Coarsening Techniques Research Articles

Coarsening effects on k-partite network classification

The growing data size poses challenges for storage and computational processing time in semi-supervised models, making their practical application difficult; researchers have explored the use of reduced network versions as a potential solution. Real-world networks contain diverse types of vertices and edges, leading to using k-partite network representation. However, the existing methods primarily reduce uni-partite networks with a single type of vertex and edge. We develop a new coarsening method applicable to the k-partite networks that maintain classification performance. The empirical analysis of hundreds of thousands of synthetically generated networks demonstrates the promise of coarsening techniques in solving large networks’ storage and processing problems. The findings indicate that the proposed coarsening algorithm achieved significant improvements in storage efficiency and classification runtime, even with modest reductions in the number of vertices, leading to over one-third savings in storage and twice faster classifications; furthermore, the classification performance metrics exhibited low variation on average.

On adaptive mesh coarsening procedures for the virtual element method for two-dimensional elastic problems

In the context of adaptive remeshing, the virtual element method provides significant advantages over the finite element method. The attractive features of the virtual element method, such as the permission of arbitrary element geometries, and the seamless permission of ‘hanging’ nodes, have inspired many works concerning error estimation and adaptivity. However, these works have primarily focused on adaptive refinement techniques with little attention paid to adaptive coarsening (i.e. de-refinement) techniques that are required for the development of fully adaptive remeshing procedures. In this work novel indicators are proposed for the identification of patches/clusters of elements to be coarsened, along with a novel procedure to perform the coarsening. The indicators are computed over prospective patches of elements rather than on individual elements to identify the most suitable combinations of elements to coarsen. The coarsening procedure is robust and suitable for meshes of structured and unstructured/Voronoi elements. Numerical results demonstrate the high degree of efficacy of the proposed coarsening procedures and sensible mesh evolution during the coarsening process. It is demonstrated that critical mesh geometries, such as non-convex corners and holes, are preserved during coarsening, and that meshes remain fine in regions of interest to engineers, such as near singularities.

A Robust Grid‐Based Meshing Algorithm for Embedding Self‐Intersecting Surfaces

AbstractThe creation of a volumetric mesh representing the interior of an input polygonal mesh is a common requirement in graphics and computational mechanics applications. Most mesh creation techniques assume that the input surface is not self‐intersecting. However, due to numerical and/or user error, input surfaces are commonly self‐intersecting to some degree. The removal of self‐intersection is a burdensome task that complicates workflow and generally slows down the process of creating simulation‐ready digital assets. We present a method for the creation of a volumetric embedding hexahedron mesh from a self‐intersecting input triangle mesh. Our method is designed for efficiency by minimizing use of computationally expensive exact/adaptive precision arithmetic. Although our approach allows for nearly no limit on the degree of self‐intersection in the input surface, our focus is on efficiency in the most common case: many minimal self‐intersections. The embedding hexahedron mesh is created from a uniform background grid and consists of hexahedron elements that are geometrical copies of grid cells. Multiple copies of a single grid cell are used to resolve regions of self‐intersection/overlap. Lastly, we develop a novel topology‐aware embedding mesh coarsening technique to allow for user‐specified mesh resolution as well as a topology‐aware tetrahedralization of the hexahedron mesh.

Efficient simulation of the heat transfer in fused filament fabrication

Heat transfer simulations of the fused filament fabrication process are an important tool to predict bonding, residual stresses and strength of 3D printed parts. But in order to capture the significant thermal gradients that occur in the FFF printing process, a fine mesh discretization and short time steps are required, leading to extensive computational efforts. In this work a simulation framework is presented which combines several efficiency measures with the objective of reducing the computational efforts required in simulating the FFF printing process without simplifying the deposition physics or reducing the overall accuracy. Thus, the material deposition has been modeled with a hybrid element activation approach and elements are adaptively coarsened through an error-based coarsening condition. Additionally, an appropriate coarsening technique is presented for geometries with air-filled infill patterns. The accuracy of the numerical framework is experimentally validated and the efficiency of the framework is validated numerically by comparing the performance of models with and without any efficiency measures. Finally, its effectiveness is shown by simulating the printing process of a larger geometry.

