Coarsening Scheme Research Articles

Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture

SUMMARYAdaptive mesh refinement and coarsening schemes are proposed for efficient computational simulation of dynamic cohesive fracture. The adaptive mesh refinement consists of a sequence of edge‐split operators, whereas the adaptive mesh coarsening is based on a sequence of vertex‐removal (or edge‐collapse) operators. Nodal perturbation and edge‐swap operators are also employed around the crack tip region to improve crack geometry representation, and cohesive surface elements are adaptively inserted whenever and wherever they are needed by means of an extrinsic cohesive zone model approach. Such adaptive mesh modification events are maintained in conjunction with a topological data structure (TopS). The so‐called PPR potential‐based cohesive model (J. Mech. Phys. Solids 2009; 57:891–908) is utilized for the constitutive relationship of the cohesive zone model. The examples investigated include mode I fracture, mixed‐mode fracture and crack branching problems. The computational results using mesh adaptivity (refinement and coarsening) are consistent with the results using uniform mesh refinement. The present approach significantly reduces computational cost while exhibiting a multiscale effect that captures both global macro‐crack and local micro‐cracks. Copyright © 2012 John Wiley & Sons, Ltd.

Improving multigrid performance for unstructured mesh drift–diffusion simulations on 147,000 cores

SUMMARYThis study considers the scaling of three algebraic multigrid aggregation schemes for a finite element discretization of a drift–diffusion system, specifically the drift–diffusion model for semiconductor devices. The approach is more general and can be applied to other systems of partial differential equations. After discretization on unstructured meshes, a fully coupled multigrid preconditioned Newton–Krylov solution method is employed. The choice of aggregation scheme for generating coarser levels has a significant impact on the performance and scalability of the multigrid preconditioner. For the test cases considered, the uncoupled aggregation scheme, which aggregates/combines the immediate neighbors, followed by repartitioning and data redistribution for the coarser level matrices on a subset of the Message Passing Interface (MPI) processes, outperformed the two other approaches, including the baseline aggressive coarsening scheme. Scaling results are presented up to 147,456 cores on an IBM Blue Gene/P platform. A comparison of the scaling of a multigrid V‐cycle and W‐cycle is provided. Results for 65,536 cores demonstrate that a factor of 3.5 × reduction in time between the uncoupled aggregation and baseline aggressive coarsening scheme can be obtained by significantly reducing the iteration count due to the increased number of multigrid levels and the generation of better quality aggregates. Copyright © 2012 John Wiley & Sons, Ltd.

A New Coarsening Strategy for Multilevel Graph Partitioning

When applying multilevel scheme to solve the graph partitioning problem, shortcomings and limitations exist in the state-of-the-art coarsening schemes depend mainly on finding maximal matchings to obtain the coarse graphs, which can cause the multilevel algorithms to produce poor-quality solutions. This paper proposes an improved coarsening scheme by improving vertex combining strategy and edge ordering criteria. The new coarsening scheme is more effective in quality, which is proved by both theoretical analysis and experimental results.

Image Segmentation by Probabilistic Bottom-Up Aggregation and Cue Integration

We present a bottom-up aggregation approach to image segmentation. Beginning with an image, we execute a sequence of steps in which pixels are gradually merged to produce larger and larger regions. In each step, we consider pairs of adjacent regions and provide a probability measure to assess whether or not they should be included in the same segment. Our probabilistic formulation takes into account intensity and texture distributions in a local area around each region. It further incorporates priors based on the geometry of the regions. Finally, posteriors based on intensity and texture cues are combined using “a mixture of experts” formulation. This probabilistic approach is integrated into a graph coarsening scheme, providing a complete hierarchical segmentation of the image. The algorithm complexity is linear in the number of the image pixels and it requires almost no user-tuned parameters. In addition, we provide a novel evaluation scheme for image segmentation algorithms, attempting to avoid human semantic considerations that are out of scope for segmentation algorithms. Using this novel evaluation scheme, we test our method and provide a comparison to several existing segmentation algorithms.

