We propose primal–dual mesh optimization algorithms that overcome shortcomings of the standard algorithm while retaining some of its desirable features. “Hodge-Optimized Triangulations” defines the “HOT energy” as a bound on the discretization error of the diagonalized Delaunay Hodge star operator. HOT energy is a natural choice for an objective function, but unstable for both mathematical and algorithmic reasons: it has minima for collapsed edges, and its extrapolation to non-regular triangulations is inaccurate and has unbounded minima. We propose a different extrapolation with a stronger theoretical foundation, and avoid extrapolation by recalculating the objective just beyond the flip threshold. We propose new objectives, based on normalizations of the HOT energy, with barriers to edge collapses and other undesirable configurations. We propose mesh improvement algorithms coupling these. When HOT optimization nearly collapses an edge, we actually collapse the edge. Otherwise, we use the barrier objective to update positions and weights and remove vertices. By combining discrete connectivity changes with continuous optimization, we more fully explore the space of possible meshes and obtain higher quality solutions.
This report details the results of a three-fold investigation of sensitivity analysis (SA) for machine learning (ML) explainability (MLE): (1) the mathematical assessment of the fidelity of an explanation with respect to a learned ML model, (2) quantifying the trustworthiness of a prediction, and (3) the impact of MLE on the efficiency of end-users through multiple users studies. We focused on the cybersecurity domain as the data is inherently non-intuitive. As ML is being using in an increasing number of domains, including domains where being wrong can elicit high consequences, MLE has been proposed as a means of generating trust in a learned ML models by end users. However, little analysis has been performed to determine if the explanations accurately represent the target model and they themselves should be trusted beyond subjective inspection. Current state-of-the-art MLE techniques only provide a list of important features based on heuristic measures and/or make certain assumptions about the data and the model which are not representative of the real-world data and models. Further, most are designed without considering the usefulness by an end-user in a broader context. To address these issues, we present a notion of explanation fidelity based on Shapley values from cooperative game theory. We find that all of the investigated MLE explainability methods produce explanations that are incongruent with the ML model that is being explained. This is because they make critical assumptions about feature independence and linear feature interactions for computational reasons. We also find that in deployed, explanations are rarely used due to a variety of reason including that there are several other tools which are trusted more than the explanations and there is little incentive to use the explanations. In the cases when the explanations are used, we found that there is the danger that explanations persuade the end users to wrongly accept false positives and false negatives. However, ML model developers and maintainers find the explanations more useful to help ensure that the ML model does not have obvious biases. In light of these findings, we suggest a number of future directions including developing MLE methods that directly model non-linear model interactions and including design principles that take into account the usefulness of explanations to the end user. We also augment explanations with a set of trustworthiness measures that measure geometric aspects of the data to determine if the model output should be trusted.
Interval Assignment (IA) is the problem of selecting the number of mesh edges (intervals) for each curve for conforming quad and hex meshing. The intervals x is fundamentally integer-valued, yet many approaches perform floating-point optimization and convert a floating-point solution into an integer solution. We avoid such steps: we start integer, stay integer. Incremental Interval Assignment (IIA) uses integer linear algebra (Hermite normal form) to find an initial solution to the matrix equation Ax = b satisfying the meshing constraints. Solving for reduced row echelon form provides integer vectors spanning the nullspace of A. We add vectors from the nullspace to improve the initial solution. Compared to floating-point optimization approaches, IIA is faster and always produces an integer solution. The potential drawback is that there is no theoretical guarantee that the solution is optimal, but in practice we achieve solutions close to the user goals. The software is freely available.
This report summarizes the accomplishments and challenges of a two year LDRD effort focused on improving design-to-simulation agility. The central bottleneck in most solid mechanics simulations is the process of taking CAD geometry and creating a discretization of suitable quality, i.e., the "meshing" effort. This report revisits meshfree methods and documents some key advancements that allow their use on problems with complex geometries, low quality meshes, nearly incompressible materials or that involve fracture. The resulting capability was demonstrated to be an effective part of an agile simulation process by enabling rapid discretization techniques without increasing the time to obtain a solution of a given accuracy. The first enhancement addressed boundary-related challenges associated with meshfree methods. When using point clouds and Euclidean metrics to construct approximation spaces, the boundary information is lost, which results in low accuracy solutions for non-convex geometries and mate rial interfaces. This also complicates the application of essential boundary conditions. The solution involved the development of conforming window functions which use graph and boundary information to directly incorporate boundaries into the approximation space.
Interval Assignment (IA) means selecting the number of mesh edges for each CAD curve. IIA is a discrete algorithm over integers. A priority queue iteratively selects compatible sets of intervals to increase in lock-step by integers. In contrast, the current capability in Cubit is floating-point Linear Programming with Branch-and-Bound for integerization (BBIA).
