Toward Accelerating the Matrix Inversion Computation of Symmetric Positive-Definite Matrices on Heterogenous GPU-Based Systems

​Block vs. Tile Algorithms

Block Algorithm

• Computation proceeds with two phases:

• Panel

• Update

• Bottlenecks

• Unnecessary synchronization points preventing lookaheads

• The fork-join model

Tile Algorithm

• Split the original column-major matrix into tiles using block data layout

• Use tiles as the fundamental unit of computations

• Out-of-order execution of tasks

Cholesky-based Matrix Invverse Computation​

1.     Compute the Cholesky factorization

                       A=LLT

    2.      Invert teh Cholesky factor
                       T=L-1
    3.      Form the product of the inverted Cholesky factor with its transpose to get the final inverted matrix
                        TTT=(LLT​)-1=A-1​

StarPU Runtime System

• Dynamic, out-of-order task scheduling on accelerator-based platforms

• Ensures data availability and coherency between the memories of different units

Mixing PLASMA and MAGMA with StarPU

• PLASMA kernels on CPUs, MAGMA kernels on GPUs

• Scheduling tasks with StarPU

• Advantage: Programmability and Productivity

Experimental Results

Compared our implementation with state-of-the-art, high performance dense linear algebra software libraries: LAPACK, PLASMA, and MAGMA

Experimental Platform

• Dual-socket quad-core Intel Xeon (24GB)

• Enhanced with two Fermi GPUs 2070 (6GB)

​Results

Our high performance implementation achieves almost half a Tflop/s (448 Gflop/s), which corresponds to 5 and 6-fold improvement compared to the equivalent routines from MAGMA adn PLASMA, respectively, and 10-fold improvement compared to LAPACK

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References