George A. Gravvanis

Explicit approximate inverse preconditioning techniques

SummaryThe numerical treatment and the production of related software for solving large sparse linear systems of algebraic equations, derived mainly from the discretization of partial differential equation, by preconditioning techniques has attracted the attention of many researchers. In this paper we give an overview of explicit approximate inverse matrix techniques for computing explicitly various families of approximate inverses based on Choleski and LU—type approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference, finite element and the domain decomposition discretization of elliptic and parabolic partial differential equations. Composite iterative schemes, using inner-outer schemes in conjunction with Picard and Newton method, based on approximate inverse matrix techniques for solving non-linear boundary value problems, are presented. Additionally, isomorphic iterative methods are introduced for the efficient solution of non-linear systems. Explicit preconditioned conjugate gradient—type schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear system of algebraic equations. Theoretical estimates on the rate of convergence and computational complexity of the explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.

The rate of convergence of explicit approximate inverse preconditioning

The rate of convergence of the explicit preconditioned methods, based on approximate inverses, is shown to be based on lower and upper bounds of the extreme eigenvalues of the iteration matrix. Estimates of the computational work required to reduce the Lr-norm by a factor e are given. The application of the method on a 2D elliptic boundary value problem is discussed and numerical results are given.

An approximate inverse matrix technique for arrowhead matrices

A new class of approximate inverse matrix techniques based on the concept of sparse LU-type factorization procedures is introduced for computing explicitly inverses of arrowhead matrices without inverting the decomposition factors. Explicit preconditioned iterative schemes in conjunction with AIM techniques are presented for the efficient solution of linear systems. Applications of the method on a linear system are discussed and numerical results are given.

Explicit preconditioned iterative methods for solving large unsymmetric finite element systems

A class of Generalized Approximate Inverse Matrix (GAIM) techniques, based on the concept of LU-sparse factorization procedures, is introduced for computing explicitly approximate inverses of large sparse unsymmetric matrices of irregular structure, without inverting the decomposition factors. Explicit preconditioned iterative methods, in conjunction with modified forms of the GAIM techniques, are presented for solving numerically initial/boundary value problems on multiprocessor systems. Application of the new methods on linear boundary-value problems is discussed and numerical results are given.ZusammenfassungEs wird eine Methode zur Approximation der verallgemeinerten inversen Matrix (GAIM) diskutiert, die auf dem Konzept der schwachbesetzten LU-Faktorisierung basiert und explizite Inverse großer schwachbesetzter unsymmetrischer Matrizen auf irregulären Strukturen approximiert, ohne die Zerlegungsfaktoren zu invertieren. In Verbindung mit Modifikationen der GAIM-Technik werden explizite Präkonditionierungsmethoden zur numerischen Lösung von Anfangsrandwertproblemen auf Multiprozessorsystemen vorgestellt. Anwendungen der neuen Methoden auf lineare Randwertaufgaben werden diskutiert und numerische Resultate präsentiert.

Explicit isomorphic iterative methods for solving arrow-type linear systems

A new class of approximate inverse arrow-type matrix techniques based on the concept of sparse approximate LU-type factorization procedures is introduced for computing explicitly approximate inverses without inverting the decomposition factors. Isomorphic methods in conjunction with explicit preconditioned schemes based on approximate inverse matrix techniques are presented for the efficient solution of arrow-type linear systems. Applications of the proposed method on linear systems is discussed and numerical results are given

A class of explicit preconditioned conjugate gradient methods for solving large finite element systems

A class of Explicit Preconditioned Conjugate Gradient (EPCG) methods for solving large sparse linear systems of algebraic equations resulting from the Finite Element discretization of Elliptic and Parabolic PDEs is introduced. The EPCG methods are based on explicit Approximate Inverse Matrix techniques and are particularly suitable for solving numerically initial/boundary-value problems on multiprocessor systems. The application of the new methods on 2D-linear boundary-value problems is discussed and numerical results are given.

Solving Symmetric Arrowhead and Special Tridiagonal Linear Systems by Fast Approximate Inverse Preconditioning

A new class of approximate inverses for arrowhead and special tridiagonal linear systems, based on the concept of sparse approximate Choleski-type factorization procedures, are introduced for computing fast explicit approximate inverses. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of symmetric linear systems. A theorem on the rate of convergence of the explicit preconditioned conjugate gradient scheme is given and estimates of the computational complexity are presented. Applications of the proposed method on linear and nonlinear systems are discussed and numerical results are given.

A three-dimensional explicit preconditioned solver

A new class of Generalized Approximate Inverse Matrix (GAIM) techniques, based on the concept of LU-sparse factorization procedures, is introduced for computing explicitly generalized approximate inverses of large sparse unsymmetric matrices of regular structure, without inverting the decomposition factors. Explicit preconditioned iterative methods, in conjunction with modified forms of the GAIM techniques, are presented for solving numerically boundary value problems in three dimensions. The numerical implementation of these algorithms is presented and Fortran subroutines are given

An explicit sparse unsymmetric finite element solver

A new class of explicit generalized approximate inverse finite element matrix algorithmic methods, based on the concept of LU-sparse factorization procedures, without inverting the decomposition factors, has recently been introduced. The large sparse unsymmetric coefficient matrix of irregular structure is factorized approximately and, in conjuction with approximate inverse matrix techniques, yields explicit preconditioned methods for the finite element (FE) and finite difference (FD) method. The numerical implementation of these algorithms is presented and Fortran subroutines for the efficient solution of the sparse unsymmetric linear systems are given.

Explicit preconditioned methods for solving 3d boundary-value problems by approximate inverse finite element matrix techniques

A new class of factorization algorithmic procedures and approximate inverse finite element matrix techniques for solving large symmetric matrices of irregular structure, without inverting the decomposition factors, are presented. Explicit preconditioned iterative methods, based on these approximate finite element inverse matrix techniques, are used for the efficient numerical solution of large linear systems resulting from the finite element discretization of boundary value problems in three space dimensions. Application of the new methods on a 3D linear boundary value problem is discussed and numerical results are given.