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    Hierarchical Matrix Operations on GPUs: Matrix-Vector Multiplication and Compression

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    Type
    Preprint
    Authors
    Boukaram, Wajih Halim
    Turkiyyah, George
    Keyes, David E. cc
    KAUST Department
    Extreme Computing Research Center (ECRC), King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia.
    Applied Mathematics and Computational Science Program
    Extreme Computing Research Center
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Office of the President
    Date
    2019-02-05
    Permanent link to this record
    http://hdl.handle.net/10754/660686
    
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    Abstract
    Hierarchical matrices are space and time efficient representations of dense matrices that exploit the low rank structure of matrix blocks at different levels of granularity. The hierarchically low rank block partitioning produces representations that can be stored and operated on in near-linear complexity instead of the usual polynomial complexity of dense matrices. In this paper, we present high performance implementations of matrix vector multiplication and compression operations for the $\mathcal{H}^2$ variant of hierarchical matrices on GPUs. This variant exploits, in addition to the hierarchical block partitioning, hierarchical bases for the block representations and results in a scheme that requires only $O(n)$ storage and $O(n)$ complexity for the mat-vec and compression kernels. These two operations are at the core of algebraic operations for hierarchical matrices, the mat-vec being a ubiquitous operation in numerical algorithms while compression/recompression represents a key building block for other algebraic operations, which require periodic recompression during execution. The difficulties in developing efficient GPU algorithms come primarily from the irregular tree data structures that underlie the hierarchical representations, and the key to performance is to recast the computations on flattened trees in ways that allow batched linear algebra operations to be performed. This requires marshaling the irregularly laid out data in a way that allows them to be used by the batched routines. Marshaling operations only involve pointer arithmetic with no data movement and as a result have minimal overhead. Our numerical results on covariance matrices from 2D and 3D problems from spatial statistics show the high efficiency our routines achieve---over 550GB/s for the bandwidth-limited mat-vec and over 850GFLOPS/s in sustained performance for the compression on the P100 Pascal GPU.
    Publisher
    arXiv
    arXiv
    1902.01829
    Additional Links
    https://arxiv.org/pdf/1902.01829
    Collections
    Preprints; Applied Mathematics and Computational Science Program; Extreme Computing Research Center; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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