Coarsening Operations Research Articles

Automatic and User-Assisted Sphere-Mesh Construction.

A sphere-mesh is a class of geometric proxies defined as the volume swept by spheres with linearly interpolated centers and radii, potentially striking a good balance between conciseness of representation, simplicity of spatial queries, and expressive power. We investigate the semiautomatic generation of sphere-meshes from standard triangular meshes. We improve one known automatic construction algorithm, based on iterative local coarsening operation, by introducing a mechanism to prevent operations that would result in spheres exceeding the target shape; then, we propose a 3-D interface designed to permit users to easily and intuitively modify the automatically generated sphere-meshes. The two phases, an improved automatic algorithm and a novel interactive tool, used in cascade, constitute a viable semiautomatic way to produce high-quality sphere-meshes. We test our method on several inputs tri-meshes, assess their quality, and finally exemplify the usability of our results by testing them in a few downstream applications.

The Squish Map and the $SL_2$ Double Dimer Model

A plane partition, whose 3D Young diagram is made of unit cubes, can be approximated by a "coarser" plane partition, made of cubes of side length 2. Indeed, there are two such approximations obtained by "rounding up" or "rounding down" to the nearest cube. We relate this coarsening (or downsampling) operation to the squish map introduced by the second author in earlier work. We exhibit a related measure-preserving map between the dimer model on the honeycomb graph, and the SL2 double dimer model on a coarser honeycomb graph; we compute the most interesting special case of this map, related to plane partition q-enumeration with 2-periodic weights. As an application, we specialize the weights to be certain roots of unity, obtain novel generating functions (some known, some new, and some conjectural) that (-1)-enumerate certain classes of pairs of plane partitions according to how their dimer configurations interact.

ResGEM: Multi-scale Graph Embedding Network for Residual Mesh Denoising.

Mesh denoising is a crucial technology that aims to recover a high-fidelity 3D mesh from a noise-corrupted one. Deep learning methods, particularly graph convolutional networks (GCNs) based mesh denoisers, have demonstrated their effectiveness in removing various complex real-world noises while preserving authentic geometry. However, it is still a quite challenging work to faithfully regress uncontaminated normals and vertices on meshes with irregular topology. In this paper, we propose a novel pipeline that incorporates two parallel normal-aware and vertex-aware branches to achieve a balance between smoothness and geometric details while maintaining the flexibility of surface topology. We introduce ResGEM, a new GCN, with multi-scale embedding modules and residual decoding structures to facilitate normal regression and vertex modification for mesh denoising. To effectively extract multi-scale surface features while avoiding the loss of topological information caused by graph pooling or coarsening operations, we encode the noisy normal and vertex graphs using four edge-conditioned embedding modules (EEMs) at different scales. This allows us to obtain favorable feature representations with multiple receptive field sizes. Formulating the denoising problem into a residual learning problem, the decoder incorporates residual blocks to accurately predict true normals and vertex offsets from the embedded feature space. Moreover, we propose novel regularization terms in the loss function that enhance the smoothing and generalization ability of our network by imposing constraints on normal consistency. Comprehensive experiments have been conducted to demonstrate the superiority of our method over the state-of-the-art on both synthetic and real-scanned datasets.

Multiscale graph neural network autoencoders for interpretable scientific machine learning

The goal of this work is to address two limitations in autoencoder-based models: latent space interpretability and compatibility with unstructured meshes. This is accomplished here with the development of a novel graph neural network (GNN) autoencoding architecture with demonstrations on complex fluid flow applications. To address the first goal of interpretability, the GNN autoencoder achieves reduction in the number nodes in the encoding stage through an adaptive graph reduction procedure. This reduction procedure essentially amounts to flowfield-conditioned node sampling and sensor identification, and produces interpretable latent graph representations tailored to the flowfield reconstruction task in the form of so-called masked fields. These masked fields allow the user to (a) visualize where in physical space a given latent graph is active, and (b) interpret the time-evolution of the latent graph connectivity in accordance with the time-evolution of unsteady flow features (e.g. recirculation zones, shear layers) in the domain. To address the goal of unstructured mesh compatibility, the autoencoding architecture utilizes a series of multi-scale message passing (MMP) layers, each of which models information exchange among node neighborhoods at various lengthscales. The MMP layer, which augments standard single-scale message passing with learnable coarsening operations, allows the decoder to more efficiently reconstruct the flowfield from the identified regions in the masked fields. Analysis of latent graphs produced by the autoencoder for various model settings are conducted using unstructured snapshot data sourced from large-eddy simulations in a backward-facing step (BFS) flow configuration with an OpenFOAM-based flow solver at high Reynolds numbers.