The university of Florida sparse matrix collection

We describe the University of Florida Sparse Matrix Collection, a large and actively growing set of sparse matrices that arise in real applications. The Collection is widely used by the numerical linear algebra community for the development and performance evaluation of sparse matrix algorithms. It allows for robust and repeatable experiments: robust because performance results with artificially generated matrices can be misleading, and repeatable because matrices are curated and made publicly available in many formats. Its matrices cover a wide spectrum of domains, include those arising from problems with underlying 2D or 3D geometry (as structural engineering, computational fluid dynamics, model reduction, electromagnetics, semiconductor devices, thermodynamics, materials, acoustics, computer graphics/vision, robotics/kinematics, and other discretizations) and those that typically do not have such geometry (optimization, circuit simulation, economic and financial modeling, theoretical and quantum chemistry, chemical process simulation, mathematics and statistics, power networks, and other networks and graphs). We provide software for accessing and managing the Collection, from MATLAB™, Mathematica™, Fortran, and C, as well as an online search capability. Graph visualization of the matrices is provided, and a new multilevel coarsening scheme is proposed to facilitate this task.

A comparative study of three-dimensional hydraulic conductivity upscaling at the macro-dispersion experiment (MADE) site, Columbus Air Force Base, Mississippi (USA)

Simple averaging, simple-Laplacian, Laplacian-with-skin, and non-uniform coarsening are the techniques investigated in this comparative study of three-dimensional hydraulic conductivity upscaling. The reference is a fine scale conditional realization of the hydraulic conductivities at the MAcro-Dispersion Experiment site on Columbus Air Force Base in Mississippi (USA). This realization was generated using a hole-effect variogram model and it was shown that flow and transport modeling in this realization (at this scale) can reproduce the observed non-Fickian spreading of the tritium plume. The purpose of this work is twofold, first to compare the effectiveness of different upscaling techniques in yielding upscaled models able to reproduce the observed transport behavior, and second to demonstrate and analyze the conditions under which flow upscaling can provide a coarse model in which the standard advection–dispersion equation can be used to model transport in seemingly non-Fickian scenarios. Specifically, the use of Laplacian-based upscaling technique coupled with a non-uniform coarsening scheme yields the best results both in terms of flow and transport reproduction, for this case study in which the coarse blocks are smaller than the correlation ranges of the fine scale conductivities.

Truly monolithic algebraic multigrid for fluid–structure interaction

AbstractThe coupling of flexible structures to incompressible fluids draws a lot of attention during the last decade. Many different solution schemes have been proposed. In this contribution, we concentrate on the strong coupling fluid–structure interaction by means of monolithic solution schemes. Therein, a Newton–Krylov method is applied to the monolithic set of nonlinear equations. Such schemes require good preconditioning to be efficient. We propose two preconditioners that apply algebraic multigrid techniques to the entire fluid–structure interaction system of equations. The first is based on a standard block Gauss–Seidel approach, where approximate inverses of the individual field blocks are based on a algebraic multigrid hierarchy tailored for the type of the underlying physical problem. The second is based on a monolithic coarsening scheme for the coupled system that makes use of prolongation and restriction projections constructed for the individual fields. The resulting nonsymmetric monolithic algebraic multigrid method therefore involves coupling of the fields on coarse approximations to the problem yielding significantly enhanced performance. Copyright © 2010 John Wiley & Sons, Ltd.

Adaptive triangular mesh coarsening with centroidal Voronoi tessellations

We present a novel algorithm for adaptive triangular mesh coarsening. The algorithm has two stages. First, the input triangular mesh is refined by iteratively applying the adaptive subdivision operator that performs a so-called red-green split. Second, the refined mesh is simplified by a clustering algorithm based on centroidal Voronoi tessellations (CVTs). The accuracy and good quality of the output triangular mesh are achieved by combining adaptive subdivision and the CVTs technique. Test results showed the mesh coarsening scheme to be robust and effective. Examples are shown that validate the method.

LibMesh : a C++ library for parallel adaptive mesh refinement/coarsening simulations

In this paper we describe the libMesh (http://libmesh.sourceforge.net) framework for parallel adaptive finite element applications. libMesh is an open-source software library that has been developed to facilitate serial and parallel simulation of multiscale, multiphysics applications using adaptive mesh refinement and coarsening strategies. The main software development is being carried out in the CFDLab (http://cfdlab.ae.utexas.edu) at the University of Texas, but as with other open-source software projects; contributions are being made elsewhere in the US and abroad. The main goals of this article are: (1) to provide a basic reference source that describes libMesh and the underlying philosophy and software design approach; (2) to give sufficient detail and references on the adaptive mesh refinement and coarsening (AMR/C) scheme for applications analysts and developers; and (3) to describe the parallel implementation and data structures with supporting discussion of domain decomposition, message passing, and details related to dynamic repartitioning for parallel AMR/C. Other aspects related to C++ programming paradigms, reusability for diverse applications, adaptive modeling, physics-independent error indicators, and similar concepts are briefly discussed. Finally, results from some applications using the library are presented and areas of future research are discussed.

