1. H–Cholesky Factorization on Many-Core
Accelerators
Gang Liao
August 2, 2015

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Background
If A is a positive definite matrix, Cholesky factorization: A = 𝐿𝐿%
Data matrices representing some numerical
observations such as proximity matrix or
correlation matrix are often huge and hard to
analyze, therefore to decompose the data
matrices into some lower-order or lower-rank
canonical forms will reveal the inherent
characteristic and structure of the matrices and
help to interpret their meaning readily.

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Hierarchical Matrix
Hierarchical matrices (H-matrices) are a powerful tool to represent dense
matrices coming from integral equations or partial differential equations in a
hierarchical, block-oriented, data-sparse way with log-linear memory costs.

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Hierarchical Matrix
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Implementation: Inadmissible Leaves:
The product index set resolves into admissible and inadmissible leaves of the tree. The
assembly, storage and matrix-vector multiplication differs for the corresponding two classes
of sub matrices.
Inadmissible Leaves:

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Implementation:Admissible Leaves:
The product index set resolves into admissible and inadmissible leaves of the tree. The
assembly, storage and matrix-vector multiplication differs for the corresponding two classes
of sub matrices.
Admissible Leaves:

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Hierarchical Matrix Representation

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Profiling

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Compiler Optimization – Full matrix
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For icc opt1, icc with optimizations
like -O2.
For icc opt2, icc with default
optimizations like -msse4.2 -O3.
For icc mkl, icc opt2 + mkl function.

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Numerical Libraries Optimization – Full matrix
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dpotrf_ vs plasma_dpotrf vs
magma_dpotrf
MKL: Intel Math Kernel Library
(Intel MKL) accelerates math
processing routines.
PLASMA: Parallel Linear Algebra
for Scalable Multi-core
Architectures
MAGMA: Matrix Algebra on GPU
and Multicore Architectures

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Parallel Optimization
The concept of task-based DAG computations is used to split the H-Cholesky
factorization into single tasks and to define corresponding dependencies to form
a DAG.

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CodeAnalysis

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Multicore Optimization – H-Cholesky Factorization
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Example 1:
Example 2:

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Manycore Optimization – H-Cholesky Factorization
1. Allocate & Copy r->a[row_offset] and r->b[col_offset] into accelerators.
2. Copy result ft->e from accelerators into CPU host memory.

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Result & Conclusion
0 500 1000 1500 2000 2500 3000 3500 4000 4500
0
2
4
6
8
10
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H−Cholesky Decomposition where the problem size (vertices) is 10002
nmin (leaf size)
Time(sec)
MKL
Hybrid
H - Cholesky factorization
on many-core accelerators
is extremely efficient,
which also can be well
scaled on large-scaled H-
matrix.