Fully hyperbolic convolutional neural networks

Convolutional neural networks (CNN) have recently seen tremendous success in various computer vision tasks. However, their application to problems with high dimensional input and output, such as high-resolution image and video segmentation or 3D medical imaging, has been limited by various factors. Primarily, in the training stage, it is necessary to store network activations for back-propagation. In these settings, the memory requirements associated with storing activations can exceed what is feasible with current hardware, especially for problems in 3D. Motivated by the propagation of signals over physical networks, that are governed by the hyperbolic Telegraph equation, in this work we introduce a fully conservative hyperbolic network for problems with high-dimensional input and output. We introduce a coarsening operation that allows completely reversible CNNs by using a learnable discrete wavelet transform and its inverse to both coarsen and interpolate the network state and change the number of channels. We show that fully reversible networks are able to achieve results comparable to the state of the art in 4D time-lapse hyper-spectral image segmentation and full 3D video segmentation, with a much lower memory footprint that is a constant independent of the network depth. We also extend the use of such networks to variational auto-encoders, where optimization begins from an exact recovery and we discover the level of compression through optimization.

3-D nonlinear magnetic field analysis with a novel adaptive finite element method

In this paper, an adaptive degrees-of-freedom finite element method is extended to three-dimensional (3-D) nonlinear magnetic field analysis. The error distribution of the discrete solution changes along with the iterative process, and mesh coarsening or other operations with the same effect is needed to keep the scale of the problem small. In this proposed adaptive method, dispensable degrees of freedom (DoFs) are eliminated from the unknown list by constraining them with supplementary interpolation functions, which are formulated with master DoFs. Compared with mesh coarsening, the administration of DoFs and geometric data are no longer required, while the topology of the mesh is maintained. To extend to 3-D problems, a novel constraint, which produces accurate coefficients for an alterable number of master DoFs, is presented. The constraint is integrated into the element algebraic equation, followed by a conventional assembly. Other techniques for the adaptive algorithm are also included in this method. Several numerical examples are tested to showcase the effectiveness of this method in 3-D problems.

Graph Wavelet-Based Multilevel Graph Coarsening and Its Application in Graph-CNN for Alzheimer’s Disease Detection

Along with the classical applications like graph partitioning, graph visualization, etc., graph coarsening has been recently applied in graph convolutional neural network (GCNN) architectures to perform the pooling operation in the graph domain. In this paper, we propose a novel two-stage graph coarsening method rooted on the graph signal processing with its application in the GCNN architecture. In the first stage of coarsening, the graph wavelet transform (GWT) based features are used to obtain a coarsened graph which preserves the topological characteristics of the original graph. In the second stage, the coarsening problem is formulated as an optimization problem where the reduced Laplacian operator at each level is obtained as a restriction of the original Laplacian operator to a specified subspace that also maximizes the topological similarity. The performance of the proposed coarsening algorithm is quantified in the general coarsening context using different graph coarsening quality measures. Its effectiveness as a pooling operator in GCNN is validated by applying it for the graph coarsening operation in the GCNN architecture. This modified GCNN architecture is then used as a graph signal classifier for the early detection of Alzheimer's disease. The results show that the proposed coarsening method outperforms state-of-the-art methods, both in the general coarsening context and as a pooling operator in the GCNN architecture.

