatics problems as well as in some composite materials simulation. Received December 27, 1994