Sanderson L. Gonzaga de Oliveira

A Systematic Review of Heuristics for Profile Reduction of Symmetric Matrices

In this work, a systematic review of heuristics for profile reduction of symmetric matrices is presented. 74 heuristics tested for reduction profile were found. Researchers compared results of their heuristics with results of other heuristics. In this review, these comparisons were analyzed and 8 heuristics were identified as the possible best for the problem. In addition, exchange methods, a form of local search, were identified that can benefit heuristics identified as the best ones for the task.

Metaheuristic-based Heuristics for Symmetric-matrix Bandwidth Reduction: A Systematic Review☆

Computational and storage costs of resolution of large sparse linear systems Ax = b can be performed by reducing the bandwidth of A. Bandwidth reduction consists of carrying out permutations of lines and columns so that they allow coefficients to remain near the main diagonal. When considering an adjacency matrix of a graph, bandwidth reduction can be considered in the sense of modifying the order in which the graph vertices are numbered. Heuristics for bandwidth reduction are revised in this study, aiming at determining which of them offers the higher bandwidth reduction with a reasonable computational cost. Specifically, metaheuristic-based heuristics are reviewed in this systematic review. Moreover, 29 metaheuristic-based heuristics tested for bandwidth reduction were found. Among them, 4 are recommended as possible state-of-the-art heuristics for addressing the problem.

Convergence analysis of the Hopmoc method

The Hopmoc method combines concepts of the modified method of characteristics (MMOC) and the Hopscotch method. First, Hopmoc resembles Hopscotch because it decomposes the set of grid points into two subsets. Namely, both subsets have their unknowns separately updated within one semi-step. Furthermore, each subset undergoes one explicit and one implicit update of its unknowns in order to lead to a symmetrical procedure. Such decomposition inspired the use of a convergence analysis similar to the one used in alternating direction implicit methods. Secondly, the steps are evaluated along characteristic lines in a semi-Lagrangian approach similar to the MMOC. In this work, both consistency and stability analysis are discussed for Hopmoc applied to a convection–diffusion equation. The analysis produces sufficient conditions for the consistency analysis and proves that the Hopmoc method presents unconditional stability. In addition, numerical results confirm the conducted convergence analysis.

A New Heuristic for Bandwidth and Profile Reductions of Matrices Using a Self-organizing Map

In this work, a heuristic for bandwidth and profile reductions of symmetric and asymmetric matrices using a one-dimensional self-organizing map is proposed. Experiments and comparisons of results obtained were performed in relation to results of the Variable neighborhood search for bandwidth reduction. Simulations with these two heuristics were performed with 113 instances of the Harwell-Boeing sparse matrix collection and with 2 sets of instances with linear systems composed of sparse symmetric positive-definite matrices. The linear systems were solved using the Jacobi-preconditioned Conjugate Gradient Method. According to the results presented here, the best heuristic in the simulations performed was the Variable neighborhood search for bandwidth reduction. On the other hand, when the vertices of the corresponding graph were originally ordered in a sequence given by a space-filling curve, no gain was obtained when applying a heuristic for reordering the graph vertices.

A Novel Approach to the Weighted Laplacian Formulation Applied to 2D Delaunay Triangulations

In this work, a novel smoothing method based on weighted Laplacian formulation is applied to resolve the heat conduction equation by finite-volume discretizations with Voronoi diagram. When a minimum number of vertices is obtained, the mesh is smoothed by means of a new approach to the weighted Laplacian formulation. The combination of techniques allows to solve the resulting linear system by the Conjugate Gradient Method. The new approach to the weighted Laplacian formulation within the set of techniques is compared to other 4 approaches to the weighted Laplacian formulation. Comparative analysis of the results shows that the proposed approach allows to maintain the approximation and presents smaller number of vertices than any of the other 4 approaches. Thus, the computational cost of the resolution is lower when using the proposed approach than when applying any of the other approaches and it is also lower than using only Delaunay refinements.

