Algebraic Multilevel Iteration Method Research Articles

Robust algebraic multilevel preconditioning in [formula omitted] and [formula omitted

An algebraic multilevel iteration method for solving a system of linear algebraic equations arising in H(curl) and H(div) spaces is presented. The algorithm is developed for the discrete problem obtained by using the space of lowest-order Nedelec and Raviart–Thomas–Nedelec elements. The theoretical analysis of the method is based only on some algebraic sequences and generalized eigenvalues of local (element-wise) problems. In the hierarchical basis framework, explicit recursion formulas are derived to compute the element matrices and the constant γ (which measures the quality of the space splitting) at any given level. It is proved that the proposed method is robust with respect to the problem parameters and is of optimal order complexity. Supporting numerical results, including the case when the parameters have jumps, are also presented.

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.

Robust multilevel methods for quadratic finite element anisotropic elliptic problems

SUMMARYThis paper discusses a class of multilevel preconditioners based on approximate block factorization for conforming finite element methods employing quadratic trial and test functions. The main focus is on diffusion problems governed by a scalar elliptic partial differential equation with a strongly anisotropic coefficient tensor. The proposed method provides a high robustness with respect to non‐grid‐aligned anisotropy, which is achieved by the interaction of the following components: (i) an additive Schur complement approximation to construct the coarse‐grid operator; (ii) a global block (Jacobi or Gauss–Seidel) smoother complementing the coarse‐grid correction based on (i); and (iii) utilization of an augmented coarse grid, which enhances the efficiency of the interplay between (i) and (ii). The performed analysis indicates the high robustness of the resulting two‐level method. Moreover, numerical tests with a nonlinear algebraic multilevel iteration method demonstrate that the presented two‐level method can be applied successfully in the recursive construction of uniform multilevel preconditioners of optimal or nearly optimal order of computational complexity. Copyright © 2013 John Wiley & Sons, Ltd.

On the robustness and optimality of algebraic multilevel methods for reaction–diffusion type problems

This paper is on preconditioners for reaction---diffusion problems that are both, uniform with respect to the reaction---diffusion coefficients, and optimal in terms of computational complexity. The considered preconditioners belong to the class of so-called algebraic multilevel iteration (AMLI) methods, which are based on a multilevel block factorization and polynomial stabilization. The main focus of this work is on the construction and on the analysis of a hierarchical splitting of the conforming finite element space of piecewise linear functions that allows to meet the optimality conditions for the related AMLI preconditioner in case of second-order elliptic problems with non-vanishing zero-order term. The finite element method (FEM) then leads to a system of linear equations with a system matrix that is a weighted sum of stiffness and mass matrices. Bounds for the constant $$\gamma $$? in the strengthened Cauchy---Bunyakowski---Schwarz inequality are computed for both mass and stiffness matrices in case of a general $$m$$m-refinement. Moreover, an additive preconditioner is presented for the pivot blocks that arise in the course of the multilevel block factorization. Its optimality is proven for the case $$m=3$$m=3. Together with the estimates for $$\gamma $$? this shows that the construction of a uniformly convergent AMLI method with optimal complexity is possible (for $$m \ge 3$$m?3). Finally, we discuss the practical application of this preconditioning technique in the context of time-periodic parabolic optimal control problems.

Additive Schur Complement Approximation and Application to Multilevel Preconditioning

This paper introduces an algorithm for additive Schur complement approximation (ASCA), which can be applied in various iterative methods for solving systems of linear algebraic equations arising from finite element (FE) discretization of partial differential equations (PDE). It is shown how the ASCA can be used to set up a nonlinear algebraic multilevel iteration (AMLI) method. This requires the construction of a linear (multiplicative) two-level preconditioner at each level. The latter is computed in the course of a simultaneous exact two-by-two block factorization of local (stiffness) matrices associated with a covering of the entire domain by overlapping subdomains. Unlike in Schwarz type domain decomposition methods, this method does not require a global coarse problem but instead uses local coarse problems to provide global communication. A robust condition number bound is proved for a particular partitioning of the nodes of a uniform grid. The presented numerical tests demonstrate that the ASCA, when combined with a proper AMLI-cycle, results in a multilevel method of optimal order of computational complexity.

Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems

Abstract While a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study the behavior of the constant in the strengthened CBS inequality for semi-coarsening mesh refinement which is a quality measure for hierarchical two-level splittings of the considered biquadratic FEM stiffness matrices. The presented new theoretical estimates are confirmed by numerically computed CBS constants for a rich set of parameters (coarsening factor and anisotropy ratio). In the paper we consider also the problem of solving efficiently systems with the pivot block matrices arising in the hierarchical basis two-level splittings. Combining the proven uniform estimates with the theory of the Algebraic MultiLevel Iteration (AMLI) methods we obtain an optimal order multilevel algorithm whose total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio.

Algebraic multilevel iteration method for lowest order Raviart–Thomas space and applications

AbstractAn optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems, that have their weak formulation in the space H(div), is presented. The algorithm is developed for the discrete problem obtained by using the lowest‐order Raviart–Thomas space. The method is theoretically analyzed and supporting numerical examples are presented. Furthermore, as a particular application, the algorithm is used for the solution of the discrete minimization problem which arises in the functional‐type a posteriori error estimates for the discontinuous Galerkin approximation of elliptic problems. Copyright © 2011 John Wiley & Sons, Ltd.

Robust AMLI methods for parabolic Crouzeix–Raviart FEM systems

For the iterative solution of linear systems of equations arising from finite element discretization of elliptic problems there exist well-established techniques to construct numerically efficient and computationally optimal preconditioners. Among those, most often preferred choices are Multigrid methods (geometric or algebraic), Algebraic MultiLevel Iteration (AMLI) methods, Domain Decomposition techniques. In this work, the method in focus is AMLI. We extend its construction and the underlying theory over to systems arising from discretizations of parabolic problems, using non-conforming finite element methods (FEM). The AMLI method is based on an approximated block two-by-two factorization of the original system matrix. A key ingredient for the efficiency of the AMLI preconditioners is the quality of the utilized block two-by-two splitting, quantified by the so-called Cauchy–Bunyakowski–Schwarz (CBS) constant, which measures the abstract angle between the two subspaces, associated with the two-by-two block splitting of the matrix. The particular choice of space discretization for the parabolic equations, used in this paper, is Crouzeix–Raviart non-conforming elements on triangular meshes. We describe a suitable splitting of the so-arising matrices and derive estimates for the associated CBS constant. The estimates are uniform with respect to discretization parameters in space and time as well as with respect to coefficient and mesh anisotropy, thus providing robustness of the method.

A Survey of Algebraic Multilevel Iteration (AMLI) Methods

For the solution by preconditioned conjugate gradient methods of symmetric positive definite equations as arising in boundary value problems we consider preconditioning methods of AMLI type. Particular attention is devoted to providing methods of optimal order of computational complexity which in addition promise to be robust, i.e. with a convergence rate which is bounded above independently of size of discretization parameter h, jumps in problem coefficients, and shape of finite elements or, equivalently, anisotropy of problem coefficients. In addition, the computational cost per iteration step must have optimal order.

An algebraic preconditioning method for M‐matrices: linear versus non‐linear multilevel iteration

AbstractIn a recent work, the author introduced a robust multilevel incomplete factorization algorithm using spanning trees of matrix graphs (Proceedings of the 1999 International Conference on Preconditioning Techniques for Large Sparse Matrix Problems in Industrial Applications, Hubert H. Humphrey Center, University of Minnesota, 1999, 251–257). Based on this idea linear and non‐linear algebraic multilevel iteration (AMLI) methods are investigated in the present paper.In both cases, the preconditioner is constructed recursively from the coarsest to finer and finer levels. The considered W‐cycles only need diagonal solvers on all levels and additionally evaluate a second‐degree matrix polynomial (linear case), or, perform ν inner GCG‐type iterations (non‐linear case) on every other level. This involves the same type of preconditioner for the corresponding Schur complement. The non‐linear variant has the additional benefit of being free from any method parameters to be estimated. Based on the same type of approximation property similar convergence rates are obtained for linear and non‐linear AMLI, even for a very small number ν of inner iterations, e.g. ν =2,3. The presented methods are robust with respect to anisotropy and discontinuities in the coefficients of the PDEs and can also be applied to unstructured‐grid problems. Copyright © 2002 John Wiley & Sons, Ltd.

