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Efficient methods for out-of-core sparse Cholesky factorization

SIAM JOURNAL ON SCIENTIFIC COMPUTING 1999年第1期21卷 129-144页

作者： Rothberg, E Schreiber, R ILOG Inc Mountain View CA 94043 USA Hewlett Packard Corp Palo Alto CA 94304 USA

We consider the problem of sparse Cholesky factorization with limited main memory. The goal is to efficiently factor matrices whose Cholesky factors essentially fill the available disk storage, using very little memory (as little as 16 Megabytes (MBytes)). This would enable very large industrial problems to be solved with workstations of very modest cost. We consider three candidate algorithms. Each is based on a partitioning of the matrix into panels, The first is a robust, out-of-core multifrontal method chat keeps the factor, the stack, and the large frontal matrices on disk. The others are left-looking methods. We find that straightforward implementations of all of them suffer from excessive disk I/O for large problems that arise in interior-point algorithms for linear programming. We introduce several improvements to these simple out-of-core methods and find that a left-looking method that nevertheless uses the multifrontal algorithm for portions of the matrix (subtrees of the supernodal elimination tree whose multifrontal stack fits in memory) is very effective. With 32 Mbytes of main memory, it achieves over 77% of its in-core performance on all but one of our 12 test matrices (67% in that one case), even though the size of the factor is, in all cases, hundreds of millions or even billions of bytes.

关键词： out-of-core methods Cholesky factorization sparse matrices

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Isosurface Extraction and View-Dependent Filtering from Time-Varying Fields Using Persistent Time-Octree (PTOT)

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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 2009年第6期15卷 1367-1374页

作者： Wang, Cong Chiang, Yi-Jen NYU Polytech Inst CSE Dept Brooklyn NY USA

We develop a new algorithm for isosurface extraction and view-dependent filtering from large time-varying fields, by using a novel Persistent Time-Octree (PTOT) indexing structure. Previously, the Persistent Octree (POT) was proposed to perform isosurface extraction and view-dependent filtering, which combines the advantages of the interval tree (for optimal searches of active cells) and of the Branch-On-Need Octree (BONO, for view-dependent filtering), but it only works for steady-state (i.e., single time step) data. For time-varying fields, a 4D version of POT, 4D-POT, was proposed for 4D isocontour slicing, where slicing on the time domain gives all active cells in the queried time step and isovalue. However, such slicing is not output sensitive and thus the searching is sub-optimal. Moreover, it was not known how to support view-dependent filtering in addition to time-domain slicing. In this paper, we develop a novel Persistent Time-Octree (PTOT) indexing structure, which has the advantages of POT and performs 4D isocontour slicing on the time domain with an output-sensitive and optimal searching. In addition, when we query the same isovalue q over m consecutive time steps, there is no additional searching overhead (except for reporting the additional active cells) compared to querying just the first time step. Such searching performance for finding active cells is asymptotically optimal, with asymptotically optimal space and preprocessing time as well. Moreover, our PTOT supports view-dependent filtering in addition to time-domain slicing. We propose a simple and effective out-of-core scheme, where we integrate our PTOT with implicit occluders, batched occlusion queries and batched CUDA computing tasks, so that we can greatly reduce the I/O cost as well as increase the amount of data being concurrently computed in GPU. This results in an efficient algorithm for isosurface extraction with view-dependent filtering utilizing a state-of-the-art programmable GPU for

关键词： Isosurface extraction time-varying fields persistent data structure view-dependent filtering out-of-core methods

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Isosurface Extraction and View-Dependent Filtering from Time-Varying Fields Using Persistent Time-Octree (PTOT)

Isosurface Extraction and View-Dependent Filtering from Time...

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IEEE Information Visualization Conference/IEEE Visualization Conference

作者： Wang, Cong Chiang, Yi-Jen NYU Polytech Inst CSE Dept Brooklyn NY USA

关键词： Isosurface extraction time-varying fields persistent data structure view-dependent filtering out-of-core methods

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