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Register a new userParallel and Scalable Heat Methods for Geodesic Distance Computation
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Added by
Bailin Deng
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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In this paper, we propose a parallel and scalable approach for geodesic distance computation on triangle meshes. Our key observation is that the recovery of geodesic distance with the heat method from [Crane et al. 2013] can be reformulated as optimization of its gradients subject to integrability, which can be solved using an efficient first-order method that requires no linear system solving and converges quickly. Afterward, the geodesic distance is efficiently recovered by parallel integration of the optimized gradients in breadth-first order. Moreover, we employ a similar breadth-first strategy to derive a parallel Gauss-Seidel solver for the diffusion step in the heat method. To further lower the memory consumption from gradient optimization on faces, we also propose a formulation that optimizes the projected gradients on edges, which reduces the memory footprint by about 50%. Our approach is trivially parallelizable, with a low memory footprint that grows linearly with respect to the model size. This makes it particularly suitable for handling large models. Experimental results show that it can efficiently compute geodesic distance on meshes with more than 200 million vertices on a desktop PC with 128GB RAM, outperforming the original heat method and other state-of-the-art geodesic distance solvers.
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In this paper, we develop a novel method for fast geodesic distance queries. The key idea is to embed the mesh into a high-dimensional space, such that the Euclidean distance in the high-dimensional space can induce the geodesic distance in the original manifold surface. However, directly solving the high-dimensional embedding problem is not feasible due to the large number of variables and the fact that the embedding problem is highly nonlinear. We overcome the challenges with two novel ideas. First, instead of taking all vertices as variables, we embed only the saddle vertices, which greatly reduces the problem complexity. We then compute a local embedding for each non-saddle vertex. Second, to reduce the large approximation error resulting from the purely Euclidean embedding, we propose a cascaded optimization approach that repeatedly introduces additional embedding coordinates with a non-Euclidean function to reduce the approximation residual. Using the precomputation data, our approach can determine the geodesic distance between any two vertices in near-constant time. Computational testing results show that our method is more desirable than previous geodesic distance queries methods.
This paper presents a new curved layer volume decomposition method for multi-axis support-free printing of freeform solid parts. Given a solid model to be printed that is represented as a tetrahedral mesh, we first establish a geodesic distance field embedded on the mesh, whose value at any vertex is the geodesic distance to the base of the model. Next, the model is naturally decomposed into curved layers by interpolating a number of iso-geodesic distance surfaces (IGDSs). These IGDSs morph from bottom-up in an intrinsic and smooth way owing to the nature of geodesics, which will be used as the curved printing layers that are friendly to multi-axis printing. In addition, to cater to the collision-free requirement and to improve the printing efficiency, we also propose a printing sequence optimization algorithm for determining the printing order of the IGDSs, which helps reduce the air-move path length. Ample experiments in both computer simulation and physical printing are performed, and the experimental results confirm the advantages of our method.
A new method of solution to the local spin density approximation to the electronic Schr{o}dinger equation is presented. The method is based on an efficient, parallel, adaptive multigrid eigenvalue solver. It is shown that adaptivity is both necessary and sufficient to accurately solve the eigenvalue problem near the singularities at the atomic centers. While preliminary, these results suggest that direct real space methods may provide a much needed method for efficiently computing the forces in complex materials.
Particle-in-cell methods couple mesh-based methods for the solution of continuum mechanics problems, with the ability to advect and evolve particles. They have a long history and many applications in scientific computing. However, they have most often only been implemented for either sequential codes, or parallel codes with static meshes that are statically partitioned. In contrast, many mesh-based codes today use adaptively changing, dynamically partitioned meshes, and can scale to thousands or tens of thousands of processors. Consequently, there is a need to revisit the data structures and algorithms necessary to use particle methods with modern, mesh-based methods. Here we review commonly encountered requirements of particle-in-cell methods, and describe efficient ways to implement them in the context of large-scale parallel finite-element codes that use dynamically changing meshes. We also provide practical experience for how to address bottlenecks that impede the efficient implementation of these algorithms and demonstrate with numerical tests both that our algorithms can be implemented with optimal complexity and that they are suitable for very large-scale, practical applications. We provide a reference implementation in ASPECT, an open source code for geodynamic mantle-convection simulations built on the deal.II library.
Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if the original manifold is not Euclidean. Note that the distance function on a manifold can always be well-defined. In this paper, we propose to learn the distance function directly on the manifold without embedding. We first provide a theoretical characterization of the distance function by its gradient field. Based on our theoretical analysis, we propose to first learn the gradient field of the distance function and then learn the distance function itself. Specifically, we set the gradient field of a local distance function as an initial vector field. Then we transport it to the whole manifold via heat flow on vector fields. Finally, the geodesic distance function can be obtained by requiring its gradient field to be close to the normalized vector field. Experimental results on both synthetic and real data demonstrate the effectiveness of our proposed algorithm.
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