31 research outputs found
Locality-aware parallel block-sparse matrix-matrix multiplication using the Chunks and Tasks programming model
We present a method for parallel block-sparse matrix-matrix multiplication on
distributed memory clusters. By using a quadtree matrix representation, data
locality is exploited without prior information about the matrix sparsity
pattern. A distributed quadtree matrix representation is straightforward to
implement due to our recent development of the Chunks and Tasks programming
model [Parallel Comput. 40, 328 (2014)]. The quadtree representation combined
with the Chunks and Tasks model leads to favorable weak and strong scaling of
the communication cost with the number of processes, as shown both
theoretically and in numerical experiments.
Matrices are represented by sparse quadtrees of chunk objects. The leaves in
the hierarchy are block-sparse submatrices. Sparsity is dynamically detected by
the matrix library and may occur at any level in the hierarchy and/or within
the submatrix leaves. In case graphics processing units (GPUs) are available,
both CPUs and GPUs are used for leaf-level multiplication work, thus making use
of the full computing capacity of each node.
The performance is evaluated for matrices with different sparsity structures,
including examples from electronic structure calculations. Compared to methods
that do not exploit data locality, our locality-aware approach reduces
communication significantly, achieving essentially constant communication per
node in weak scaling tests.Comment: 35 pages, 14 figure
Higher order derivatives of matrix functions
We present theory for general partial derivatives of matrix functions on the
form where is a matrix path of several variables
(). Building on results by Mathias [SIAM J. Matrix Anal.
Appl., 17 (1996), pp. 610-620] for the first order derivative, we develop a
block upper triangular form for higher order partial derivatives. This block
form is used to derive conditions for existence and a generalized
Dalecki\u{i}-Kre\u{i}n formula for higher order derivatives. We show that
certain specializations of this formula lead to classical formulas of quantum
perturbation theory. We show how our results are related to earlier results for
higher order Fr\'echet derivatives. Block forms of complex step approximations
are introduced and we show how those are related to evaluation of derivatives
through the upper triangular form. These relations are illustrated with
numerical examples.Comment: 24 pages, 2 figure
Canonical density matrix perturbation theory
Density matrix perturbation theory [Niklasson and Challacombe, Phys. Rev.
Lett. 92, 193001 (2004)] is generalized to canonical (NVT) free energy
ensembles in tight-binding, Hartree-Fock or Kohn-Sham density functional
theory. The canonical density matrix perturbation theory can be used to
calculate temperature dependent response properties from the coupled perturbed
self-consistent field equations as in density functional perturbation theory.
The method is well suited to take advantage of sparse matrix algebra to achieve
linear scaling complexity in the computational cost as a function of system
size for sufficiently large non-metallic materials and metals at high
temperatures.Comment: 21 pages, 3 figure