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Sparse grid

About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.


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Journal ArticleDOI
TL;DR: The Taichi programming language is proposed, which exposes a high-level interface for developing and processing spatially sparse multi-level data structures, and an optimizing compiler that automatically reduces data structure overhead.
Abstract: 3D visual computing data are often spatially sparse. To exploit such sparsity, people have developed hierarchical sparse data structures, such as multi-level sparse voxel grids, particles, and 3D hash tables. However, developing and using these high-performance sparse data structures is challenging, due to their intrinsic complexity and overhead. We propose Taichi, a new data-oriented programming language for efficiently authoring, accessing, and maintaining such data structures. The language offers a high-level, data structure-agnostic interface for writing computation code. The user independently specifies the data structure. We provide several elementary components with different sparsity properties that can be arbitrarily composed to create a wide range of multi-level sparse data structures. This decoupling of data structures from computation makes it easy to experiment with different data structures without changing computation code, and allows users to write computation as if they are working with a dense array. Our compiler then uses the semantics of the data structure and index analysis to automatically optimize for locality, remove redundant operations for coherent accesses, maintain sparsity and memory allocations, and generate efficient parallel and vectorized instructions for CPUs and GPUs. Our approach yields competitive performance on common computational kernels such as stencil applications, neighbor lookups, and particle scattering. We demonstrate our language by implementing simulation, rendering, and vision tasks including a material point method simulation, finite element analysis, a multigrid Poisson solver for pressure projection, volumetric path tracing, and 3D convolution on sparse grids. Our computation-data structure decoupling allows us to quickly experiment with different data arrangements, and to develop high-performance data structures tailored for specific computational tasks. With 110 th as many lines of code, we achieve 4.55× higher performance on average, compared to hand-optimized reference implementations.

119 citations

Journal ArticleDOI
TL;DR: In this article, an adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parameterized functions with high-dimensional parameters is proposed, where the weights are determined by the probability densities of the random variables.
Abstract: We propose an adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parameterized functions with high-dimensional parameters. Leja sequences are arbitrarily granular (any number of nodes may be added to a current sequence, producing a new sequence) and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions. When undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables. This paper establishes that a certain weighted formulation of one-dimensional Leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the Gauss quadrature nodes associated with the weight function. This property is true even for unbounded domains. We apply the Leja sparse grid approach to several high-dimensional problems and...

113 citations

Journal ArticleDOI
TL;DR: Results show that the use of sparse grid methods works better than popular counterparts, and the automatic sampling, special interpolation process, and dimension-adaptivity feature make SGI more flexible and efficient than using the uniform sample based metamodeling techniques.
Abstract: Current methods for uncertainty propagation suffer from their limitations in providing accurate and efficient solutions to high-dimension problems with interactions of random variables. The sparse grid technique, originally invented for numerical integration and interpolation, is extended to uncertainty propagation in this work to overcome the difficulty. The concept of Sparse Grid Numerical Integration (SGNI) is extended for estimating the first two moments of performance in robust design, while the Sparse Grid Interpolation (SGI) is employed to determine failure probability by interpolating the limit-state function at the Most Probable Point (MPP) in reliability analysis. The proposed methods are demonstrated by high-dimension mathematical examples with notable variate interactions and one multidisciplinary rocket design problem. Results show that the use of sparse grid methods works better than popular counterparts. Furthermore, the automatic sampling, special interpolation process, and dimension-adaptivity feature make SGI more flexible and efficient than using the uniform sample based metamodeling techniques.

111 citations

Journal ArticleDOI
TL;DR: The stochastic collocation method combined with the proposed material and geometry uncertainty models provides robust designs by utilizing already developed deterministic solvers and is utilized in the design of compliant mechanisms.
Abstract: The aim of this paper is to introduce the stochastic collocation methods in topology optimization for mechanical systems with material and geometric uncertainties. The random variations are modeled by a memory-less transformation of spatially varying Gaussian random fields which ensures their physical admissibility. The stochastic collocation method combined with the proposed material and geometry uncertainty models provides robust designs by utilizing already developed deterministic solvers. The computational cost is discussed in details and solutions to decrease it, like sparse grids and discretization refinement are proposed and demonstrated as well. The method is utilized in the design of compliant mechanisms.

107 citations

Journal ArticleDOI
19 Nov 2014
TL;DR: A new method for fluid simulation on high-resolution adaptive grids which rivals the throughput and parallelism potential of methods based on uniform grids is introduced and an adaptive multigrid-preconditioned Conjugate Gradient solver is demonstrated that achieves resolution-independent convergence rates while admitting a lightweight implementation with a modest memory footprint.
Abstract: We introduce a new method for fluid simulation on high-resolution adaptive grids which rivals the throughput and parallelism potential of methods based on uniform grids. Our enabling contribution is SPGrid, a new data structure for compact storage and efficient stream processing of sparsely populated uniform Cartesian grids. SPGrid leverages the extensive hardware acceleration mechanisms inherent in the x86 Virtual Memory Management system to deliver sequential and stencil access bandwidth comparable to dense uniform grids. Second, we eschew tree-based adaptive data structures in favor of storing simulation variables in a pyramid of sparsely populated uniform grids, thus avoiding the cost of indirect memory access associated with pointer-based representations. We show how the costliest algorithmic kernels of fluid simulation can be implemented as a composition of two kernel types: (a) stencil operations on a single sparse uniform grid, and (b) structured data transfers between adjacent levels of resolution, even when modeling non-graded octrees. Finally, we demonstrate an adaptive multigrid-preconditioned Conjugate Gradient solver that achieves resolution-independent convergence rates while admitting a lightweight implementation with a modest memory footprint. Our method is complemented by a new interpolation scheme that reduces dissipative effects and simplifies dynamic grid adaptation. We demonstrate the efficacy of our method in end-to-end simulations of smoke flow.

105 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202314
202242
202157
202040
201960
201872