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DOE OSTI2022

Distributed Hierarchical Contour Trees

Contour trees are a significant tool for data analysis as they capture both local and global variation. However, their utility has been limited by scalability, in particular for distributed computation and storage. We report a distributed data structure for storing the contour tree of a data set distributed on a cluster, based on a fan-in hierarchy, and an algorithm for computing it based on the boundary tree that represents only the superarcs of a contour tree that involve contours that cross boundaries between blocks. This allows us to limit the communication cost for contour tree computation to the complexity of the block boundaries rather than of the entire data set.

Carr, Hamish A↗

DOE OSTI2021

Optimization and Augmentation for Data Parallel Contour Trees

Contour trees are used for topological data analysis in scientific visualization. While originally computed with serial algorithms, recent work has introduced a vector-parallel algorithm. Furthermore, this algorithm is relatively slow for fully augmented contour trees which are needed for many practical data analysis tasks. We therefore introduce a representation called the hyperstructure that enables efficient searches through the contour tree and use it to construct a fully augmented contour tree in data parallel, with performance on average 6 times faster than the state-of-the-art parallel algorithm in the TTK topological toolkit.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Distributed Augmentation, Hypersweeps, and Branch Decomposition of Contour Trees for Scientific Exploration

Contour trees describe the topology of level sets in scalar fields and are widely used in topological data analysis and visualization. A main challenge of utilizing contour trees for large-scale scientific data is their computation at scale using highperformance computing. To address this challenge, recent work has introduced distributed hierarchical contour trees for distributed computation and storage of contour trees. However, effective use of these distributed structures in analysis and visualization requires subsequent computation of geometric properties and branch decomposition to support contour extraction and exploration. In this work, we introduce distributed algorithms for augmentation, hypersweeps, and branch decomposition that enable parallel computation of geometric properties, and support the use of distributed contour trees as query structures for scientific exploration. Finally, we evaluate the parallel performance of these algorithms and apply them to identify and extract important contours for scientific visualization.

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 OSTI2019

Scalable Contour Tree Computation by Data Parallel Peak Pruning

As data sets grow to exascale, automated data analysis and visualization are increasingly important, to intermediate human understanding and to reduce demands on disk storage via in situ analysis. Trends in architecture of high performance computing systems necessitate analysis algorithms to make effective use of combinations of massively multicore and distributed systems. One of the principal analytic tools is the contour tree, which analyses relationships between contours to identify features of more than local importance. Unfortunately, the predominant algorithms for computing the contour tree are explicitly serial, and founded on serial metaphors, which has limited the scalability of this form of analysis. While there is some work on distributed contour tree computation, and separately on hybrid GPU-CPU computation, there is no efficient algorithm with strong formal guarantees on performance allied with fast practical performance. Furthermore, we report the first shared SMP algorithm for fully parallel contour tree computation, with formal guarantees of O(lg V lg t) parallel steps and O(V lg V) work for data with V samples and t contour tree supernodes, and implementations with more than 30× parallel speed up on both CPU using TBB and GPU using Thrust and up 70× speed up compared to the serial sweep and merge algorithm.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

A Mountaintop View Requires Minimal Sorting: A Faster Contour Tree Algorithm

Consider a scalar field f : M → R, where M is a triangulated simplicial mesh in R d . A level set, or contour, at value v is a connected component of f –1 (v). As v is changed, these contours change topology, merge into each other, or split. Contour trees are concise representations of f that track this contour behavior. The vertices of these trees are the critical points of f, where the gradient is zero. The edges represent changes in the topology of contours. It is a fundamental data structure in data analysis and visualization, and there is significant previous work (both theoretical and practical) on algorithms for constructing contour trees. Suppose M has n vertices, N facets, and t critical points. A classic result of Carr, Snoeyink, and Axen (2000) gives an algorithm that takes O(n log n+Nα(N)) time (where α(·) is the inverse Ackermann function). A further improvement to O(t log t + N) time was given by Chiang et al. All these algorithms involve a global sort of the critical points, a significant computational bottleneck. Unfortunately, lower bounds of Ω(t log t) also exist. We present the first algorithm that can avoid the global sort and has a refined time complexity that depends on the contour tree structure. Intuitively, if the tree is short and fat, we get significant improvements in running time. For a partition of the contour tree into a set of descending paths, P, our algorithm runs in O($\Sigma$ pϵP |p| log |p| + tα(t) + N). This is at most O(t log D + N), where D is the diameter of the contour tree. Moreover, it is O(tα(t) + N) for balanced trees, a significant improvement over the previous complexity. Our algorithm requires numerous ideas: partitioning the contour tree into join and split trees, a local growing procedure to iteratively build contour trees, and the use of heavy path decompositions for the time complexity analysis. There is a crucial use of a family of binomial heaps to maintain priorities, ensuring that any comparison made is between comparable nodes of the contour tree. We also prove lower bounds showing that the $\Sigma$ pϵP |p| log |p| complexity is inherent to computing contour trees.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

