461 research outputs found
A Parallel Solver for Graph Laplacians
Problems from graph drawing, spectral clustering, network flow and graph
partitioning can all be expressed in terms of graph Laplacian matrices. There
are a variety of practical approaches to solving these problems in serial.
However, as problem sizes increase and single core speeds stagnate, parallelism
is essential to solve such problems quickly. We present an unsmoothed
aggregation multigrid method for solving graph Laplacians in a distributed
memory setting. We introduce new parallel aggregation and low degree
elimination algorithms targeted specifically at irregular degree graphs. These
algorithms are expressed in terms of sparse matrix-vector products using
generalized sum and product operations. This formulation is amenable to linear
algebra using arbitrary distributions and allows us to operate on a 2D sparse
matrix distribution, which is necessary for parallel scalability. Our solver
outperforms the natural parallel extension of the current state of the art in
an algorithmic comparison. We demonstrate scalability to 576 processes and
graphs with up to 1.7 billion edges.Comment: PASC '18, Code: https://github.com/ligmg/ligm
Fine-grained Locality-aware Parallel Scheme for Anisotropic Mesh Adaptation
AbstractIn this paper, we provide a fine-grained parallel scheme for anisotropic mesh adaptation on NUMA11Non-Uniform Memory Access architectures.Data dependencies are expressed by a graph for each kernel, and concurrency is extracted through fine-grained graph coloring. Tasks are structured into bulk-synchronous steps to avoid data races and to aggregate shared-data accesses.To ensure performance prediction, time cost and load imbalance are theoretically characterized.The devised scheme was evaluated on a 4 NUMA node (2-socket) machine, and a mean efficiency of 70% was reached on 32 cores for 3 kernels out of 4. The impact of irregular degree distribution and data layout on scalability is highlighted
Variational Barycentric Coordinates
We propose a variational technique to optimize for generalized barycentric
coordinates that offers additional control compared to existing models. Prior
work represents barycentric coordinates using meshes or closed-form formulae,
in practice limiting the choice of objective function. In contrast, we directly
parameterize the continuous function that maps any coordinate in a polytope's
interior to its barycentric coordinates using a neural field. This formulation
is enabled by our theoretical characterization of barycentric coordinates,
which allows us to construct neural fields that parameterize the entire
function class of valid coordinates. We demonstrate the flexibility of our
model using a variety of objective functions, including multiple smoothness and
deformation-aware energies; as a side contribution, we also present
mathematically-justified means of measuring and minimizing objectives like
total variation on discontinuous neural fields. We offer a practical
acceleration strategy, present a thorough validation of our algorithm, and
demonstrate several applications.Comment: https://anadodik.github.io
Opt: A Domain Specific Language for Non-linear Least Squares Optimization in Graphics and Imaging
Many graphics and vision problems can be expressed as non-linear least
squares optimizations of objective functions over visual data, such as images
and meshes. The mathematical descriptions of these functions are extremely
concise, but their implementation in real code is tedious, especially when
optimized for real-time performance on modern GPUs in interactive applications.
In this work, we propose a new language, Opt (available under
http://optlang.org), for writing these objective functions over image- or
graph-structured unknowns concisely and at a high level. Our compiler
automatically transforms these specifications into state-of-the-art GPU solvers
based on Gauss-Newton or Levenberg-Marquardt methods. Opt can generate
different variations of the solver, so users can easily explore tradeoffs in
numerical precision, matrix-free methods, and solver approaches. In our
results, we implement a variety of real-world graphics and vision applications.
Their energy functions are expressible in tens of lines of code, and produce
highly-optimized GPU solver implementations. These solver have performance
competitive with the best published hand-tuned, application-specific GPU
solvers, and orders of magnitude beyond a general-purpose auto-generated
solver
TetCNN: Convolutional Neural Networks on Tetrahedral Meshes
Convolutional neural networks (CNN) have been broadly studied on images,
videos, graphs, and triangular meshes. However, it has seldom been studied on
tetrahedral meshes. Given the merits of using volumetric meshes in applications
like brain image analysis, we introduce a novel interpretable graph CNN
framework for the tetrahedral mesh structure. Inspired by ChebyNet, our model
exploits the volumetric Laplace-Beltrami Operator (LBO) to define filters over
commonly used graph Laplacian which lacks the Riemannian metric information of
3D manifolds. For pooling adaptation, we introduce new objective functions for
localized minimum cuts in the Graclus algorithm based on the LBO. We employ a
piece-wise constant approximation scheme that uses the clustering assignment
matrix to estimate the LBO on sampled meshes after each pooling. Finally,
adapting the Gradient-weighted Class Activation Mapping algorithm for
tetrahedral meshes, we use the obtained heatmaps to visualize discovered
regions-of-interest as biomarkers. We demonstrate the effectiveness of our
model on cortical tetrahedral meshes from patients with Alzheimer's disease, as
there is scientific evidence showing the correlation of cortical thickness to
neurodegenerative disease progression. Our results show the superiority of our
LBO-based convolution layer and adapted pooling over the conventionally used
unitary cortical thickness, graph Laplacian, and point cloud representation.Comment: Accepted as a conference paper to Information Processing in Medical
Imaging (IPMI 2023) conferenc
An interactive analysis of harmonic and diffusion equations on discrete 3D shapes
AbstractRecent results in geometry processing have shown that shape segmentation, comparison, and analysis can be successfully addressed through the spectral properties of the Laplace–Beltrami operator, which is involved in the harmonic equation, the Laplacian eigenproblem, the heat diffusion equation, and the definition of spectral distances, such as the bi-harmonic, commute time, and diffusion distances. In this paper, we study the discretization and the main properties of the solutions to these equations on 3D surfaces and their applications to shape analysis. Among the main factors that influence their computation, as well as the corresponding distances, we focus our attention on the choice of different Laplacian matrices, initial boundary conditions, and input shapes. These degrees of freedom motivate our choice to address this study through the executable paper, which allows the user to perform a large set of experiments and select his/her own parameters. Finally, we represent these distances in a unified way and provide a simple procedure to generate new distances on 3D shapes
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