A UNIFIED FRAMEWORK FOR THE CONSTRUCTION OF VARIOUS ALGEBRAIC MULTILEVEL PRECONDITIONING METHODS

1.IntroductionThediscretizationofmanysecondorderselfadjointellipticboundaryvalueproblemsbythefiniteelementmethodleadstolargesparsesystemsoflinearequationswithsymmetricpositivedefinite(SPD)coefficientmatrices.Fortheselinearsystems,algebraicmultilevelp...     

HYBRID ALGEBRAIC MULTILEVEL PRECONDITIONING METHODS

ＨＹＢＲＩＤＡＬＧＥＢＲＡＩＣＭＵＬＴＩＬＥＶＥＬＰＲＥＣＯＮＤＩＴＩＯＮＩＮＧＭＥＴＨＯＤＳ￥ＢａｉＺｈｏｎｇｚｈｉ（白中治）（ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，复旦大学，邮编：２００４３３）Ａｂｓｔｒａｃｔ：Ａｃｌａｓｓｏｆｈｙｂｒｉｄａｌｇｅｂｒ...     

A CLASS OF NEW PARALLEL HYBRID ALGEBRAIC MULTILEVEL ITERATIONS

1. IntroductionConsider the large sparse system of linear equationsAx = b, (1.1)where, for a fixed positive integer cr, A e L(R") is a symmetric positive definite (SPD) matrir,having the bloCked formx,b E R" are the uDknwn and the known vectors, respectively, having the correspondingblocked formsni(ni S n, i = 1, 2,', a) are a given positthe integers, satisfying Z ni = n. This systemi= 1of linear equations often arises in sultable finite element discretizations of many secondorderseifad…     

A survey of multilevel preconditioned iterative methods

We survey multilevel iterative methods applied for solving large sparse systems with matrices, which depend on a level parameter, such as arise by the discretization of boundary value problems for partial differential equations when successive refinements of an initial discretization mesh is used to construct a sequence of nested difference or finite element meshes.We discuss various two-level (two-grid) preconditioning techniques, including some for nonsymmetric problems. The generalization of these techniques to the multilevel case is a nontrivial task. We emphasize several ways this can be done including classical multigrid methods and a recently proposed algebraic multilevel preconditioning method. Conditions for which the methods have an optimal order of computational complexity are presented.On leave from the Institute of Mathematics, and Center for Informatics and Computer Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria. The research of the second author reported here was partly supported by the Stichting Mathematisch Centrum, Amsterdam.     

Spectral optimization of explicit iterative methods. I

Methods of constructing preconditioning of explicit iterative methods of solving systems of linear, algebraic equations with sparse matrices are considered in the work. The techniques considered can, first of all, be realized within the framework of the simplest data structures; secondly, the graph structures of the corresponding algorithms are well adapted to realization on parallel computers; thirdly, in conjunction with modifications of Chebyshev methods they make it possible to construct rather effective computational algorithms. Experimental data are presented which demonstrate the effect of the proposed techniques of preconditioning of the distribution of the eigenvalues of matrices of systems arising in discretization of two-dimensional elliptic boundary-value problems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 51–60, 1984.     

Scalable solver software

Towards Optimal Petascale Simulations (TOPS) is a scalable solver software project based on domain decomposed parallelization to research, implement, and support in collaborations with users an open-source package for large-scale discretized PDE problems. Optimal complexity methods, such as multigrid/multilevel preconditioners, keep the time spent in dominant algebraic kernels close to linear in discrete problem size as the applications scale on massively parallel computers. Krylov accelerators and Jacobian-free variants of Newton's method, as appropriate, are wrapped around the multilevel methods to deliver robustness in multirate, multiscale coupled systems, which are solved either implicitly or in more traditional forms of operator splitting. The TOPS software framework is being extended beyond direct computational simulation to computational optimization, including design, control, and inverse problems. We outline and illustrate the philosophy of TOPS. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Application of auxiliary space preconditioning in field-scale reservoir simulation

We study a class of preconditioners to solve large-scale linear systems arising from fully implicit reservoir simulation.These methods are discussed in the framework of the auxiliary space preconditioning method for generality.Unlike in the case of classical algebraic preconditioning methods,we take several analytical and physical considerations into account.In addition,we choose appropriate auxiliary problems to design the robust solvers herein.More importantly,our methods are user-friendly and general enough to be easily ported to existing petroleum reservoir simulators.We test the efciency and robustness of the proposed method by applying them to a couple of benchmark problems and real-world reservoir problems.The numerical results show that our methods are both efcient and robust for large reservoir models.     

Multilevel Preconditioners for Mixed Methods for Second Order Elliptic Problems

A new approach for constructing algebraic multilevel preconditioners for mixed finite element methods for second order elliptic problems with tensor coefficients on general geometry is proposed. The linear system arising from the mixed methods is first algebraically condensed to a symmetric, positive definite system for Lagrange multipliers, which corresponds to a linear system generated by standard nonconforming finite element methods. Algebraic multilevel preconditioners for this system are then constructed based on a triangulation of the domain into tetrahedral substructures. Explicit estimates of condition numbers and simple computational schemes are established for the constructed preconditioners. Finally, numerical results for the mixed finite element methods are presented to illustrate the present theory.     