Parallel peak pruning for scalable SMP contour tree computation

As data sets grow to exascale, automated data analysis and visualisation are increasingly important, to intermediate human understanding and to reduce demands on disk storage via in situ analysis. Trends in architecture of high performance computing systems necessitate analysis algorithms to make effective use of combinations of massively multicore and distributed systems. One of the principal analytic tools is the contour tree, which analyses relationships between contours to identify features of more than local importance. Unfortunately, the predominant algorithms for computing the contour tree are explicitly serial, and founded on serial metaphors, which has limited the scalability of this form of analysis. While there is some work on distributed contour tree computation, and separately on hybrid GPU-CPU computation, there is no efficient algorithm with strong formal guarantees on performance allied with fast practical performance. Here in this paper, we report the first shared SMP algorithm for fully parallel contour tree computation, withfor-mal guarantees of O(lgnlgt) parallel steps and O(n lgn) work, and implementations with up to 10x parallel speed up in OpenMP and up to 50x speed up in NVIDIA Thrust.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

Hybrid Parallel Contour Trees, Version 1.0

A common operation in scientific visualization is to compute and render a contour of a data set. Given a function of the form f : R^d -> R, a level set is defined as an inverse image f^-1(h) for an isovalue h, and a contour is a single connected component of a level set. The Reeb graph can then be defined to be the result of contracting each contour to a single point, and is well defined for Euclidean spaces or for general manifolds. For simple domains, the graph is guaranteed to be a tree, and is called the contour tree. Analysis can then be performed on the contour tree in order to identify isovalues of particular interest, based on various metrics, and render the corresponding contours, without having to know such isovalues a priori. This code is intended to be the first data-parallel algorithm for computing contour trees. Our implementation will use the portable data-parallel primitives provided by Nvidia’s Thrust library, allowing us to compile our same code for both GPUs and multi-core CPUs. Native OpenMP and purely serial versions of the code will likely also be included. It will also be extended to provide a hybrid data-parallel / distributed algorithm, allowing scaling beyond a single GPU or CPU.

Sewell, Christopher↗

DOE OSTI2017

Hybrid Parallel Contour Trees, Version 1.0

Sewell, Christopher↗

DOE OSTI2017

Data-Parallel Algorithm for Contour Tree Construction

The goal of this project is to develop algorithms for additional visualization and analysis filters in order to expand the functionality of the VTK-m toolkit to support less critical but commonly used operators.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization

In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse–Smale complexes play an essential role in capturing the shape of scalar field data. Herein we present a state–of–the–art report on scalar field comparison using topological descriptors. We provide a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time–varying fields, and ensembles. These tasks include symmetry detection, periodicity detection, key event/feature detection, feature tracking, clustering, and structure statistics. Our main contributions include the formulation of a set of desirable mathematical and computational properties of comparative measures, and the classification of visualization tasks and applications that are enabled by these measures.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

TopoSZ: Preserving Topology in Error-Bounded Lossy Compression

Existing error-bounded lossy compression techniques control the pointwise error during compression to guarantee the integrity of the decompressed data. However, they typically do not explicitly preserve the topological features in data. When performing post hoc analysis with decompressed data using topological methods, preserving topology in the compression process to obtain topologically consistent and correct scientific insights is desirable. In this paper, we introduce TopoSZ, an error-bounded lossy compression method that preserves the topological features in 2D and 3D scalar fields. Specifically, we aim to preserve the types and locations of local extrema as well as the level set relations among critical points captured by contour trees in the decompressed data. The main idea is to derive topological constraints from contour-tree-induced segmentation from the data domain, and incorporate such constraints with a customized error-controlled quantization strategy from the SZ compressor (version 1.4). In conclusion, our method allows users to control the pointwise error and the loss of topological features during the compression process with a global error bound and a persistence threshold.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Lorentzian contours for tree-level string amplitudes

We engineer compact contours on the moduli spaces of genus-zero Riemann surfaces that achieve analytic continuation from Euclidean to Lorentzian worldsheets. These generalized Pochhammer contours are based on the combinatorics of associahedra and make the analytic properties of tree-level amplitudes entirely manifest for any number and type of external strings. We use them in practice to perform first numerical computations of open and closed string amplitudes directly in the physical kinematics for n=4,5,6,7,8,9 n = 4 , 5 , 6 , 7 , 8 , 9 . We provide a code that allows anyone to do such computations.

Physics↗

DOE OSTI2016

Topological Methods for Visualization

This slide presentation describes basic topological concepts, including topological spaces, homeomorphisms, homotopy, betti numbers. Scalar field topology explores finding topological features and scalar field visualization, and vector field topology explores finding topological features and vector field visualization.

