Class Of Preconditioners Research Articles

Preconditioning the Mortar Method by Substructuring: The High Order Case.

AbstractWe analyze a class of preconditioners for the mortar method, based on substructuring. After splitting in a suitable way the degrees of freedom in interior, edge and vertex, we study the performance of a block Jacobi type preconditioner for which the condition number of the preconditioned matrix only grows polylogarithmically. Unlike the previous work by Achdou, Maday and Widlund [1], which is restricted to the case of first order finite element, this paper relies on abstract assumptions and therefore applies to finite element of any order. Moreover, the use of a suitable coarse preconditioner (whose effect we analyze) makes this technique more efficient. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hybrid preconditioned algorithms for boundary hypersingular integral equations

Three preconditioners for solving boundary hypersingular integral equations of the first kind are proposed. The first and the second preconditioners are based on circulant operators. The third is a hybrid of iterative substructuring techniques and circulant operators. Although this class of preconditioners are composition of circulant operators but we have shown that the condition number of the third preconditioner dose not exceed the condition number of two preconditioners. Numerical results are included.

Preconditioned GMRES methods for discretization equation of nonsymmetric and indefinite elliptic problem

In this paper, a class of preconditioners for GMRES method are proposed for solving linear systems arising from discretization second-order nonsymmetric and indefinite elliptic problems by finite element method. The convergence of GMRES method is verified and the rate of convergence is shown. The generality of our result enables us to apply not only any known preconditioners designed for symmetric positive definite problems, but also many other preconditioners based on original operator to nonsymmetric and indefinite problems without losing optimality.

Electromagnetic induction in a generalized 3D anisotropic earth, Part 2: The LIN preconditioner

A practical limitation in the use of generalized 3D forward modeling algorithms for inversion of electromagnetic data is the high computational cost of solving large, ill‐conditioned systems of linear equations arising from the discretization of the governing Maxwell equations. To address this problem, a new class of preconditioners has recently been proposed which is based on a Helmholtz decomposition of the electric field in the low induction number (LIN) regime. This paper further develops that idea and introduces a LIN preconditioner which can be applied to problems characterized by a fully generalized anisotropic medium. Included are sample calculations demonstrating a reduction by two orders of magnitude in the number of “quasi‐minimal residual” iterates and a speedup by a factor of approximately four in the solution time for one forward calculation. Also included are results previously unobtainable by standard Jacobi preconditioning for simulating multicomponent induction sonde response in a horizontal well within a crossbedded formation.

Circulant Wavelet Preconditioners for Solving Elliptic Differential Equations and Boundary Integral Equations

In this paper is discussed solving an elliptic equation and a boundary integral equation of the second kind by representation of compactly supported wavelets. By using wavelet bases and the Galerkin method for these equations, we obtain a stiff sparse matrix that can be ill-conditioned. Therefore, we have to introduce an operator which maps every sparse matrix to a circulant sparse matrix. This class of circulant matrices is a class of preconditioners in a Banach space. Based on having some properties in the spectral theory for this class of matrices, we conclude that the circulant matrices are a good class of preconditioners for solving these equations. We called them circulant wavelet preconditioners (CWP). Therefore, a class of algorithms is introduced for rapid numerical application.

Support Theory for Preconditioning

We present support theory, a set of techniques for bounding extreme eigenvalues and condition numbers for matrix pencils. Our intended application of support theory is to enable proving condition number bounds for preconditioners for symmetric, positive definite systems. One key feature sets our approach apart from most other works: We use support numbers instead of generalized eigenvalues. Although closely related, we believe support numbers are more convenient to work with algebraically. This paper provides the theoretical foundation of support theory and describes a set of analytical tools and techniques. For example, we present a new theorem for bounding support numbers (generalized eigenvalues) where the matrices have a known factorization (not necessarily square or triangular). This result generalizes earlier results based on graph theory. We demonstrate the utility of this approach by a simple example: block Jacobi preconditioning on a model problem. Also, our analysis of a new class of preconditioners, maximum-weight basis preconditioners, in [E. G. Boman, D. Chen, B. Hendrickson, and S. Toledo, Numer. Linear Algebra Appl., to appear] is based on results contained in this paper.

Boundary Element Preconditioners for Problems Defined on Slender Domains

Within the framework of two-dimensional potential theory, we consider problems defined on slender domains. The boundary element matrices for such problems are poorly conditioned due to the presence of two disparate length scales. A new class of preconditioners is proposed to remedy this problem so that the preconditioned boundary element matrices become insensitive to the domain slenderness. The preconditioners are constructed using the spectral properties of the boundary integral operators defined on the elliptical domains.

Fast wavelet iterative solvers applied to the Neumann problem

The effectiveness of the resolution of linear systems, arising from a finite element or a finite difference formulation of the Neumann problem, depends on both the treatment of the singularity and the choice of a preconditioner. In this paper, we apply a Fast Wavelet Transform (FWT) to the linear system. In this discretization we propose a new class of preconditioners as an alternative to the diagonal wavelet preconditioner. Moreover we present two methods to deal with the singularity of the new system. Numerical results show that the number of matrix-vector products is almost independent of the size of the system.