Combining ensemble Kalman filter and multiresolution analysis for efficient assimilation into adaptive mesh models

A new approach is developed for data assimilation into adaptive mesh models with the ensemble Kalman filter (EnKF). The EnKF is combined with a wavelet-based multiresolution analysis (MRA) scheme to enable robust and efficient assimilation in the context of reduced-complexity adaptive spatial discretization. The wavelet representation of the solution enables the use of different meshes that are individually adapted to the corresponding members of the EnKF ensemble. The analysis step of the EnKF is then performed by involving coarsening, refinement, and projection operations on its ensemble members. Depending on the choice of these operations, five variants of the MRA-EnKF are introduced, and tested on the one-dimensional Burgers equation with periodic boundary condition. The numerical results suggest that, given an appropriate tolerance value for the coarsening operation, four out of the five proposed schemes significantly reduce the computational complexity of the assimilation system, with marginal accuracy loss compared to the reference, full resolution, and EnKF solution. Overall, the proposed framework offers the possibility of capitalizing on the advantages of adaptive mesh techniques, and the flexibility of choosing suitable context-oriented criteria for efficient data assimilation.

An approach to feature moving of hexahedral mesh

In the product design process, the finite element mesh regeneration is an inevitable step, but it is time consuming. The mesh editing is a kind of effective mesh regeneration technique. In this paper, a method for large scale feature movement of hexahedral mesh is proposed to satisfy application requirement. After the feature moving operation is applied to CAD model, the mesh deformation based on the mean value coordinates is carried out in the corresponding deformed region of the hex mesh to decrease the possible inverted elements caused by a large scale deformation. Then, the parameters of generating the density field are extracted from the initial hexahedral mesh, and used to calculate the local target density field by considering the geometric and topological information on the deformed region. Finally, based on difference between current density field and target density field of deformed region, the local refining and coarsening operations are devised and conducted to make the deformed region compatible with the target density field. Experimental results for the mechanical parts verify the effectiveness of proposed method.

Adaptive floating node method for modelling cohesive fracture of composite materials

The cohesive element has been widely employed to model delamination and matrix cracking in composite materials. However, an extremely fine mesh along the potential crack path is required to achieve accurate predictions of stresses within the cohesive zone. A sufficient number of cohesive elements must be present within the cohesive zone ahead of the crack tip, resulting in very high computational cost and time for application to practical composite structures. To mitigate this problem, an adaptive floating node method (A-FNM) with potential to reduce model size and computational effort is proposed. An element with adaptive partitioning capabilities is designed such that it can be formulated as a master element, a refined element and a coarsened element, depending on the damage state in the progressive damage process. A relatively coarse overall mesh may be used initially, and by transforming the element configurations adaptively, the local refinement and coarsening schemes are applied in the analysis. The localized stress gradient ahead of the crack front within the refinement zone is captured by the refined elements. The refinement and coarsening operations are performed at the elemental level with fixed nodal connectivity, so that global successive remeshing in adaptive mesh refinement (AMR) techniques is avoided; this is the key difference between AMR and A-FNM. It is demonstrated that, without loss of accuracy, the present method simplifies the modelling procedure and reduces computational cost.

M-Channel Perfect Recovery of Coarsened Graphs and Graph Signals With Spectral Invariance and Topological Preservation

In this paper, an M -channel perfect reconstruction filter bank based on a coarsening algorithm is proposed. Compared with most of the designs of graph filter banks that do not consider the graph reconstruction, our proposed design can provide the perfect graph reconstruction as well as perfect graph signal reconstruction. In the analysis part, the proposed filter bank provides a coarse version of the input graph as well as coarsened graph signal spectral invariant to the input signal. In the synthesis part, the filter bank perfectly reconstructs the input graph as well as input signal from their coarse version. The spirit of the proposed design is to partition the spectral information of the input graph and input graph signal into every channel. The partition method is adjustable and can be nonuniform. Two intuitive schemes named sort-by-eigenvalue and sort-by-intensity for uniform partition are introduced. In the proposed design, the coarsening operators are obtained through an existing coarsening algorithm, while the recovery operators are defined by taking the conjugate transpose of the coarsening operators. Besides, we address the relation between the proposed design and a framework of sampling graph signals based on the discrete sampling theory. It is shown that the coarsening operator in the proposed design is actually a special case of the sampling operator in the framework based on the discrete sampling theory. Experimental results are presented to demonstrate the effectiveness of the proposed design of filter banks.