Random and fuzzy sets in coarse data analysis

The theoretical aspects of statistical inference with imprecise data, with focus on random sets, are considered. On the setting of coarse data analysis imprecision and randomness in observed data are exhibited, and the relationship between probability and other types of uncertainty, such as belief functions and possibility measures, is analyzed. Coarsening schemes are viewed as models for perception-based information gathering processes in which random fuzzy sets appear naturally. As an implication, fuzzy statistics is statistics with fuzzy data. That is, fuzzy sets are a new type of data and as such, complementary to statistical analysis in the sense that they enlarge the domain of applications of statistical science.

Coarse grid classification: a parallel coarsening scheme for algebraic multigrid methods

AbstractIn this paper, we present a new approach to the parallelization of algebraic multigrid (AMG), i.e. to the parallel coarse‐grid selection in AMG. Our approach involves (almost) no special treatment of processor subdomain boundaries and hence avoids a number of drawbacks of other AMG parallelization techniques. The key idea is to select an appropriate (local) coarse grid on each processor from a set of admissible grids such that the composed coarse grid forms a suitable coarse grid for the whole domain, i.e. there is no need for a special boundary treatment. To this end, we first construct multiple equivalent coarse grids on each processor subdomain. In a second step we then select exactly one grid per processor by a graph clustering technique. The results of our numerical experiments clearly indicate that this approach results in coarse grids of high quality which are very close to those obtained with sequential AMG. Furthermore, the operator and grid complexities of our parallel AMG are mostly smaller than those obtained by other parallel AMG methods, whereas the scale‐up behaviour of the proposed algorithm is similar to that of other parallel AMG techniques. However a significant improvement with respect to the speed‐up performance is achieved. Copyright © 2006 John Wiley & Sons, Ltd.

Reducing Complexity in Parallel Algebraic Multigrid Preconditioners

Algebraic multigrid (AMG) is a very efficient iterative solver and preconditioner for large unstructured sparse linear systems. Traditional coarsening schemes for AMG can, however, lead to computational complexity growth as problem size increases, resulting in increased memory use and execution time, and diminished scalability. Two new parallel AMG coarsening schemes are proposed that are based solely on enforcing a maximum independent set property, resulting in sparser coarse grids. The new coarsening techniques remedy memory and execution time complexity growth for various large three-dimensional (3D) problems. If used within AMG as a preconditioner for Krylov subspace methods, the resulting iterative methods tend to converge fast. This paper discusses complexity issues that can arise in AMG, describes the new coarsening schemes, and examines the performance of the new preconditioners for various large 3D problems.

Real-time optimal adaptation for planetary geometry and texture: 4-8 tile hierarchies

The real-time display of huge geometry and imagery databases involves view-dependent approximations, typically through the use of precomputed hierarchies that are selectively refined at runtime. A classic motivating problem is terrain visualization in which planetary databases involving billions of elevation and color values are displayed on PC graphics hardware at high frame rates. This paper introduces a new diamond data structure for the basic selective-refinement processing, which is a streamlined method of representing the well-known hierarchies of right triangles that have enjoyed much success in real-time, view-dependent terrain display. Regular-grid tiles are proposed as the payload data per diamond for both geometry and texture. The use of 4-8 grid refinement and coarsening schemes allows level-of-detail transitions that are twice as gradual as traditional quadtree-based hierarchies, as well as very high-quality low-pass filtering compared to subsampling-based hierarchies. An out-of-core storage organization is introduced based on Sierpinski indices per diamond, along with a tile preprocessing framework based on fine-to-coarse, same-level, and coarse-to-fine gathering operations. To attain optimal frame-to-frame coherence and processing-order priorities, dual split and merge queues are developed similar to the Realtime Optimally Adapting Meshes (ROAM) Algorithm, as well as an adaptation of the ROAM frustum culling technique. Example applications of lake-detection and procedural terrain generation demonstrate the flexibility of the tile processing framework.