58 GEOSCIENCES↗

DOE OSTI2016

SU-C-BRA-03: An Automated and Quick Contour Errordetection for Auto Segmentation in Online Adaptive Radiotherapy

Purpose: To develop a tool that can quickly and automatically assess contour quality generated from auto segmentation during online adaptive replanning. Methods: Due to the strict time requirement of online replanning and lack of ‘ground truth’ contours in daily images, our method starts with assessing image registration accuracy focusing on the surface of the organ in question. Several metrics tightly related to registration accuracy including Jacobian maps, contours shell deformation, and voxel-based root mean square (RMS) analysis were computed. To identify correct contours, additional metrics and an adaptive decision tree are introduced. To approve in principle, tests were performed with CT sets, planned and daily CTs acquired using a CT-on-rails during routine CT-guided RT delivery for 20 prostate cancer patients. The contours generated on daily CTs using an auto-segmentation tool (ADMIRE, Elekta, MIM) based on deformable image registration of the planning CT and daily CT were tested. Results: The deformed contours of 20 patients with total of 60 structures were manually checked as baselines. The incorrect rate of total contours is 49%. To evaluate the quality of local deformation, the Jacobian determinant (1.047±0.045) on contours has been analyzed. In an analysis of rectum contour shell deformed, the higher rate (0.41) of error contours detection was obtained compared to 0.32 with manual check. All automated detections took less than 5 seconds. Conclusion: The proposed method can effectively detect contour errors in micro and macro scope by evaluating multiple deformable registration metrics in a parallel computing process. Future work will focus on improving practicability and optimizing calculation algorithms and metric selection.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2025

Cuts and contours

The traditional formulation of string amplitudes via worldsheet integrals provides a parametrization of the moduli space that fails to expose the complete singularity structure of the amplitudes. This problem is solved by the positive parametrization of string amplitudes given by surfaceology. In this work, we use this formalism to study a number of properties of string amplitudes at tree-level and one-loop. We introduce several global prescriptions for an integration contour for which the integrals are finite everywhere in kinematic space. At tree-level, this is done in two ways: one directly implements the Feynman iε to analytically continue from Euclidean to Lorentzian worldsheets; the other is a generalization of the closed Pochhammer contour to arbitrary number of points. At loop-level, we present a systematic way of extracting cuts directly from the worldsheet integrand. This provides a powerful set of unitarity constraints, which we use to test the consistency of different “stringy” UV regularizations of field theory amplitudes. In addition, we identify the massive threshold expansion of the integrand, which allows us to reduce the problem to a finite set of Feynman integrals in Schwinger parametrization and provide a straightforward contour prescription reminiscent of its field-theory version.

Bosonic Strings↗

DOE OSTI2016

In situ and in-transit analysis of cosmological simulations

Abstract Modern cosmological simulations have reached the trillion-element scale, rendering data storage and subsequent analysis formidable tasks. To address this circumstance, we present a new MPI-parallel approach for analysis of simulation data while the simulation runs, as an alternative to the traditional workflow consisting of periodically saving large data sets to disk for subsequent ‘offline’ analysis. We demonstrate this approach in the compressible gasdynamics/ N -body code Nyx, a hybrid $\mbox{MPI}+\mbox{OpenMP}$ MPI + OpenMP code based on the BoxLib framework, used for large-scale cosmological simulations. We have enabled on-the-fly workflows in two different ways: one is a straightforward approach consisting of all MPI processes periodically halting the main simulation and analyzing each component of data that they own (‘ in situ ’). The other consists of partitioning processes into disjoint MPI groups, with one performing the simulation and periodically sending data to the other ‘sidecar’ group, which post-processes it while the simulation continues (‘in-transit’). The two groups execute their tasks asynchronously, stopping only to synchronize when a new set of simulation data needs to be analyzed. For both the in situ and in-transit approaches, we experiment with two different analysis suites with distinct performance behavior: one which finds dark matter halos in the simulation using merge trees to calculate the mass contained within iso-density contours, and another which calculates probability distribution functions and power spectra of various fields in the simulation. Both are common analysis tasks for cosmology, and both result in summary statistics significantly smaller than the original data set. We study the behavior of each type of analysis in each workflow in order to determine the optimal configuration for the different data analysis algorithms.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Smooth splitting and zeros from on-shell recursion

We describe a new approach to understanding the origins of recently discovered “hidden zeros” and “smooth splitting” of tree-level amplitudes in Tr ϕ 3 , Non-Linear Sigma Model (NLSM), Yang-Mill-Scalar (YMS) and the special Galileon. Introducing a new type of linear shift in kinematic space we demonstrate that the mysterious splitting formulae follow from a simple contour integration argument in the style of on-shell recursion. The argument makes use of only standard notions of tree-level factorization on propagators, but assumes improved UV behavior in the form of the absence of a residue at infinity. In the case of Tr ϕ 3 and NLSM this is proven by identifying our shift as a special case of a more general construction called a g-vector shift; in the case of YMS it remains an unproven conjecture. This recursive perspective leads to numerous new results: we derive generalizations of the splitting formulae on more relaxed near-zero kinematics, including interesting new kinematic limits in which the amplitude splits into a triple-product; we also demonstrate that the uncolored special Galileon model has improved UV scaling and hence also splits. We also investigate the possible realization of hidden zeros in four dimensions. The conditions under which the dimensionality constraints are compatible with zero kinematics is investigated in detail for Tr ϕ 3 and YMS; for the latter we find they can be realized only with certain restrictions on external helicity states. The realizable 4d zeros are proven by a similar recursive argument based on BCFW and is found to generalize to a new class of intrinsically 4d “helicity zeros” present in all sectors of YM and also gravity.

effective field theories↗