Adaptive Numerical Treatment of Elliptic Systems on Manifolds

Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2- and 3-manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2- and 3-manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unity-based method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4-component covariant elliptic system on a Riemannian 3-manifold which arises in general relativity. A number of operator properties and solvability results recently established in [55] are first summarized, making possible two quasi-optimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented; more detailed examples using MC for this application may be found in [26].     

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 preconditioning

Summary This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the Bramble-Pasciak-Xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shift-invariant spaces, in particular, covering those induced by various types of wavelets.The work of this author was partially supported by the Air Force Office of Scientific Research (Contract No. 89/0455) and by the Office of Naval Research (Contract No. N00014/90/1343) during her stay at the Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.     

CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems

The goal of this work is to derive and justify a multilevel preconditioner of optimal arithmetic complexity for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. Our approach is based on the following simple idea given in [R.D. Lazarov, P.S. Vassilevski, L.T. Zikatanov, Multilevel preconditioning of second order elliptic discontinuous Galerkin problems, Preprint, 2005]. The finite element space of piece-wise polynomials, discontinuous on the partition , is projected onto the space of piece-wise constant functions on the same partition that constitutes the largest space in the multilevel method. The discontinuous Galerkin finite element system on this space is associated to the so-called “graph-Laplacian”. In 2-D this is a sparse M-matrix with -1 as off diagonal entries and nonnegative row sums. Under the assumption that the finest partition is a result of multilevel refinement of a given coarse mesh, we develop the concept of hierarchical splitting of the unknowns. Then using local analysis we derive estimates for the constants in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality, which are uniform with respect to the levels. This measure of the angle between the spaces of the splitting was used by Axelsson and Vassilevski in [Algebraic multilevel preconditioning methods II, SIAM J. Numer. Anal. 27 (1990) 1569–1590] to construct an algebraic multilevel iteration (AMLI) for finite element systems. The main contribution in this paper is a construction of a splitting that produces new estimates for the CBS constant for graph-Laplacian. As a result we have a preconditioner for the system of the discontinuous Galerkin finite element method of optimal arithmetic complexity.     

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.     

Multilevel preconditioning — Appending boundary conditions by Lagrange multipliers

For saddle point problems stemming from appending essential boundary conditions in connection with Galerkin methods for elliptic boundary value problems, a class of multilevel preconditioners is developed. The estimates are based on the characterization of Sobolev spaces on the underlying domain and its boundary in terms of weighted sequence norms relative to corresponding multilevel expansions. The results indicate how the various ingredients of a typical multilevel framework affect the growth rate of the condition numbers. In particular, it is shown how to realize even condition numbers that are uniformly bounded independently of the discretization.These investigations are motivated by the idea of employing nested refinable shift-invariant spaces as trial spaces covering various types of wavelets that are of advantage for the solution of boundary value problems from other points of view. Instead of incorporating the boundary conditions into the approximation spaces in the Galerkin formulation, they are appended by means of Lagrange multipliers leading to a saddle point problem.The work of the author is partially supported by the Deutsche Forschungsgemeinschaft under grant numbers Ku1028/1-1 and Pr336/4-1.     

Convergence analysis for parallel‐in‐time solution of hyperbolic systems

Parallel‐in‐time algorithms have been successfully employed for reducing time‐to‐solution of a variety of partial differential equations, especially for diffusive (parabolic‐type) equations. A major failing of parallel‐in‐time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel‐in‐time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (a) space–time local Fourier analysis, using a Fourier ansatz in space and time; (b) semi‐algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time; and (c) a two‐level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi‐algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.     

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Preconditioned Krylov subspace (KSP) methods are widely used for solving large‐scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely, Gauss–Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in two and three dimensions with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts. BoomerAMG with a proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill‐conditioned or saddle‐point problems, whereas multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.     

Algebraic multigrid preconditioning for iterative eigensolvers

The objective is a comparative study of iterative solvers for eigenproblems arising from elliptic and self–adjoint partial differential operators. Typically only a few of the smallest eigenvalues of these problems are to be computed. We discuss various gradient based preconditioned eigensolvers which make use of algebraic multigrid preconditioning. We present some algorithms together with numerical results. Performance characteristics are derived by comparison with the solutions of standard problems. We show that known advantages of algebraic multigrid preconditioning (e.g. for boundary–value problems with large jumps in the coefficients) transfer to AMG–preconditioned eigensolvers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parameterized preconditioning for generalized saddle point problems arising from the Stokes equation

A parameterized preconditioning framework is proposed to improve the conditions of the generalized saddle point problems. Based on the eigenvalue estimates for the generalized saddle point matrices, a strategy to minimize the upper bounds of the spectral condition numbers of the matrices is given, and the explicit expression of the quasi-optimal preconditioning parameter is obtained. In numerical experiment, parameterized preconditioning techniques are applied to the generalized saddle point problems derived from the mixed finite element discretization of the stationary Stokes equation. Numerical results demonstrate that the involved preconditioning procedures are efficient.     

Parameterized preconditioning for generalized saddle point problems arising from the Stokes equation

A parameterized preconditioning framework is proposed to improve the conditions of the generalized saddle point problems. Based on the eigenvalue estimates for the generalized saddle point matrices, a strategy to minimize the upper bounds of the spectral condition numbers of the matrices is given, and the explicit expression of the quasi-optimal preconditioning parameter is obtained. In numerical experiment, parameterized preconditioning techniques are applied to the generalized saddle point problems derived from the mixed finite element discretization of the stationary Stokes equation. Numerical results demonstrate that the involved preconditioning procedures are efficient.