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14 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

A Practical Solver for Scalar Data Topological Simplification

This paper presents a practical approach for the optimization of topological simplification, a central pre-processing step for the analysis and visualization of scalar data. Given an input scalar field f and a set of “signal” persistence pairs to maintain, our approaches produces an output field g that is close to f and which optimizes (i) the cancellation of “non-signal” pairs, while (ii) preserving the “signal” pairs. In contrast to pre-existing simplification algorithms, our approach is not restricted to persistence pairs involving extrema and can thus address a larger class of topological features, in particular saddle pairs in three-dimensional scalar data. Our approach leverages recent generic persistence optimization frameworks and extends them with tailored accelerations specific to the problem of topological simplification. Extensive experiments report substantial accelerations over these frameworks, thereby making topological simplification optimization practical for real-life datasets. Our approach enables a direct visualization and analysis of the topologically simplified data, e.g., via isosurfaces of simplified topology (fewer components and handles). We apply our approach to the extraction of prominent filament structures in three-dimensional data. Specifically, we show that our pre-simplification of the data leads to practical improvements over standard topological techniques for removing filament loops. Here, we also show how our approach can be used to repair genus defects in surface processing. Finally, we provide a C++ implementation for reproducibility purposes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Topological Simplifications of Hypergraphs

Here we study hypergraph visualization via its topological simplification. We explore both vertex simplification and hyperedge simplification of hypergraphs using tools from topological data analysis. In particular, we transform a hypergraph into its graph representations, known as the line graph and clique expansion. A topological simplification of such a graph representation induces a simplification of the hypergraph. In simplifying a hypergraph, we allow vertices to be combined if they belong to almost the same set of hyperedges, and hyperedges to be merged if they share almost the same set of vertices. Our proposed approaches are general and mathematically justifiable, and put vertex simplification and hyperedge simplification in a unifying framework.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening

This work introduces a number of algebraic topology approaches, including multi-component persistent homology, multi-level persistent homology, and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. In contrast to the conventional persistent homology, multi-component persistent homology retains critical chemical and biological information during the topological simplification of biomolecular geometric complexity. Multi-level persistent homology enables a tailored topological description of inter- and/or intra-molecular interactions of interest. Electrostatic persistence incorporates partial charge information into topological invariants. These topological methods are paired with Wasserstein distance to characterize similarities between molecules and are further integrated with a variety of machine learning algorithms, including k-nearest neighbors, ensemble of trees, and deep convolutional neural networks, to manifest their descriptive and predictive powers for protein-ligand binding analysis and virtual screening of small molecules. Extensive numerical experiments involving 4,414 protein-ligand complexes from the PDBBind database and 128,374 ligand-target and decoy-target pairs in the DUD database are performed to test respectively the scoring power and the discriminatory power of the proposed topological learning strategies. It is demonstrated that the present topological learning outperforms other existing methods in protein-ligand binding affinity prediction and ligand-decoy discrimination.

Biochemistry & Molecular Biology↗
DOE OSTI2018

Electric Power Distribution System Model Simplification Using Segment Substitution

Quasi-static time-series (QSTS) simulation is used to simulate the behavior of distribution systems over long periods of time (typically hours to years). The technique involves repeatedly solving the load-flow problem for a distribution system model and is useful for distributed energy resource (DER) planning. When a QSTS simulation has a small time step and a long duration, the computational burden of the simulation can be a barrier to integration into utility workflows. One way to relieve the computational burden is to simplify the system model. The segment substitution method of simplifying distribution system models introduced in this paper offers model bus reduction of up to 98% with a simplification error as low as 0.2% (0.002 pu voltage). In contrast to existing methods of distribution system model simplification, which rely on topological inspection and linearization, the segment substitution method uses black-box segment data and an assumed simplified topology.