The classical problem of calculating the volume of the union of d-dimensional balls is known as "Union Volume." We present line-sampling approximation algorithms for Union Volume. Our methods may be extended to other Boolean operations, such as setminus; or to other shapes, such as hyper-rectangles. The deterministic, exact approaches for Union Volume do not scale well to high dimensions. However, we adapt several of these exact approaches to approximation algorithms based on sampling. We perform local sampling within each ball using lines. We have several variations, depending on how the overlapping volume is partitioned, and depending on whether radial, axis-aligned, or other line patterns are used. Our variations fall within the family of Monte Carlo sampling, and hence have about the same theoretical convergence rate, 1 /$\sqrt{M}$, where M is the number of samples. In our limited experiments, line-sampling proved more accurate per unit work than point samples, because a line sample provides more information, and the analytic equation for a sphere makes the calculation almost as fast. We performed a limited empirical study of the efficiency of these variations. We suggest a more extensive study for future work. We speculate that different ball arrangements, differentiated by the distribution of overlaps in terms of volume and degree, will benefit the most from patterns of line samples that preferentially capture those overlaps. Acknowledgement We thank Karl Bringman for explaining his BF-ApproxUnion (ApproxUnion) algorithm [3] to us. We thank Josiah Manson for pointing out that spoke darts oversample the center and we might get a better answer by uniform sampling. We thank Vijay Natarajan for suggesting random chord sampling. The authors are grateful to Brian Adams, Keith Dalbey, and Vicente Romero for useful technical discussions. This work was sponsored by the Laboratory Directed Research and Development (LDRD) Program at Sandia National Laboratories. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR), Applied Mathematics Program. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.
Blue noise sampling has proved useful for many graphics applications, but remains underexplored in high-dimensional spaces due to the difficulty of generating distributions and proving properties about them. We present a blue noise sampling method with good quality and performance across different dimensions. The method, spoke-dart sampling, shoots rays from prior samples and selects samples from these rays. It combines the advantages of two major high-dimensional sampling methods: the locality of advancing front with the dimensionality-reduction of hyperplanes, specifically line sampling. We prove that the output sampling is saturated with high probability, with bounds on distances between pairs of samples and between any domain point and its nearest sample. We demonstrate spoke-dart applications for approximate Delaunay graph construction, global optimization, and robotic motion planning. Both the blue-noise quality of the output distribution and the adaptability of the intermediate processes of our method are useful in these applications.
Over the past decade, polyhedral meshing has been gaining popularity as a better alternative to tetrahedral meshing in certain applications. Within the class of polyhedral elements, Voronoi cells are particularly attractive thanks to their special geometric structure. What has been missing so far is a Voronoi mesher that is sufficiently robust to run automatically on complex models. In this video, we illustrate the main ideas behind the VoroCrust algorithm, highlighting both the theoretical guarantees and the practical challenges imposed by realistic inputs.
Leibniz International Proceedings in Informatics, LIPIcs
Abdelkader, Ahmed; Bajaj, Chandrajit L.; Ebeida, Mohamed S.; Mahmoud, Ahmed H.; Mitchell, Scott A.; Owens, John D.; Rushdi, Ahmad A.
We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from α-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an ϵ-sample on the bounding surface, with a weak σ-sparsity condition, we work with the balls of radius δ times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of ϵ, σ and δ, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generating a bounded number of sites in its interior. Compared to state-of-the-art methods based on clipping, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured grids or random samples.
Over the past decade, polyhedral meshing has been gaining popularity as a better alternative to tetrahedral meshing in certain applications. Within the class of polyhedral elements, Voronoi cells are particularly attractive thanks to their special geometric structure. What has been missing so far is a Voronoi mesher that is sufficiently robust to run automatically on complex models. In this video, we illustrate the main ideas behind the VoroCrust algorithm, highlighting both the theoretical guarantees and the practical challenges imposed by realistic inputs.
We consider the problem of selecting a small set (mosaic) of sensor images (footprints) whose union covers a two-dimensional Region Of Interest (ROI) on Earth. We take the approach of modeling the mosaic problem as a Mixed-Integer Linear Program (MILP). This allows solutions to this subproblem to feed into a larger remote-sensor collection-scheduling MILP. This enables the scheduler to dynamically consider alternative mosaics, without having to perform any new geometric computations. Our approach to set up the optimization problem uses maximal disk sampling and point-in-polygon geometric calculations. Footprints may be of any shape, even non-convex, and we show examples using a variety of shapes that may occur in practice. The general integer optimization problem can become computationally expensive for large problems. In practice, the number of placed footprints is within an order of magnitude of ten, making the time to solve to optimality on the order of minutes. This is fast enough to make the approach relevant for near real-time mission applications. We provide open source software for all our methods, "GeoPlace."
We propose a porous materials analysis pipeline using persistent homology. We rst compute persistent homology of binarized 3D images of sampled material subvolumes. For each image we compute sets of homology intervals, which are represented as summary graphics called persistence diagrams. We convert persistence diagrams into image vectors in order to analyze the similarity of the homology of the material images using the mature tools for image analysis. Each image is treated as a vector and we compute its principal components to extract features. We t a statistical model using the loadings of principal components to estimate material porosity, permeability, anisotropy, and tortuosity. We also propose an adaptive version of the structural similarity index (SSIM), a similarity metric for images, as a measure to determine the statistical representative elementary volumes (sREV) for persistence homology. Thus we provide a capability for making a statistical inference of the uid ow and transport properties of porous materials based on their geometry and connectivity.