Research on Light Extraction of GaN-based LED with Surface/Interface Coarsening

Light-emitting diodes (LEDs) are a highly versatile light source that have been widely employed in various fields. In order to maximize its quantum efficiency, it is essential to boost the efficiency of light extraction. Internal photon reflection can have a distinct negative impact on the luminous output power of GaN-based LEDs, which can be addressed by surface and interface texturing of different LED layers. This article discusses the mechanisms of light extraction losses in GaN-based LEDs and summarizes various surface and interface coarsening techniques proposed for improving the light extraction rate, including a range of physical and chemical etching methods.

Improving deep learning performance for predicting large-scale geological {{CO}_{2}} sequestration modeling through feature coarsening

Physics-based reservoir simulation for fluid flow in porous media is a numerical simulation method to predict the temporal-spatial patterns of state variables (e.g. pressure p) in porous media, and usually requires prohibitively high computational expense due to its non-linearity and the large number of degrees of freedom (DoF). This work describes a deep learning (DL) workflow to predict the pressure evolution as fluid flows in large-scale 3-dimensional(3D) heterogeneous porous media. In particular, we develop an efficient feature coarsening technique to extract the most representative information and perform the training and prediction of DL at the coarse scale, and further recover the resolution at the fine scale by spatial interpolation. We validate the DL approach to predict pressure field against physics-based simulation data for a field-scale 3D geologic CO_2 sequestration reservoir model. We evaluate the impact of feature coarsening on DL performance, and observe that the feature coarsening not only decreases the training time by >74% and reduces the memory consumption by >75%, but also maintains temporal error 0.63% on average. Besides, the DL workflow provides predictive efficiency with 1406 times speedup compared to physics-based numerical simulation. The key findings from this research significantly improve the training and prediction efficiency of deep learning model to deal with large-scale heterogeneous reservoir models, and thus it can also be further applied to accelerate workflows of history matching and reservoir optimization for close-loop reservoir management.

Multi-level adaptive mesh refinement technique for phase-field method

In order to solve the contradiction between the calculation efficiency and accuracy of the phase-field method (PFM), a multi-level adaptive mesh refinement technique suitable for PFM was developed and improved. A multi-level mesh refinement method of 2-D quadrilateral elements including three-layer transition elements without hanging nodes is designed, and the automatic refinement crack tip elements is realized by combining the phase field value. The local element coarsening technique is adopted to solve the problem of rapid elements number growth and reduce elements number in the calculation process. Combined with backtracking algorithm and “tree structure” node information storage technique, crack propagation is simulated. Through the benchmark example, the accuracy of the multi-level adaptive mesh refinement technique and the applicability to non-orthogonal mesh are verified, and the efficiency is greatly improved. For more complex crack propagation, the simulation results are also in good agreement with the experimental results. The critical gradient method proposed at last can find the refinement range more accurately and reduce elements number, which shows a great prospect in engineering applications.

Facilitating program comprehension with call graph multilevel hierarchical abstractions

Program comprehension is a fundamental prerequisite for software maintenance and evolution. In order to understand a software structure, developers often read its codebase or documentation—if available and not outdated. Both approaches are tedious, time-consuming, and inefficient. Recent methods and tools have emerged to facilitate program comprehension, such as static call graphs, which depict the structure of the software system as a directed graph. However, the usage of call graphs still faces two main challenges: (1) large call graphs can be difficult to understand, and (2) they are limited to a single level of granularity, such as function calls. In this paper, we introduce a coarsening technique to create multi-level, hierarchical representations of the call graph. Specifically, we propose a hierarchical clustering approach of the execution paths to visualize the call graph at different granularity levels and for different software units, including packages, classes, and functions. Our overarching goal is to assist software developers in understanding the software system from a high-level of abstraction to the low-level of implementation with the ability to focus on particular parts of the system individually. To validate our approach and tool support, we conducted a user study of 18 software engineers from more than 11 industries who carried out several tasks using our system and then answered a survey. The results demonstrate that our approach is feasible to automatically construct multi-level abstractions of the call graph and hierarchically cluster them into meaningful abstractions. A video demo of the tool is available at https://rakanalanazi.github.io/CodEx/.

An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids

In this article we design and analyze a class of two-level non-overlapping additive Schwarz preconditioners for the solution of the linear system of equations stemming from discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polytopic meshes. The preconditioner is based on a coarse space and a non-overlapping partition of the computational domain where local solvers are applied in parallel. In particular, the coarse space can potentially be chosen to be non-embedded with respect to the finer space; indeed it can be obtained from the fine grid by employing agglomeration and edge coarsening techniques. We investigate the dependence of the condition number of the preconditioned system with respect to the diffusion coefficient and the discretization parameters, i.e., the mesh size and the polynomial degree of the fine and coarse spaces. Numerical examples are presented which confirm the theoretical bounds.