Reiman, Andrew P. (ORCID:0000000302042960)↗
DOE OSTI2016

Critical point cancellation in 3D vector fields: Robustness and discussion

Vector field topology has been successfully applied to represent the structure of steady vector fields. Critical points, one of the essential components of vector field topology, play an important role in describing the complexity of the extracted structure. Simplifying vector fields via critical point cancellation has practical merit for interpreting the behaviors of complex vector fields such as turbulence. However, there is no effective technique that allows direct cancellation of critical points in 3D. This work fills this gap and introduces the first framework to directly cancel pairs or groups of 3D critical points in a hierarchical manner with a guaranteed minimum amount of perturbation based on their robustness, a quantitative measure of their stability. In addition, our framework does not require the extraction of the entire 3D topology, which contains non-trivial separation structures, and thus is computationally effective. Furthermore, our algorithm can remove critical points in any subregion of the domain whose degree is zero and handle complex boundary configurations, making it capable of addressing challenging scenarios that may not be resolved otherwise. Here, we apply our method to synthetic and simulation datasets to demonstrate its effectiveness.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Extremely Scalable Distributed Computation of Contour Trees via Pre-Simplification

Contour trees offer an abstract representation of the level set topology in scalar fields and are widely used in topological data analysis and visualization. However, applying contour trees to large-scale scientific datasets remains challenging due to scalability limitations. Recent developments in distributed hierarchical contour trees have addressed these challenges by enabling scalable computation across distributed systems. Building on these structures, advanced analytical tasks—such as volumetric branch decomposition and contour extraction—have been introduced to facilitate large-scale scientific analysis. Despite these advancements, such analytical tasks substantially increase memory usage, which hampers scalability. In this paper, we propose a pre-simplification strategy to significantly reduce the memory overhead associated with analytical tasks on distributed hierarchical contour trees. We demonstrate enhanced scalability through strong scaling experiments, constructing the largest known contour tree—comprising over half a trillion nodes with complex topology—in under 15 minutes on a dataset containing 550 billion elements.

Li, Mingzhe [University of Utah]↗
DOE OSTI2024

Localized Evaluation for Constructing Discrete Vector Fields

Topological abstractions offer a method to summarize the behavior of vector fields, but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. Here, we couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Topological Regularization via Persistence-Sensitive Optimization

Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have emerged as a way to provide a more precise and expressive control over the solution, relying on persistent homology to quantify and reduce its roughness. All such existing techniques back-propagate gradients through the persistence diagram, which is a summary of the topological features of a function. Their downside is that they provide information only at the critical points of the function. We propose a method that instead builds on persistence-sensitive simplification and translates the required changes to the persistence diagram into changes on large subsets of the domain, including both critical and regular points. This approach enables a faster and more precise topological regularization, the benefits of which we illustrate with experimental evidence.

Nigmetov, Arnur↗
DOE OSTI2017

Eigenvector centrality for geometric and topological characterization of porous media

Solving flow and transport through complex geometries such as porous media is computationally difficult. Such calculations usually involve the solution of a system of discretized differential equations, which could lead to extreme computational cost depending on the size of the domain and the accuracy of the model. Geometric simplifications like pore-networks, where the pores are represented by nodes and pore throats by edges connecting pores, have been proposed. These models, despite their ability to preserve the connectivity of the medium, have difficulties in capturing preferential paths (high velocity) and stagnation zones (low velocity) as they do not consider the specific relations between nodes. Nonetheless, network theory approaches, where a complex network is a graph, can help to simplify and better understand fluid dynamics and transport in porous media. Here we present a novel method to address these issues based on eigenvector centrality, which has been corrected to overcome the centralization problem, and modified to introduce a bias in the centrality distribution along a particular direction to address the flow and transport anisotropy in porous media. We compare the model predictions with millifluidic transport experiments, which shows that, albeit simple, this technique is computationally efficient and has potential for predicting preferential paths and stagnation zones for flow and transport in porous media. We propose to use eigenvector centrality probability distribution to compute entropy as an indicator of the ‘mixing capacity’ of the system.

58 GEOSCIENCES↗
DOE OSTI2021

A Kinetic Monte Carlo Approach for Simulating Cascading Transmission Line Failure

In this work, cascading transmission line failures are studied through a dynamical model of the power system operating under fixed conditions. The power grid is modeled as a stochastic dynamical system where first-principles electromechanical dynamics are excited by small Gaussian disturbances in demand and generation around a specified operating point. In this context, a single line failure is interpreted in a large deviation context as a first escape event across a surface in phase space defined by line security constraints. The resulting system of stochastic differential equations admits a transverse decomposition of the drift, which leads to considerable simplification in evaluating the quasipotential (rate function) and, consequently, computation of exit rates. Tractable expressions for the rate of transmission line failure in a restricted network are derived from large deviation theory arguments and validated against numerical simulations. Extensions to realistic settings are considered, and individual line failure models are aggregated into a Markov model of cascading failure inspired by chemical kinetics. Cascades are generated by traversing a graph composed of weighted edges representing transitions to degraded network topologies. Numerical results indicate that the Markov model can produce cascades with qualitative power-law properties similar to those observed in empirical cascades.