Feature moving operation of hexahedral mesh

In the product design process, the finite element mesh regeneration is an inevitable step, but it is time consuming. The mesh editing is a kind of an effective mesh regeneration technique. In this paper, a method for widely moving features of hexahedral mesh is proposed to satisfy application requirement. After the feature moving operation is applied to CAD model, the mesh deformation based on the mean value coordinates is carried out in the corresponding deformed region of the hex mesh to decrease the possible inverted elements caused by a wide range deformation. Then, the parameters of generating the density field are extracted from the initial hexahedral mesh, and used to calculate the local target density field by considering the geometric and topological information on the deformed region. Finally, based on difference between current density field and target density field of deformed region, the local refining and coarsening operations are devised and conducted to make the deformed region compatible with the target density field. Experimental results for the mechanical parts verify the effectiveness of proposed method.

Subgraph-Based Filterbanks for Graph Signals

We design a critically-sampled compact-support biorthogonal transform for graph signals, via graph filterbanks. Instead of partitioning the nodes in two sets so as to remove one every two nodes in the filterbank downsampling operations, the design is based on a partition of the graph in connected subgraphs. Coarsening is achieved by defining one "supernode" for each subgraph and the edges for this coarsened graph derives from the connectivity between the subgraphs. Unlike the "one every two nodes" downsampling on bipartite graphs, this coarsening operation does not have an exact formulation in the graph Fourier domain. Instead, we rely on the local Fourier bases of each subgraph to define filtering operations. We apply successfully this method to decompose graph signals, and show promising performance on compression and denoising.

Coarsening in ergodic theory

This paper deals with the coarsening operation in dynamical systems where the phase space with a finite invariant measure is partitioned into measurable pieces and the summable function transferred by the phase flow is averaged over these pieces at each instant of time. Letting the time tend to infinity and then refining the partition, we arrive at a modernization of the von Neumann ergodic theorem, which is useful for the purposes of nonequilibrium statistical mechanics. In particular, for fine-grained partitions, we obtain the law of increment of coarse entropy for systems approaching the state of statistical equilibrium.

Adaptive Mesh Refinement for 2D Non-Isothermal Three-Phase Flows in Porous Media Comprised of Different Rock Types

In this paper we develop an adaptive mesh refinement (AMR) algorithm for solving the non-isothermal three-phase flows in heterogeneous porous media such as steam flooding in heavy oil reservoir. This porous media are made up of different rock types. The refinement criteria are proposed to overcome the difficulty that the saturation is discontinuous across the interface of two different rock types. Since the ratios of relative permeability between different rock types are found to be continuous, they are chosen as the control parameters to perform the refining and coarsening operations. The numerical examples show that the proposed AMR algorithm is fast with good accuracy.

AN ADAPTIVE DOMAIN DECOMPOSITION PRECONDITIONER FOR CRACK PROPAGATION PROBLEMS MODELED BY XFEM

Application of an algebraic multigrid (AMG) solver to linear systems arising from fracture problems modeled by extended finite elements (XFEM) will often result in poor convergence. This is due to coarsening operators in AMG that disregard the discontinuous enrichment functions and automatically coarsen across cracks. To overcome the AMG coarsening limitation, we propose a multiplicative-Schwarz domain decomposition preconditioner to the generalized minimum residual method. In this approach the domain is decomposed into one uncracked subdomain and multiple cracked subdomains. A cracked subdomain is the domain containing the crack and its enrichment functions and the uncracked subdomain contains the rest of the domain with a one-element-layer overlap between the two. Within the preconditioning scheme, one AMG V-cycle is applied to the uncracked subdomain to obtain an approximate solution while the cracked subdomains (often much smaller compared to the uncracked part) are solved concurrently by a direct solver, thus resolving the error from the discontinuous fields exactly. Hence any black box AMG solver can be used for XFEM, and the need for development of special coarsening procedures that handle enriched degrees of freedom can be avoided. We consider multiple propagating cracks and develop an algorithm that adaptively updates the subdomains, following the cracks. This adaptive scheme can be obtained directly from level set values which are updated with crack growth or from close neighbor search algorithms. The level set update scheme is fast but does not guarantee tight subdomains, while a neighbor search is slower but gives optimal subdomains. The preconditioner is tested on structured and unstructured meshes with multiple propagating cracks and shows convergence rates that are significantly better than a brute force application of AMG to the entire domain.

