On the multigrid cycle strategy with approximate inverse smoothing

2014 ◽  
Vol 31 (1) ◽  
pp. 110-122 ◽  
Author(s):  
George A. Gravvanis ◽  
Christos K. Filelis-Papadopoulos
Keyword(s):  
Finite Difference ◽  
Linear Systems ◽  
Multigrid Method ◽  
Multigrid Methods ◽  
Inverse Matrix ◽  
Sparse Linear Systems ◽  
Approximate Inverse ◽  
Content Type ◽  
Sparsity Pattern ◽  
Approximate Inverses

Purpose – The purpose of this paper is to propose multigrid methods in conjunction with explicit approximate inverses with various cycles strategies and comparison with the other smoothers. Design/methodology/approach – The main motive for the derivation of the various multigrid schemes lies in the efficiency of the multigrid methods as well as the explicit approximate inverses. The combination of the various multigrid cycles with the explicit approximate inverses as smoothers in conjunction with the dynamic over/under relaxation (DOUR) algorithm results in efficient schemes for solving large sparse linear systems derived from the discretization of partial differential equations (PDE). Findings – Application of the proposed multigrid methods on two-dimensional boundary value problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than the V-cycle multigrid schemes presented in a recent report (Filelis-Papadopoulos and Gravvanis). Research limitations/implications – The limitations of the proposed scheme lie in the fact that the explicit finite difference approximate inverse matrix used as smoother in the multigrid method is a preconditioner for specific sparsity pattern. Further research is carried out in order to derive a generic explicit approximate inverse for any type of sparsity pattern. Originality/value – A novel smoother for the geometric multigrid method is proposed, based on optimized banded approximate inverse matrix preconditioner, the Richardson method in conjunction with the DOUR scheme, for solving large sparse linear systems derived from finite difference discretization of PDEs. Moreover, the applicability and convergence behavior of the proposed scheme is examined based on various cycles and comparative results are given against the damped Jacobi smoother.

scholarly journals Parallel Preconditioned Conjugate Gradient Square Method Based on Normalized Approximate Inverses

2005 ◽  
Vol 13 (2) ◽  
pp. 79-91 ◽  
Author(s):  
George A. Gravvanis ◽  
Konstantinos M. Giannoutakis
Keyword(s):  
Linear Systems ◽  
Conjugate Gradient ◽  
Message Passing ◽  
Message Passing Interface ◽  
Distributed Memory ◽  
Inverse Matrix ◽  
Preconditioned Conjugate Gradient ◽  
Sparse Linear Systems ◽  
Approximate Inverse ◽  
Approximate Inverse Matrix Techniques

A new class of normalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. A new parallel normalized explicit preconditioned conjugate gradient square method in conjunction with normalized approximate inverse matrix techniques for solving efficiently sparse linear systems on distributed memory systems, using Message Passing Interface (MPI) communication library, is also presented along with theoretical estimates on speedups and efficiency. The implementation and performance on a distributed memory MIMD machine, using Message Passing Interface (MPI) is also investigated. Applications on characteristic initial/boundary value problems in three dimensions are discussed and numerical results are given.

GENERIC APPROXIMATE SPARSE INVERSE MATRIX TECHNIQUES

2014 ◽  
Vol 11 (06) ◽  
pp. 1350084 ◽  
Author(s):  
CHRISTOS K. FILELIS-PAPADOPOULOS ◽  
GEORGE A. GRAVVANIS
Keyword(s):  
Linear Systems ◽  
Krylov Subspace ◽  
Coefficient Matrix ◽  
Sparse Linear Systems ◽  
Approximate Inverse ◽  
Matrix Algorithm ◽  
Sparsity Pattern ◽  
Gradient Type ◽  
Comparative Results ◽  
Conjugate Gradient Type Methods

During the last decades explicit preconditioning methods have gained interest among the scientific community, due to their efficiency for solving large sparse linear systems in conjunction with Krylov subspace iterative methods. The effectiveness of explicit preconditioning schemes relies on the fact that they are close approximants to the inverse of the coefficient matrix. Herewith, we propose a Generic Approximate Sparse Inverse (GenASPI) matrix algorithm based on ILU(0) factorization. The proposed scheme applies to matrices of any structure or sparsity pattern unlike the previous dedicated implementations. The new scheme is based on the Generic Approximate Banded Inverse (GenAbI), which is a banded approximate inverse used in conjunction with Conjugate Gradient type methods for the solution of large sparse linear systems. The proposed GenASPI matrix algorithm, is based on Approximate Inverse Sparsity patterns, derived from powers of sparsified matrices and is computed with a modified procedure based on the GenAbI algorithm. Finally, applicability and implementation issues are discussed and numerical results along with comparative results are presented.

scholarly journals Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

2012 ◽  
Vol 2012 ◽  
pp. 1-49 ◽  
Author(s):  
Massimiliano Ferronato
Keyword(s):  
Linear Systems ◽  
Specific Problem ◽  
General Purpose ◽  
Sparse Linear Systems ◽  
System Matrix ◽  
Preconditioning Techniques ◽  
Key Factor ◽  
Recent Developments ◽  
Approximate Inverses ◽  
Linear Systems Of Equations

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper.