Preconditioned Conjugate Gradient Methods for the Refined FEM Discretizations of Nearly Incompressible Elasticity Problems in Three Dimensions

Nearly incompressible problems in three dimensions are the important problems in practical engineering computation. The volume-locking phenomenon will appear when the commonly used finite elements such as linear elements are applied to the solution of these problems. There are many efficient approaches to overcome this locking phenomenon, one of which is the higher-order conforming finite element method. However, we often use the lower-order nonconforming elements as Wilson elements by considering the computational complexity for three-dimensional (3D) problems considered. In general, the convergence of Wilson elements will heavily rely on the quality of the meshes. It will greatly deteriorate or no longer converge when the mesh distortion is very large. In this paper, the refined element method based on Wilson element is first applied to solve nearly incompressible elasticity problems, and the influence of mesh quality on the refined element is tested numerically. Its validity is verified by some numerical examples. By using the internal condensation method, the refined element discrete system of equations is deduced into the one which is spectrally equivalent to an 8-node hexahedral element discrete system of equations. And then, a type of efficient algebraic multigrid (AMG) preconditioner is presented by combining both the coarsening techniques based on the distance matrix and the effective smoothing operators. The resulting preconditioned conjugate gradient (PCG) method is efficient for 3D nearly incompressible problems. The numerical results verify the efficiency and robustness of the proposed method.

Equivalent method of coarsening permeability in low permeability reservoirs

In view of considering reservoir segregation, it is preferred that the permeability coarsening method is of great importance. Based on the mesh coarsening technique, the conceptual model of low permeability reservoir segregation distribution is established, and different permeability coarsening methods are selected to simulate the daily production of oil field and other parameters and the error analysis. The most appropriate permeability equivalent method and the grid step size are selected at last. It is concluded that the harmonic-arithmetic mean method is the best method to simulate errors of the parameters, and it is also the best way to calculate the equivalent permeability in low permeability reservoir considering the reservoir segregation distribution. After the coarsening, the grid step on horizon is 20 m or so and on vertical is 4-6m.

Residual stress prediction in selective laser melting

This review focuses on the analysis of numerical models aimed at predicting the residual stress-strain field produced by the selective laser melting process. Our first intent is to favor an intuitive understanding of the underlying physics and then to provide an overview of the available simulation strategies specifying their field of application. In fact, given the complexity and the multi-scale nature of the process, various tailored models are needed for the assessment and prediction of defects and manufacturing issues arising during the building phase. Regarding the estimation of residual stresses, the available models were reviewed and classified on the basis of the dimensional scale of the simulated phenomena. Meso-scale models perform the detailed simulation of the scanning process, but the high computational cost currently prevents their application on the whole build volume (the current limit on the scanning volume is approximately 100 mm3 with dynamic mesh coarsening techniques). Macro-scale models have been developed to overcome this limit by introducing deep simplifications of the thermo-structural problem. From our review, it appears that meso-scale modeling has reached a significant maturity, while a widely adopted strategy for macro-scale simulations has not emerged yet.

Spectral coarsening of geometric operators

We introduce a novel approach to measure the behavior of a geometric operator before and after coarsening. By comparing eigenvectors of the input operator and its coarsened counterpart, we can quantitatively and visually analyze how well the spectral properties of the operator are maintained. Using this measure, we show that standard mesh simplification and algebraic coarsening techniques fail to maintain spectral properties. In response, we introduce a novel approach for spectral coarsening. We show that it is possible to significantly reduce the sampling density of an operator derived from a 3D shape without affecting the low-frequency eigenvectors. By marrying techniques developed within the algebraic multigrid and the functional maps literatures, we successfully coarsen a variety of isotropic and anisotropic operators while maintaining sparsity and positive semi-definiteness. We demonstrate the utility of this approach for applications including operatorsensitive sampling, shape matching, and graph pooling for convolutional neural networks.