Scale- and resolution-invariance of suitable geographic range for shorebird metapopulations

The habitat suitability of a shorebird metapopulation is studied as a function of the scale (extent) and resolution (grain-size) of the environmental covariates with a maximum entropy species distribution model ( MaxEnt) for integration with climate change simulations. For this study, the species considered is the Snowy Plover ( Charadrius a. nivosus), which is a threatened shorebird whose geographic range spans the northwest and southwestern gulf coasts of Florida. The habitat suitability is analyzed at different resolutions by coarsening the classes of the ecogeographical variables with two algorithms: (i) preserving the information of each class at the finer resolution (conservative algorithm); and (ii) considering the most frequent class (majority algorithm). Ultimately, the most suitable habitat is found to be estuarine and ocean beaches made of alkaline medium and fine white sand and silt. The model fit to the observed species distribution decreases with the resolution. Due to the loss of the physical habitat (barrier islands) resulting from the coarsening operation, there is a threshold below which the model fails to predict the species distribution. As a result, the suitable geographic range is resolution-invariant below a resolution-threshold that is determined by the geomorphological features of the landscape rather than by biological constrains. This result holds for both the coarsening algorithms, however the conservative algorithm allows a continuous mapping of the habitat suitability. The suitable geographical range is found to be scale- and resolution-invariant, yet the habitat suitability at-a-point appears scale-dependent due to the strong heterogeneity of the ecogeographical variables. The scale- and resolution-invariance of the suitable geographic range appears a universal result for metapopulations of different species. This is important for reducing the uncertainty of population viability models that are based also on the choice of extent and grain-size of habitat predictions. However, attention must be paid in choosing a resolution that is not too large in order to correctly represent the physical habitat of the species. This implies a potential increase in the effectiveness of conservation campaigns to face the threats of climate change, such as sea-level rise for the Snowy Plover.

Maximal independent set graph partitions for representations of body-centered cubic lattices

A maximal independent set graph data structure for a body-centered cubic lattice is presented. Refinement and coarsening operations are defined in terms of set-operations resulting in robust and easy implementation compared to a quad-tree-based implementation. The graph only stores information corresponding to the leaves of a quad-tree thus has a smaller memory foot-print. The adjacency information in the graph relieves one from going up and down the quad-tree when searching for neighbors. This results in constant time complexities for refinement and coarsening operations.

RGB Subdivision

We introduce the RGB Subdivision: an adaptive subdivision scheme for triangle meshes, which is based on the iterative application of local refinement and coarsening operators, and generates the same limit surface of the Loop subdivision, independently on the order of application of local operators. Our scheme supports dynamic selective refinement, as in Continuous Level Of Detail models, and it generates conforming meshes at all intermediate steps. The RGB subdivision is encoded in a standard topological data structure, extended with few attributes, which can be used directly for further processing. We present an interactive tool that permits to start from a base mesh and use RGB subdivision to dynamically adjust its level of detail.

Generation of Quadrilateral Mesh from Discretized Models

Because of the inevitable tedious model repair by using CAD model as the source for mesh generation, an algorithm of generating all-quadrilateral element mesh from discretized models is proposed. By this method, discretized models, such as scattered points or STL file format model, are used as the source for mesh generation, and the original mesh generation paving method is modified to handle discretized models. Several key technologies are presented, the first one is the control of mesh size by using the discretized model as background mesh, the second one is the adjustment of mesh density distribution by mesh refinement and coarsening operation, and the third one is automatic identification of surface features in discretized models and preservation in mesh models. A new method to search intersections between paving boundaries is proposed. The effectiveness of the algorithm and the quality of the mesh generated are demonstrated by using several complex examples.