Fuzzy and random sets

The two distinct concepts of fuzzy and random sets coexist to form the basics for modeling perception-based information gathering processes in artificial intelligence. These processes, viewed as coarsening schemes, provide also justifications for various types of uncertainty measures such as belief functions and possibility measures.

A Multilevel Genetic Algorithm for the Traveling Salesman Problem

Abstract In this paper, we introduce a multilevel genetic algorithm for the traveling salesman problem. The resulting new algorithm starts by coarsening the original problem into a sequence of smaller problems using a coarsening scheme. Thereafter an initial feasible population is generated at the smallest problem and this population is projected back to the original problem by going through a refinement phase at each intermediate level using a simple genetic algorithm. Experimental results indicate that the combination of the genetic algorithm with the multilevel paradigm produces tours of better quality than those produced by the standard genetic algorithm.

BoomerAMG: A parallel algebraic multigrid solver and preconditioner

Driven by the need to solve linear systems arising from problems posed on extremely large, unstructured grids, there has been a recent resurgence of interest in algebraic multigrid (AMG). AMG is attractive in that it holds out the possibility of multigrid-like performance on unstructured grids. The sheer size of many modern physics and simulation problems has led to the development of massively parallel computers, and has sparked much research into developing algorithms for them. Parallelizing AMG is a difficult task, however. While much of the AMG method parallelizes readily, the process of coarse-grid selection, in particular, is fundamentally sequential in nature. We have previously introduced a parallel algorithm [A.J. Cleary, R.D. Falgout, V.E. Henson, J.E. Jones, in: Proceedings of the Fifth International Symposium on Solving Irregularly Structured Problems in Parallel, Springer, New York, 1998] for the selection of coarse-grid points, based on modifications of certain parallel independent set algorithms and the application of heuristics designed to insure the quality of the coarse grids, and shown results from a prototype serial version of the algorithm. In this paper we describe an implementation of a parallel AMG code, using the algorithm of A.J. Cleary, R.D. Falgout, V.E. Henson, J.E. Jones [in: Proceedings of the Fifth International Symposium on Solving Irregularly Structured Problems in Parallel, Springer, New York, 1998] as well as other approaches to parallelizing the coarse-grid selection. We consider three basic coarsening schemes and certain modifications to the basic schemes, designed to address specific performance issues. We present numerical results for a broad range of problem sizes and descriptions, and draw conclusions regarding the efficacy of the method. Finally, we indicate the current directions of the research.

Coarse-Grid Correction for Nonelliptic and Singular Perturbation Problems

The suitability of coarse-grid correction operators for multigrid solution of discretized nonelliptic partial differential equations is studied. It is found that the requirements for the orders of intergrid transfers and the order of approximation of the coarse-grid operators differ in an important way from the well-known "rules" pertaining to elliptic operators. Both Galerkin coarsening and coarsening schemes based on the differential operator are examined, and new coarse-grid correction strategies are suggested in light of this analysis.

Multilevelk-way Partitioning Scheme for Irregular Graphs

In this paper, we present and study a class of graph partitioning algorithms that reduces the size of the graph by collapsing vertices and edges, we find ak-way partitioning of the smaller graph, and then we uncoarsen and refine it to construct ak-way partitioning for the original graph. These algorithms compute ak-way partitioning of a graphG= (V,E) inO(|E|) time, which is faster by a factor ofO(logk) than previously proposed multilevel recursive bisection algorithms. A key contribution of our work is in finding a high-quality and computationally inexpensive refinement algorithm that can improve upon an initialk-way partitioning. We also study the effectiveness of the overall scheme for a variety of coarsening schemes. We present experimental results on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that this new scheme produces partitions that are of comparable or better quality than those produced by the multilevel bisection algorithm and requires substantially smaller time. Graphs containing up to 450,000 vertices and 3,300,000 edges can be partitioned in 256 domains in less than 40 s on a workstation such as SGI's Challenge. Compared with the widely used multilevel spectral bisection algorithm, our new algorithm is usually two orders of magnitude faster and produces partitions with substantially smaller edge-cut.

Adaptive grids for coupled viscous flow and transport

An adaptive grid refinement and coarsening scheme in two dimensions for biquadratic elements is developed. This has been incorporated in a two-dimensional finite element program for steady and transient generalized Newtonian flow problems. Numerical results are given for Navier-Stokes and power law non-Newtonian flow calculations as well as coupled fluid flow and transport problems including free surfaces, thermocapillary flows, die swell and electro-rheological flows.