An evaluation of four reordering algorithms to reduce the computational cost of the Jacobi-preconditioned conjugate gradient method using high-precision arithmetic

In this work, four heuristics for bandwidth and profile reductions are evaluated. Specifically, the results of a recent proposed heuristic for bandwidth and profile reductions of symmetric and asymmetric matrices using a one-dimensional self-organising map is evaluated against the results obtained from the variable neighbourhood search for bandwidth reduction heuristic, the original reverse Cuthill-McKee method, and the reverse Cuthill-McKee method with starting pseudo-peripheral vertex given by the George-Liu algorithm. These four heuristics were applied to three datasets of linear systems composed of sparse symmetric positive-definite matrices arising from discretisations of the heat conduction and Laplace equations by finite volumes. The linear systems are solved by the Jacobi-preconditioned conjugate gradient method when using high-precision numerical computations. The best heuristic in the simulations performed with one of the datasets used was the Cuthill-McKee method with starting pseudo-peripheral vertex given by the George-Liu algorithm. On the other hand, no gain was obtained in relation to the computational cost of the linear system solver when a heuristic for bandwidth and profile reduction is applied to instances contained in two of the datasets used.

Autonomous Leaves Graph Applied to the Boundary Layer Problem

In physics and fluid mechanics, the boundary layer is the fluid layer in the immediate vicinity of a bounding surface. It is important in many aerodynamic problems. This work presents a numerical simulation of the bidimensional laminar boundary-layer problem considering a steady incompressible flow with no-slip condition on the surface by Autonomous Leaves Graph based on finite volume discretizations. In addition, a Modified Hilbert Curve numbers the control volumes. Initially, the numerical solution of the flat-plate problem is compared to its analytical solution, namely Blasius Solution. Secondly, simulations of the flow along a NACA airfoil shape are presented. Computer experiments show that an adaptive mesh refinement using the Autonomous Leaves Graph with the Modified Hilbert Curve numbering is appropriate for a aerodynamic problem. Finally, results illustrate that the method provides a good trade-off between speed and accuracy.

Tuning Up TVD HOPMOC Method on Intel MIC Xeon Phi Architectures with Intel Parallel Studio Tools

This paper focuses on the parallelization of TVD Method scheme for numerical time integration of evolutionary differential equations. The Hopmoc method for numerical integration of differential equations was developed aiming at benefiting from both the concept of integration along characteristic lines as well as from the spatially decomposed Hopscotch method. The set of grid points is initially decomposed into two subsets during the implementation of the integration step. Then, two updates are performed, one explicit and one implicit, on each variable in the course of the iterative process. Each update requires an integration semi step. This is carried out along characteristic lines in a Semi-Lagrangian scheme based on the Modified Method of Characteristics. This work analises two strategies to implement the parallel version of TVD Hopmoc based on the analysis performed by Intel Tools such Parallel and Threading Advisor. A naive solution is substituted by a chunk loop strategy in order to avoid fine-grain tasks inside main loops.

An Analysis of Reordering Algorithms to Reduce the Computational Cost of the Jacobi-Preconditioned CG Solver Using High-Precision Arithmetic

Several heuristics for bandwidth and profile reductions have been proposed since the 1960s. In systematic reviews, 133 heuristics applied to these problems have been found. The results of these heuristics have been analyzed so that, among them, 13 were selected in a manner that no simulation or comparison showed that these algorithms could be outperformed by any other algorithm in the publications analyzed, in terms of bandwidth or profile reductions and also considering the computational costs of the heuristics. Therefore, these 13 heuristics were selected as the most promising low-cost methods to solve these problems. Based on this experience, this article reports that in certain cases no heuristic for bandwidth or profile reduction can reduce the computational cost of the Jacobi-preconditioned Conjugate Gradient Method when using high-precision numerical computations.

Autonomous leaves graph applied to the simulation of the boundary layer around a non-symmetric NACA airfoil

The Boundary Layer is a fluid layer in the neighborhood of a surface. It is important in many disciplines related to Physics and Fluid Mechanics, which aerodynamics is an example. This paper presents a 2D numerical simulation of this problem considering an incompressible laminar flux in steady state with non-slip condition. An adaptive mesh refinement is carried out by the Autonomous Leaves Graph with finite volume discretizations. The Modified Hilbert Curve is implemented to traverse and provide the total ordering of the finite volumes. Flux simulations are presented around a non-symmetric airfoil NACA2415 shape. The results show evidences that the adaptive mesh refinement scheme is adequate for numerical solution of this type of problem.