93 research outputs found
GPU Parallel Computation of Morse-Smale Complexes
The Morse-Smale complex is a well studied topological structure that
represents the gradient flow behavior of a scalar function. It supports
multi-scale topological analysis and visualization of large scientific data.
Its computation poses significant algorithmic challenges when considering large
scale data and increased feature complexity. Several parallel algorithms have
been proposed towards the fast computation of the 3D Morse-Smale complex. The
non-trivial structure of the saddle-saddle connections are not amenable to
parallel computation. This paper describes a fine grained parallel method for
computing the Morse-Smale complex that is implemented on a GPU. The
saddle-saddle reachability is first determined via a transformation into a
sequence of vector operations followed by the path traversal, which is achieved
via a sequence of matrix operations. Computational experiments show that the
method achieves up to 7x speedup over current shared memory implementations
The parallel computation of morse-smale complexes
pre-printTopology-based techniques are useful for multi-scale exploration of the feature space of scalar-valued functions, such as those derived from the output of large-scale simulations. The Morse-Smale (MS) complex, in particular, allows robust identification of gradient-based features, and therefore is suitable for analysis tasks in a wide range of application domains. In this paper, we develop a two-stage algorithm to construct the Morse-Smale complex in parallel, the first stage independently computing local features per block and the second stage merging to resolve global features. Our implementation is based on MPI and a distributed-memory architecture. Through a set of scalability studies on the IBM Blue Gene/P supercomputer, we characterize the performance of the algorithm as block sizes, process counts, merging strategy, and levels of topological simplification are varied, for datasets that vary in feature composition and size. We conclude with a strong scaling study using scientific datasets computed by combustion and hydrodynamics simulations
Parallel Computation of Piecewise Linear Morse-Smale Segmentations
This paper presents a well-scaling parallel algorithm for the computation of
Morse-Smale (MS) segmentations, including the region separators and region
boundaries. The segmentation of the domain into ascending and descending
manifolds, solely defined on the vertices, improves the computational time
using path compression and fully segments the border region. Region boundaries
and region separators are generated using a multi-label marching tetrahedra
algorithm. This enables a fast and simple solution to find optimal parameter
settings in preliminary exploration steps by generating an MS complex preview.
It also poses a rapid option to generate a fast visual representation of the
region geometries for immediate utilization. Two experiments demonstrate the
performance of our approach with speedups of over an order of magnitude in
comparison to two publicly available implementations. The example section shows
the similarity to the MS complex, the useability of the approach, and the
benefits of this method with respect to the presented datasets. We provide our
implementation with the paper.Comment: Journal: IEEE Transactions on Visualization and Computer Graphics /
Submitted: 22-Jun-2022 / Accepted: 13-Mar-202
Progressive Wasserstein Barycenters of Persistence Diagrams
This paper presents an efficient algorithm for the progressive approximation
of Wasserstein barycenters of persistence diagrams, with applications to the
visual analysis of ensemble data. Given a set of scalar fields, our approach
enables the computation of a persistence diagram which is representative of the
set, and which visually conveys the number, data ranges and saliences of the
main features of interest found in the set. Such representative diagrams are
obtained by computing explicitly the discrete Wasserstein barycenter of the set
of persistence diagrams, a notoriously computationally intensive task. In
particular, we revisit efficient algorithms for Wasserstein distance
approximation [12,51] to extend previous work on barycenter estimation [94]. We
present a new fast algorithm, which progressively approximates the barycenter
by iteratively increasing the computation accuracy as well as the number of
persistent features in the output diagram. Such a progressivity drastically
improves convergence in practice and allows to design an interruptible
algorithm, capable of respecting computation time constraints. This enables the
approximation of Wasserstein barycenters within interactive times. We present
an application to ensemble clustering where we revisit the k-means algorithm to
exploit our barycenters and compute, within execution time constraints,
meaningful clusters of ensemble data along with their barycenter diagram.
Extensive experiments on synthetic and real-life data sets report that our
algorithm converges to barycenters that are qualitatively meaningful with
regard to the applications, and quantitatively comparable to previous
techniques, while offering an order of magnitude speedup when run until
convergence (without time constraint). Our algorithm can be trivially
parallelized to provide additional speedups in practice on standard
workstations. [...
Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)
This paper presents a computational framework for the Principal Geodesic
Analysis of merge trees (MT-PGA), a novel adaptation of the celebrated
Principal Component Analysis (PCA) framework [87] to the Wasserstein metric
space of merge trees [92]. We formulate MT-PGA computation as a constrained
optimization problem, aiming at adjusting a basis of orthogonal geodesic axes,
while minimizing a fitting energy. We introduce an efficient, iterative
algorithm which exploits shared-memory parallelism, as well as an analytic
expression of the fitting energy gradient, to ensure fast iterations. Our
approach also trivially extends to extremum persistence diagrams. Extensive
experiments on public ensembles demonstrate the efficiency of our approach -
with MT-PGA computations in the orders of minutes for the largest examples. We
show the utility of our contributions by extending to merge trees two typical
PCA applications. First, we apply MT-PGA to data reduction and reliably
compress merge trees by concisely representing them by their first coordinates
in the MT-PGA basis. Second, we present a dimensionality reduction framework
exploiting the first two directions of the MT-PGA basis to generate
two-dimensional layouts of the ensemble. We augment these layouts with
persistence correlation views, enabling global and local visual inspections of
the feature variability in the ensemble. In both applications, quantitative
experiments assess the relevance of our framework. Finally, we provide a
lightweight C++ implementation that can be used to reproduce our results
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