Sampling and multilevel coarsening algorithms for fast matrix approximations

SummaryThis paper addresses matrix approximation problems for matrices that are large, sparse, and/or representations of large graphs. To tackle these problems, we consider algorithms that are based primarily on coarsening techniques, possibly combined with random sampling. A multilevel coarsening technique is proposed, which utilizes a hypergraph associated with the data matrix and a graph coarsening strategy based on column matching. We consider a number of standard applications of this technique as well as a few new ones. Among standard applications, we first consider the problem of computing partial singular value decomposition, for which a combination of sampling and coarsening yields significantly improved singular value decomposition results relative to sampling alone. We also consider the column subset selection problem, a popular low‐rank approximation method used in data‐related applications, and show how multilevel coarsening can be adapted for this problem. Similarly, we consider the problem of graph sparsification and show how coarsening techniques can be employed to solve it. We also establish theoretical results that characterize the approximation error obtained and the quality of the dimension reduction achieved by a coarsening step, when a proper column matching strategy is employed. Numerical experiments illustrate the performances of the methods in a few applications.

Multigrid for model reduction of power grid networks

SummaryThis paper presents a method for determining the relevant buses for reduced models of power grid networks described by systems of differential‐algebraic equations and for constructing the coarse‐grain dynamical power grid systems. To determine these buses, time integration of differential equations is not needed, but rather, a stationary system is analyzed. However, unlike stationary‐system approaches that determine only coarse generator buses by approximating the coherency of the generators, the proposed method analyzes the graph Laplacian associated with the admittance matrix. The buses for the reduced model are chosen to ensure that the graph Laplacian of the reduced model is an accurate approximation to the graph Laplacian of the full system. Both load and generator buses can be selected by this procedure since the Laplacian is defined on all the buses. The basis of this proposed approach lies in the close relationship between the synchrony of the system and the spectral properties of this Laplacian, that is, conditions on the spectrum of this Laplacian that almost surely guarantee the synchrony of the system. Thus, assuming that the full system is in synchrony, our strategy is to coarsen the full‐system Laplacian such that the coarse Laplacian possesses good approximation to these spectral conditions. Accurate approximation to these conditions then can better lead to synchronous reduced models. The coarsened Laplacian is defined on coarse degrees of freedom (DOFs), which are associated with the relevant buses to include in the reduced model. To realize this coarse DOF selection, we use multigrid coarsening techniques based on compatible relaxation. Multigrid is the natural choice since it has been extensively used to coarsen Laplacians arising from discretizations of elliptic partial differential equations and is actively being extended to graph Laplacians. With the selection of the buses for the reduced model, the reduced model is completed by constructing the coarse admittance matrix values and other physical parameters using standard power grid techniques or by using the intergrid operators constructed in the coarse DOFs selection process. Unfortunately, the selection of the coarse buses and the coarsening of the admittance matrix and physical parameters are not sufficient by themselves to produce a stable reduced system. To achieve a stable system, system structures of the fine‐grain model must be preserved in the reduced model. We analyze this to develop a multigrid methodology for constructing stable reduced models of power grid systems. Numerical examples are presented to validate this methodology.

Preconditioning Methods for the Penalty Function FEM Discretizations of 3D Nearly Incompressible Elasticity Problems

Penalty function finite element method (FEM) is an efficient method for the solution of the nearly incompressible elasticity problems in three dimensions. In order to improve the overall efficiency, some faster solvers for the corresponding FEM system of equations must be presented. Since the resulting system of equations is symmetric positive definite and highly ill-conditioned, the preconditioned conjugate gradient (PCG) method is one of the most efficient methods for solving such FEM equations. In this paper, we have first presented some types of efficient PCG methods including [Formula: see text]-PCG and [Formula: see text]-PCG with two different block diagonal inverses as a preconditioner, respectively, and including the RS-GAMG-PCG based on the global matrix for the penalty function FEM equations of three-dimensional (3D) nearly incompressible elasticity problems with mixed boundary conditions. Furthermore, we apply these methods to the solution of the penalty function quadratic discretizations for the cantilever and the Cook’s membrane problems. The numerical results have shown that much more efficient PCG methods for 3D nearly incompressible problems can be obtained by using the known information that is easily available in most FEM applications, for instance, the type of the partial differential equations (PDEs) considered and the number of physical unknowns residing in each grid, and by combining the coarsening techniques used in the classic algebraic multigrid (AMG) method. This will greatly improve the overall efficiency of the penalty function FEM analysis.

