1,720,975 research outputs found
On the Multigrid Method Based on Finite Difference Approximate Inverses
No full textDuring the last decades, multigrid methods have been extensively used in order to solve large scale linear systems derived from the discretization of partial differential equations using the finite difference method. Approximate Inverses in conjunction with Richardon’s iterative method could be used as smoothers in the multigrid method. Thus, a new class of smoothers based on approximate inverses could be derived. Effectiveness of explicit approximate inverses relies in the fact that they are close approximants to the inverse of the coefficient matrix and are fast to compute in parallel. Furthermore, the class of finite difference approximate inverses proposed in conjunction with the explicit preconditioned Richardson method present improved results against the classic smoothers such as Jacobi and Gauss – Seidel method. Moreover, a dynamic relaxation scheme is proposed based on the Dynamic Over / Under Relaxation (DOUR) algorithm. Furthermore, results for the multigrid preconditioned Bi-CGSTAB based on approximate inverse smoothing and a dynamic relaxation technique are presented for a class of model problems
Hybrid multilevel solution of sparse least-squares linear systems
No full text
Purpose
Large sparse least-squares problems arise in different scientific disciplines such as optimization, data analysis, machine learning and simulation. This paper aims to propose a two-level hybrid direct-iterative scheme, based on novel block independent column reordering, for efficiently solving large sparse least-squares linear systems.
Design/methodology/approach
Herewith, a novel block column independent set reordering scheme is used to separate the columns in two groups: columns that are block independent and columns that are coupled. The permutation scheme leads to a two-level hierarchy. Using this two-level hierarchy, the solution of the least-squares linear system results in the solution of a reduced size Schur complement-type square linear system, using the preconditioned conjugate gradient (PCG) method as well as backward substitution using the upper triangular factor, computed through sparse Q-less QR factorization of the columns that are block independent. To improve the convergence behavior of the PCG method, the upper triangular factor, computed through sparse Q-less QR factorization of the coupled columns, is used as a preconditioner. Moreover, to further reduce the fill-in, then the column approximate minimum degree (COLAMD) algorithm is used to permute the block consisting of the coupled columns.
Findings
The memory requirements for solving large sparse least-squares linear systems are significantly reduced compared to Q-less QR decomposition of the original as well as the permuted problem with COLAMD. The memory requirements are reduced further by choosing to form larger blocks of independent columns. The convergence behavior of the iterative scheme is improved due to the chosen preconditioning scheme. The proposed scheme is inherently parallel due to the introduction of block independent column reordering.
Originality/value
The proposed scheme is a hybrid direct-iterative approach for solving sparse least squares linear systems based on the implicit computation of a two-level approximate pseudo-inverse matrix. Numerical results indicating the applicability and effectiveness of the proposed scheme are given.
</jats:sec
Algebraic Multigrid Methods Based on Generic Approximate Banded Inverse Matrix Techniques
Get PDFSince the introduction of the Algebraic MultiGrid algorithm (AMG) over twenty years ago, significant progress has been made in improving the coarsening and the convergence behavior of the method. In this paper, an AMG method is introduced that utilizes a new generic approximate inverse algorithm as a smoother in conjunction with common coarsening techniques, such as classical Ruge-Stüben coarsening, CLJP and PMIS coarsening. The proposed approximate inverse scheme, namely Generic Approximate Banded Inverse (GenAbI), is a banded approximate inverse based on Incomplete LU factorization with zero fill–in (ILU(0)). The new class of Generic Approximate Banded Inverse can be computed for any sparsity pattern of the coefficient matrix, in an analogous way as the explicit approximate inverse, yielding a suitable smoother to be used in conjunction with an Algebraic Multigrid method. The proposed smoother is parameterized and thus by increasing the “retention” parameter the smoothing scheme becomes more effective in terms of required number of cycles for convergence. Finally, the applicability and effectiveness of the proposed AMG method along with implementation issues, based on the Generic Approximate Banded Inverse matrix, is demonstrated by solving two and three dimensional problems and numerical results on the convergence behavior and convergence factor are given
Parallel Multiprojection Preconditioned Methods Based on Subspace Compression
Get PDFDuring the last decades, the continuous expansion of supercomputing infrastructures necessitates the design of scalable and robust parallel numerical methods for solving large sparse linear systems. A new approach for the additive projection parallel preconditioned iterative method based on semiaggregation and a subspace compression technique, for general sparse linear systems, is presented. The subspace compression technique utilizes a subdomain adjacency matrix and breadth first search to discover and aggregate subdomains to limit the average size of the local linear systems, resulting in reduced memory requirements. The depth of aggregation is controlled by a user defined parameter. The local coefficient matrices use the aggregates computed during the formation of the subdomain adjacency matrix in order to avoid recomputation and improve performance. Moreover, the rows and columns corresponding to the newly formed aggregates are ordered last to further reduce fill-in during the factorization of the local coefficient matrices. Furthermore, the method is based on nonoverlapping domain decomposition in conjunction with algebraic graph partitioning techniques for separating the subdomains. Finally, the applicability and implementation issues are discussed and numerical results along with comparative results are presented.</jats:p
A Class of Symmetric Factored Approximate Inverses and Hybrid Two-Level Solver
No full text A new class of symmetric factored approximate inverses is proposed and used in conjunction with the Preconditioned Conjugate Gradient method for solving sparse symmetric linear systems. Additionally, a new hybrid two-level solver is proposed utilizing a block independent set reordering, in order to create the two level hierarchy. The Schur complement is formed explicitly by inverting the blocks created by reordering. Then, the preconditioned conjugate gradient method is used in conjunction with the symmetric factored approximate inverse to solve the reduced order linear system. Furthermore, numerical results on the performance and convergence behavior for solving various model problems are presented. </jats:p
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
- …
