Parallel multigrid in an adaptive PDE solver based on hashing and space-filling curves

Griebel, Michael; Zumbusch, Gerhard

doi:10.1016/s0167-8191(99)00020-4

Cited by 79 publications

(38 citation statements)

References 8 publications

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“…Figure 5 shows a twodimensional adaptive quadtree mesh ordered according to the Z-curve together with the cell tree and the linearised representation. In scientific simulation software, the combination of tree-structured grids and space-filling curves has been used in several ways, for example augmented by hashing [41], or for partial differential equation solvers with cache-optimised data administration [42,5]. Octor [43] and Dendro [6] are two examples of parallel octree libraries that have been scaled to 62,000 [44] and 32,000 [45] cores, operating on parent-child pointers and a linearised octant storage, respectively.…”

Section: Parallel Tree-structured Gridsmentioning

confidence: 99%

Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

Lahnert¹,

Burstedde²,

Holm³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. We present the minimally-invasive exchange of the regular Cartesian grid in the lattice-Boltzmann solver of ESPResSo by a dynamically-adaptive octree grid. Octree grids are favoured by computer scientists over other grid types as they are very memory-efficient. In addition, they represent a natural generalisation of regular Cartesian grids, such that most discretisation details of a regular grid solver can be maintained. Optimised codes, however, require a special tree-oriented grid traversal, which typically conflicts with existing simulation codes using various iterators, some for only parts of the grid, e.g., boundaries. ESPResSo is a large software package developed for soft-matter simulations involving fluid flow, electrostatic, and electrokinetic effects, and molecular dynamics. The currently used regular Cartesian grid hinders the simulation of realistic domain sizes and significant time periods, a problem that can be solved using grid adaptivity. In a first step, we focus on the lattice-Boltzmann flow solver in ESPResSo.p4est is a grid framework, that already provides dynamically adaptive quadtree and octree grids together with high-level interfaces for flexible grid traversals with direct neighbour access in all grid components. In this paper, we first describe extensions of p4est that were necessary to fulfill certain application requirements. The second part of our work consists of the minimally-invasive changes in ESPResSo preserving the expertise accumulated in the software's implementation over years. Our numerical results demonstrate physical correctness of the implementation, good parallel scalability and low overhead of the dynamical grid adaptivity. These are prerequisites to actually profit from grid adaptivity in terms of being able to simulate larger domains over longer time periods with limited computational resources. Thus, the current status forms a solid basis for further steps such as the development of refinement criteria, the setup of more realistic application scenarios, and a GPU implementation.

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Section: Parallel Tree-structured Gridsmentioning

confidence: 99%

Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

Lahnert¹,

Burstedde²,

Holm³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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“…Ichikawa et al [12] demonstrate an iterative data partitioning scheme for parallelizing a PDE solver. Griebel et al [5] made excellent progress using parallel multigrid to solve PDE's. More recently, we have seen research in image processing that involves large linear systems on huge images using a streaming multigrid that can benefit by running on parallel systems [8] in such a way that could be adapted to problems in GIS.…”

Section: Related Workmentioning

confidence: 99%

Parallel ODETLAP for terrain compression and reconstruction

Stookey¹,

Xie²,

Cutler³

et al. 2008

Proceedings of the 16th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems

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We introduce a parallel approximation of an Over-determined Laplacian Partial Differential Equation solver (ODETLAP) applied to the compression and restoration of terrain data used for Geographical Information Systems (GIS). ODET-LAP can be used to reconstruct a compressed elevation map, or to generate a dense regular grid from airborne Light Detection and Ranging (LIDAR) point cloud data. With previous methods, the time to execute ODETLAP does not scale well with the size of the input elevation map, resulting in running times that are prohibitively long for large data sets. Our algorithm divides the data set into patches, runs ODET-LAP on each patch, and then merges the patches together. This method gives two distinct speed improvements. First, we provide scalability by reducing the complexity such that the execution time grows almost linearly with the size of the input, even when run on a single processor. Second, we are able to calculate ODETLAP on the patches concurrently in a parallel or distributed environment. Our new patchbased implementation takes 2 seconds to run ODETLAP on an 800 × 800 elevation map using 128 processors, while the original version of ODETLAP takes nearly 10 minutes on a single processor (271 times longer). We demonstrate the effectiveness of the new algorithm by running it on data sets as large as 16000 × 16000 on a cluster of computers. We also discuss our preliminary results from running on an IBM Blue Gene/L system with 32,768 processors.

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“…Closely related issues emerge in the area of parallel and distributed computing where remote data transfer can become the primary bottleneck in the computation. In this context space filling curves are often used as a tool to determine very quickly data distribution layouts that guarantee good geometric locality [14][22] [24]. Space filling curves [27] have been also used in a wide variety of applications [3] both because of their hierarchical fractal structure as well as for their spatial locality properties.…”

Section: Previous Workmentioning

confidence: 99%

Global static indexing for real-time exploration of very large regular grids

Pascucci

Frank

2001

Proceedings of the 2001 ACM/IEEE Conference on Supercomputing

106

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In this paper we introduce a new indexing scheme for progressive traversal and visualization of large regular grids. We demonstrate the potential of our approach by providing a tool that displays at interactive rates planar slices of scalar field data with very modest computing resources. We obtain unprecedented results both in terms of absolute performance and, more importantly, in terms of scalability. On a laptop computer we provide real time interaction with a 2048 3 grid (8 Giga-nodes) using only 20MB of memory. On an SGI Onyx we slice interactively an 8192 3 grid (½ tera-nodes) using only 60MB of memory. The scheme relies simply on the determination of an appropriate reordering of the rectilinear grid data and a progressive construction of the output slice. The reordering minimizes the amount of I/O performed during the out-of-core computation. The progressive and asynchronous computation of the output provides flexible quality/speed tradeoffs and a timecritical and interruptible user interface.

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Parallel multigrid in an adaptive PDE solver based on hashing and space-filling curves

Cited by 79 publications

References 8 publications

Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

Parallel ODETLAP for terrain compression and reconstruction

Global static indexing for real-time exploration of very large regular grids

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