Approximate inverse preconditionings for sparse linear systems

1992 ◽  
Vol 44 (1-4) ◽  
pp. 91-110 ◽  
Author(s):  
J. D. F. Cosgrove ◽  
J. C. Díaz ◽  
A. Griewank
Keyword(s):  
Linear Systems ◽  
Sparse Linear Systems ◽  
Approximate Inverse

Parallel multilevel recursive approximate inverse techniques for solving general sparse linear systems

2016 ◽  
Vol 72 (6) ◽  
pp. 2259-2282 ◽  
Author(s):  
Antonios T. Makaratzis ◽  
Christos K. Filelis-Papadopoulos ◽  
George A. Gravvanis
Keyword(s):  
Linear Systems ◽  
Sparse Linear Systems ◽  
Approximate Inverse

scholarly journals A parallel preconditioner based on the approximation of an inverse matrix by power series for solving sparse linear systems on graphics processors

2019 ◽  
pp. 444-456
Author(s):  
А.В. Юлдашев ◽  
Н.В. Репин ◽  
В.В. Спеле
Keyword(s):  
Power Series ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Computing System ◽  
Neumann Series ◽  
Inverse Matrix ◽  
Reservoir Modeling ◽  
Sparse Linear Systems ◽  
Graphics Processors ◽  
Tridiagonal Matrices

Рассмотрена применимость метода AIPS, аппроксимирующего обратную матрицу на основе степенного разложения в ряд Неймана, в рамках двухступенчатого предобусловливателя CPR. Предложен ориентированный на архитектуру CUDA параллельный алгоритм решения линейных систем с трехдиагональной матрицей, состоящей из независимых блоков различного размера. Показано, что реализация предложенного алгоритма может более чем в 2 раза превосходить по быстродействию функции решения трехдиагональных систем из библиотеки cuSPARSE. Проведено тестирование метода BiCGStab с предобусловливателем CPRAIPS на современных GPU, в том числе на гибридной вычислительной системе с 4 GPU NVIDIA Tesla V100, показавшее приемлемую масштабируемость данного предобусловливателя, а также возможность ускорить решение линейных систем, характерных для задачи гидродинамического моделирования нефтегазовых месторождений, по сравнению с CPRAMG. The applicability of the AIPS method approximating an inverse matrix using Neumann series is considered in the framework of the CPR two stage preconditioner. A parallel CUDAoriented algorithm is proposed for solving linear systems with tridiagonal matrices consisting of independent blocks of different sizes. It is shown that the implementation of the proposed algorithm can be more than twice the speed of the similar functions from the cuSPARSE library. Experimental evaluation of the BiCGStab method with the CPRAIPS preconditioner on modern GPUs, including a hybrid computing system with 4 GPU NVIDIA Tesla V100, is performed. Numerical experiments show an adequate scalability of this preconditioner as well as the possibility (compared to the CPRAMG) to accelerate the solution of linear systems being typical for the reservoir modeling problems.

A class of generic factored and multi-level recursive approximate inverse techniques for solving general sparse systems

2016 ◽  
Vol 33 (1) ◽  
pp. 74-99 ◽  
Author(s):  
Christos K. Filelis-Papadopoulos ◽  
George A. Gravvanis
Keyword(s):  
Linear Systems ◽  
Krylov Subspace ◽  
Theoretical Estimate ◽  
Independent Set ◽  
Schur Complement ◽  
Approximate Inverse ◽  
Content Type ◽  
Convergence Behavior ◽  
Multi Level