Preconditined semilinear stochastic singular and hypersingular integral equations

In this paper, three classes of preconditioners are proposed for solving some stochastic integral equations with the weakly singular kernel and the hypersingular kernel. The first and the second class of preconditioners are based on circulant operators, but the third class of preconditioners is based on iterative substructuring. It is proved that substructuring preconditioners can be better than other preconditioners. Also, the spaces of solutions are discussed such that the solutions of these equations are smooth, therefore, we give special Banach spaces for these integral equations. Finally, numerical results which support our theories are presented

Estimation of conflict-free matrix product preconditioner on vector supercomputers

Two classes of preconditioners are considered for nonsymmetric linear systems arising from second order difference discretization of non-self-adjoint elliptic partial differential equations. Experimental results show that the usual incomplete LU and modified ILU factorization preconditioner have bad effect due to bank conflict on current vector supercomputers. That is, the bank conflict is an underlying factor for attaining high performance of the vector supercomputers. To avoid the effect of bank conflict, a matrix product preconditioner is introduced. In experiments with two-dimensional problems with constant and variable coefficients on a uniform mesh, the matrix product preconditioner will be shown to be more efficient than the standard incomplete factorization preconditioners.

Preconditioning convection dominated convection‐diffusion problems

This paper is concerned with the numerical solution of multi‐dimensional convection dominated convection‐diffusion problems. These problems are characterized by a large parameter, K, multiplying the convection terms. The goal of this work is the development and analysis of effective preconditioners for iteratively solving the large system of linear equations arising from various finite element and finite difference discretizations with grid size h. When centered finite difference schemes and standard Galerkin finite element methods are used, h must be related to K by the stability constraint, Kh ≤ C0, where the constant C0 is sufficiently small. A class of preconditioners is developed that significantly reduces the condition number for large K and small h. Furthermore, these preconditioners are inexpensive to implement and well suited for parallel computation. It is shown that under suitable assumptions, the number of iterations remains bounded as h ↓0 with K fixed and, at worst, grows slowly as K ↓ ∞. Numerical results are presented illustrating the theory. It is also shown how to apply the theoretical results to more general convection‐diffusion problems and alternative discretizations (including streamline diffusion methods) that remain stable as Kh ↓ ∞.

A numerical study of optimized sparse preconditioners

Preconditioning strategies based on incomplete factorizations and polynomial approximations are studied through extensive numerical experiments. We are concerned with the question of the optimal rate of convergence that can be achieved for these classes of preconditioners.

Iterative Solution Methods and Preconditioners for Block-Tridiagonal Systems of Equations

Systems of equations arising from implicit time discretizations and finite difference space discretizations of systems of partial differential equations in two space dimensions are considered. The nonsymmetric linear systems are solved using a preconditioned CG-like iterative method. A class of preconditioners, referred to as semicirculant, is examined. For a scalar hyperbolic model problem, it is shown that the number of iterations required using a preconditioner in the semicirculant class is independent of the number of unknowns, provided that the quotient $\kappa $ between the time- and space-step is held constant. Also, it is shown that the number of iterations grows no faster than $\sqrt \kappa $. This type of favorable convergence property is also observed in numerical experiments solving more complicated problems.

Preconditioning for Boundary Integral Equations

New classes of preconditioners are proposed for the linear systems arising from a boundary integral equation method. The problem under consideration is Laplace’s equation in three dimensions. The system arising in this context is dense and unsymmetric. These preconditioners, which are based on solving small linear systems at each node, reduce the number of iterations in some cases by a factor of 8. Three iterative methods are considered: conjugate gradient on the normal equations, CGS of Sonneveld, and GMRES of Saad and Schultz. For a simple model problem, the exact relationship between the preconditioners and the resulting condition number of the system is investigated. This analysis proves that the condition number of the preconditioned system is decreased by a factor asymptotically greater than any constant.

Preconditioned iterative methods for convection diffusion and related boundary value problems

We develop and analyze preconditioners for the iterative solution of the system of equations arising from the discretization of multi-dimensional singularity perturbed boundary value problems. This includes a class of convection diffusion models. The choice of preconditioner is crucial for the efficient solution of the system of equations. In particular, it is necessary to choose a preconditioner that substantially reduces the condition number κ both for small grid size h and for large values of the parameter K multiplying the convection terms. A class of preconditioners is analyzed that is inexpensive to implement and for which κ = 0(1) as h→0 and κ = (1 + K 1 2 ) as K → ∞ for some convection diffusion problems with positive definite symmetric part. This result is then used to develop an algorithm with work estimate 0(1 + K 1 2 as K → ∞ for a more general class of convection diffusion problems including those with indefinite symmetric part. Numerical experiments using a symmetric multigrid preconditioner demonstrate the effectiveness of the numerical method even for large problems.

Some fast 3D finite element solvers for the generalized Stokes problem

AbstractThis paper is devoted to a comparison of various iterative solvers for the Stokes problem, based on the preconditioned Uzawa approach. In the first section the basic equations and general results of gradient‐like methods are recalled. Then a new class of preconditioners, whose optimality will be shown, is introduced. In the last section numerical experiments and comparisons with multigrid methods prove the quality of these schemes, whose discretization is detailed.