On the Additive Version of the Algebraic Multilevel Iteration Method for Anisotropic Elliptic Problems

In this paper a recently proposed additive version of the algebraic multilevel iteration method for iterative solution of elliptic boundary value problems is studied. The method constructs a nearly optimal order parameter-free preconditioner, which is robust with respect to anisotropy and discontinuity of the problem coefficients. It uses a new strategy for approximating the blocks corresponding to "new" basis functions on each discretization level. To cope with the difficulties arising from the anisotropy, the problem on the coarsest mesh is solved using a bordering technique with a special choice of bordering vectors. The aim is to find a parameter-free "black-box" robust solver. The results are derived in the framework of a hierarchical basis, linear finite element discretization of an elliptic problem on arbitrary triangular meshes, and a hierarchical basis, bilinear finite element discretization on Cartesian meshes. A comparison of the method with some other iterative solution techniques is presented. Robustness and high efficiency of the proposed algorithm are demonstrated on several model-type problems.

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.

Semi‐coarsening AMLI algorithms for elasticity problems

The constant γ in the strengthened Cauchy-Buniakowski-Schwarc (CBS) inequality plays a key role in the convergence analysis of the multilevel iterative methods. We consider in this paper the approximation of the two-dimensional elasticity problem by bilinear rectangle finite elements. Two semi-coarsening refinement procedures are studied. We prove for both cases new estimates of the constant γ, uniformly on the Poisson ratio. As a result of the presented analysis we obtain an optimal order algebraic multiLevel iteration (AMLI) method for the case of balanced semi-coarsening mesh refinement. The total computational complexity of the algorithm is proportional to the size of the discrete problem with a proportionality constant independent of the Poisson ratio, that is, the algorithm is of optimal order for almost incompressible elasticity problems. Copyright © 1999 John Wiley & Sons, Ltd.

An algebraic multilevel iteration method for finite element matrices

To solve a sparse linear system of equations resulting from the finite element approximation of elliptic self-adjoint second-order boundary-value problems an algebraic multilevel iteration method is presented. The new method can be considered as an extension of methods, which have been defined by Axelsson and Eijkhout (1991) for nine-point matrices and later generalized by Axelsson and Neytcheva (1994) for the Stieltjes matrices, on a more wider class of sparse symmetric positive-definite matrices. The rate of convergence and the computational complexity of the method are analyzed. Experimental results on some standard test problems are presented and discussed.

On Two Ways of Stabilizing the Hierarchical Basis Multilevel Methods

A survey of two approaches for stabilizing the hierarchical basis (HB) multilevel preconditioners, both additive and multiplicative, is presented. The first approach is based on the algebraic extension of the two-level methods, exploiting recursive calls to coarser discretization levels. These recursive calls can be viewed as inner iterations (at a given discretization level), exploiting the already defined preconditioner at coarser levels in a polynomially-based inner iteration method. The latter gives rise to hybrid-type multilevel cycles. This is the so-called (hybrid) algebraic multilevel iteration (AMLI) method. The second approach is based on a different direct multilevel splitting of the finite element discretization space. This gives rise to the so-called wavelet multilevel decomposition based on $L^2$-projections, which in practice must be approximated. Both approaches---the AMLI one and the one based on approximate wavelet decompositions---lead to optimal relative condition numbers of the multilevel preconditioners.

Experience in implementing the Algebraic Multilevel Iteration method on a SIMD-type computer

In this presentation we discuss the Algebraic Multilevel Iteration method and some aspects of its implementation for massively parallel distributed memory computer systems (CM-2, CM-200). Numerical results are presented and compared with those obtained from a serial implementation of the method (IBM RS/6000, SUN SPARC work stations).

Algebraic Multilevel Preconditioning of Anisotropic Elliptic Problems

In this paper the recently proposed algebraic multilevel iteration method for iterative solution of elliptic boundary value problems with anisotropy and discontinuous coefficients is studied. Based on a special approximation of the blocks corresponding to the new nodes at every discretization level, an optimal order preconditioner with respect to the arithmetic cost independent of both the discontinuity and the anisotropy of the coefficients is constructed. The advantages of the proposed algorithms are illustrated by numerical tests.

Asymptotic work estimates for AMLI methods

Recently the authors have presented the AMLI (algebraic multilevel iteration) method to solve second-order elliptic equations discretized by finite element methods on an arbitrary triangular mesh and shown that the method has an optimal rate of convergence and an optimal order of computational complexity. In the present paper we derive sharp upper bounds and asymptotic estimates for the computational labor and apply these estimates for the above type of problems.

Condition Estimates

Condition Estimates