Purpose – The purpose of this paper is to propose novel factored approximate sparse inverse schemes and multi-level methods for the solution of large sparse linear systems. Design/methodology/approach – The main motive for the derivation of the various generic preconditioning schemes lies to the efficiency and effectiveness of factored preconditioning schemes in conjunction with Krylov subspace iterative methods as well as multi-level techniques for solving various model problems. Factored approximate inverses, namely, Generic Factored Approximate Sparse Inverse, require less fill-in and are computed faster due to the reduced number of nonzero elements. A modified column wise approach, namely, Modified Generic Factored Approximate Sparse Inverse, is also proposed to further enhance performance. The multi-level approximate inverse scheme, namely, Multi-level Algebraic Recursive Generic Approximate Inverse Solver, utilizes a multi-level hierarchy formed using Block Independent Set reordering scheme and an approximation of the Schur complement that results in the solution of reduced order linear systems thus enhancing performance and convergence behavior. Moreover, a theoretical estimate for the quality of the multi-level approximate inverse is also provided. Findings – Application of the proposed schemes to various model problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than results by other researchers for some of the model problems. Research limitations/implications – Further enhancements are investigated for the proposed factored approximate inverse schemes as well as the multi-level techniques to improve quality of the schemes. Furthermore, the proposed schemes rely on the definition of multiple parameters that for some problems require thorough testing, thus adaptive techniques to define the values of the various parameters are currently under research. Moreover, parallel schemes will be investigated. Originality/value – The proposed approximate inverse preconditioning schemes as well as multi-level schemes are efficient computational methods that are valuable for computer scientists and for scientists and engineers in engineering computations.

A NOTE ON PARALLEL FINITE DIFFERENCE APPROXIMATE INVERSE PRECONDITIONING ON MULTICORE SYSTEMS USING POSIX THREADS

2013 ◽  
Vol 10 (05) ◽  
pp. 1350032 ◽  
Author(s):  
G. A. GRAVVANIS ◽  
C. K. FILELIS-PAPADOPOULOS ◽  
K. M. GIANNOUTAKIS ◽  
E. A. LIPITAKIS
Keyword(s):  
Finite Difference ◽  
Conjugate Gradient ◽  
Fish Bone ◽  
Inverse Matrix ◽  
Computational Techniques ◽  
Multicore Systems ◽  
Approximate Inverse ◽  
Posix Threads ◽  
Dimensional Boundary ◽  
Explicit Finite Difference

New parallel computational techniques are introduced for the parallelization of explicit finite difference (FD) approximate inverse matrix methods, based on Portable Operating System Interface for UniX (POSIX) threads, for multicore systems. Parallelization of the Optimized Banded Generalized Approximate Inverse Matrix (OBGAIM) algorithm is achieved based on the concept of the "fish bone" approach with the use of a thread pool pattern. Theoretical estimates on speedups and efficiency are also presented. Additionally, new parallel computational techniques are proposed for the parallelization of explicit preconditioned biconjugate conjugate gradient type methods, based on POSIX threads, for multicore systems. For parallelization purposes a replication of the parallel explicit preconditioned biconjugate conjugate gradient-STAB (PEPBICG-STAB) method was assigned on each created thread, with different index bands and with proper synchronization points on inner products and matrix-vector multiplications. Theoretical estimates on speedups and efficiency are also presented. Finally, numerical results for the performance of the Parallel Fish Bone OBGAIM (PaFiBo-OBGAIM) algorithm and the PEPBICG-STAB method for solving classical two-dimensional boundary value problems on multicore computer systems are presented, which are favorably compared to corresponding results from multiprocessor systems. The implementation issues of the proposed method are also discussed using POSIX threads on multicore systems.

ON THE RATE OF CONVERGENCE AND COMPLEXITY OF NORMALIZED IMPLICIT PRECONDITIONING FOR SOLVING FINITE DIFFERENCE EQUATIONS IN THREE SPACE VARIABLES

2004 ◽  
Vol 01 (02) ◽  
pp. 367-386 ◽  
Author(s):  
GEORGE A. GRAVVANIS ◽  
KONSTANTINOS M. GIANNOUTAKIS
Keyword(s):  
Finite Difference ◽  
Rate Of Convergence ◽  
Linear Systems ◽  
Conjugate Gradient ◽  
Three Dimensional ◽  
Preconditioned Conjugate Gradient ◽  
Sparse Linear Systems ◽  
Approximate Factorization ◽  
Dimensional Boundary ◽  
Three Space

Normalized approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference method of partial differential equations in three space variables, are presented. Normalized implicit preconditioned conjugate gradient-type schemes in conjunction with normalized approximate factorization procedures are presented for the efficient solution of sparse linear systems. The convergence analysis with theoretical estimates on the rate of convergence and computational complexity of the normalized implicit preconditioned conjugate gradient method are also given. Application of the proposed method on characteristic three dimensional boundary value problems is discussed and numerical results are given.

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