Kinetic Monte Carlo, rare events, Cascading Failur↗
DOE OSTI2026

Modeling of injected current stream-induced 3D perturbations in local helicity injection plasmas

Solenoid-free tokamak startup techniques are essential for spherical tokamaks and offer a pathway to cost reduction and design simplification in fusion energy systems. Local helicity injection (LHI) is one such approach, employing compact edge current sources to drive open field line current that initiates and sustains tokamak plasmas. The recently commissioned Pegasus-III Experiment provides a platform for advancing this and other solenoid-free startup methods. This study investigates the effect of LHI on magnetic topology in Pegasus-III plasmas. A helical filament model represents the injected current, and the linear plasma response to its three-dimensional field is calculated with M3D-C1. Poincaré mapping reveals substantial flux surface degradation in all modeled cases. The onset of overlapping magnetic structures and large-scale surface deformation begins near Ψ N ≈ 0.37, indicating a broad region of perturbed topology extending toward the edge. In rotating plasmas, both single-fluid and two-fluid models exhibit partial screening of the n = 1 perturbation, with two-fluid calculations showing stronger suppression near the edge. In contrast, the absence of rotation leads to strong resonant field amplification in the single-fluid case, while the two-fluid case with zero electron rotation mitigates this amplification and preserves edge screening. Magnetic probe measurements indicate that modeling the stream with spatial spreading—representing distributed current and/or oscillatory motion—better reproduces measured magnetic power profiles than a rigid filament model. The results underscore the role of rotation and two-fluid physics in screening stream perturbations and point to plasma flow measurements and refined stream models as key steps toward improving predictive fidelity.

Schaefer, Carolyn E. [Univ. of Wisconsin, Madison,↗
DOE OSTI2021

Component-wise reduced order model lattice-type structure design

Lattice-type structures can provide a combination of stiffness with light weight that is desirable in a variety of applications. Design optimization of these structures must rely on approximations of the governing physics to render solution of a mathematical model feasible. In this paper, we propose a topology optimization (TO) formulation that approximates the governing physics using component-wise reduced order modeling as introduced in Huynh et al. (2013); Eftang and Patera (2013), which can reduce solution time by multiple orders of magnitude over a full-order finite element model while providing a relative error in the solution of 1%. In addition, the offline training data set from such component-wise models is reusable, allowing its application to many design problems for only the cost of a single offline training phase, and the component-wise method is nearly embarrassingly parallel. We also show how the parameterization chosen in our optimization allows a simplification of the component-wise reduced order model (CWROM) not noted in previous literature, for further speedup of the optimization process. Furthermore, the sensitivity of the compliance with respect to the particular parameterization is derived solely at the component level. In numerical examples, we demonstrate a 1000x speedup over a full-order FEM model with relative error of 1% and show minimum compliance designs for two different cantilever beam examples, one smaller and one larger. Finally, error bounds for the displacement field, compliance, and compliance sensitivity of the CWROM are derived.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Computing a Stable Distance on Merge Trees

Distances on merge trees facilitate visual comparison of collections of scalar fields. Two desirable properties for these distances to exhibit are 1) the ability to discern between scalar fields which other, less complex topological summaries cannot and 2) to still be robust to perturbations in the dataset. The combination of these two properties, known respectively as stability and discriminativity, has led to theoretical distances which are either thought to be or shown to be computationally complex and thus their implementations have been scarce. In order to design similarity measures on merge trees which are computationally feasible for more complex merge trees, many researchers have elected to loosen the restrictions on at least one of these two properties. The question still remains, however, if there are practical situations where trading these desirable properties is necessary. Here we construct a distance between merge trees which is designed to retain both discriminativity and stability. While our approach can be expensive for large merge trees, we illustrate its use in a setting where the number of nodes is small. This setting can be made more practical since we also provide a proof that persistence simplification increases the outputted distance by at most half of the simplified value. As a result, we demonstrate our distance measure on applications in shape comparison and on detection of periodicity in the von Kármán vortex street.

97 MATHEMATICS AND COMPUTING↗