A multilevel ACO approach for solving forest transportation planning problems with environmental constraints

Abstract This paper presents a multilevel ant colony optimization (MLACO) approach to solve constrained forest transportation planning problems (CFTPPs). A graph coarsening technique is used to coarsen a network representing the problem into a set of increasingly coarser level problems. Then, a customized ant colony optimization (ACO) algorithm is designed to solve the CFTPP from coarser to finer level problems. The parameters of the ACO algorithm are automatically configured by evaluating a parameter combination domain through each level of the problem. The solution obtained by the ACO for the coarser level problems is projected into finer level problem components, which are used to help the ACO search for finer level solutions. The MLACO was tested on 20 CFTPPs and solutions were compared to those obtained from other approaches including a mixed integer programming (MIP) solver, a parameter iterative local search (ParamILS) method, and an exhaustive ACO parameter search method. Experimental results showed that the MLACO approach was able to match solution qualities and reduce computing time significantly compared to the tested approaches.

Details of tetrahedral anisotropic mesh adaptation

We have implemented tetrahedral anisotropic mesh adaptation using the local operations of coarsening, swapping, refinement and smoothing in MATLAB without the use of any for-N loops, i.e. the script is fully vectorised. In the process of doing so, we have made three observations related to details of the implementation: 1. restricting refinement to a single edge split per element not only simplifies the code, it also improves mesh quality, 2. face to edge swapping is unnecessary, and 3. optimising for the Vassilevski functional tends to give a little higher value for the mean condition number functional than optimising for the condition number functional directly. These observations have been made for a uniform and a radial shock metric field, both starting from a structured mesh in a cube. Finally, we compare two coarsening techniques and demonstrate the importance of applying smoothing in the mesh adaptation loop. The results pertain to a unit cube geometry, but we also show the effect of corners and edges by applying the implementation in a spherical geometry. Program summaryProgram title: trullekrul v0.1.0Catalogue identifier: AEZC_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZC_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: New style BSD licenseNo. of lines in distributed program, including test data, etc.: 8395No. of bytes in distributed program, including test data, etc.: 92021Distribution format: tar.gzProgramming language: MATLAB/GNU Octave.Computer: Single core desktops.Operating system: Linux/Windows/Mac.RAM: Variable, from megabytes to gigabytesClassification: 4.2, 4.3, 4.12, 4.14.Nature of problem: Adaptation of tetrahedral meshes to metric fields.Solution method: Local operations, colouring and APL vectorised code.Restrictions: Region IDs/internal boundaries are not supported and 3D adaptation with 1M elements requires ≈4GB memory.Unusual features: Vectorised script implementation, signed distance functions, selection of refinement and coarsening techniques.Running time: 2D problems solve in seconds, but 3D problems can take up to an hour without running out of memory.

Adaptive refinement of hierarchical B‐spline finite elements with an efficient data transfer algorithm

SummaryWe present an adaptive B‐spline FEM that employs hierarchical refinement and coarsening techniques. Hierarchical B‐spline refinement augments the approximation space through the use of finer, more compactly supported B‐splines. With this approach compatibility is automatic because problematic T‐vertices do not arise. The process of refinement does not change the regularity of the basis functions, and can be easily generalized to arbitrary dimensions. Data transfer during both refinement and coarsening is addressed. The techniques are then applied to several time‐dependent problems for which adaptivity is advantageous, and the performance of the method is evaluated. Copyright © 2014 John Wiley & Sons, Ltd.

Investigation of Multigrid Preconditioner for Integral Equation Fast Analysis of Electromagnetic Scattering Problems

This paper proposes a multigrid preconditioner for integral equation fast analysis of electromagnetic scattering problems. The multigrid model is constructed by recursively dividing every triangular element on the coarse level into 4 small ones. The Rao-Wilton-Glisson (RWG) basis functions defined on the coarse level are expressed with linear combinations of those on the fine level, and the expansion coefficients of which are used to construct the restriction and interpolation operator. To further accelerate the convergence of iterative solution, a set of solenoidal basis functions on the coarse level are constructed additionally, which are also expressed with linear combinations of RWG basis functions on the fine level and adopted to construct the second restriction and interpolation operator. The two sets of coarsening techniques are then combined together to remove the low-frequency components of the iterative error. Numerical results show that the proposed method can reduce both the steps of iteration and CPU time compared with the iterative solver without preconditioning.