Combinatorial Scientific Computing is concerned with the development, analysis and utilization of discrete algorithms in scientific and engineering applications. Graph and geometric algorithms are the fundamental tools of combinatorial scientific

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Combinatorial Algorithms Enabling Computational
Science Tales from the Front Erik Boman, Karen
Devine and Bruce Hendrickson, Sandia National
Laboratories Sanjukta Bhowmick, Paul Hovland and
Todd Munson, Argonne National Laboratory Assefaw
Gebremedhin and Alex Pothen, Old Dominion
University
Hypergraph partitioning is superior to graph
partitioning for parallel decompositions as it
more accurately models communication cost. A
parallel hypergraph partitioner has recently been
developed as part of Sandias Zoltan toolkit for
petascale computing. The new partitioner
generates decompositions that improve the
performance of numerical operations for a wide
variety of parallel applications.
Assorted variants of graph coloring are key tools
for reducing the work required to compute
derivative matrices using automatic
differentiation. For example, the number of
function evaluations needed to compute a sparse
Jacobian can be reduced by finding structurally
orthogonal sets of columns, a problem that can be
phrased as a distance-two coloring of the column
vertices of the bipartite graph. Researchers at
Old Dominion University have developed algorithms
for this and other coloring problems. Scalable
parallel versions have been developed for the
distance-two and distance-one coloring variants.
Combinatorial Scientific Computing is concerned
with the development, analysis and utilization of
discrete algorithms in scientific and engineering
applications. Graph and geometric algorithms are
the fundamental tools of combinatorial scientific
computing. They play a crucial enabling role in
numerous areas, including sparse matrix
computation, partitioning for parallelization,
mesh generation, and automatic differentiation.
Here we report on some recent developments in
this highly interdisciplinary and rapidly
evolving field.
Exploiting symmetry in a Hessian computation can
reduce the computational cost by almost 50.
Researchers at Argonne have developed a
polynomial time algorithm for detecting symmetry
in a computation described by a directed acyclic
graph (DAG) (detecting symmetry in general graphs
is NP-hard). Symmetry in a DAG is defined as
finding a dual for every vertex such that for
each vertex v and its dual v, the successors of
v are the dual of the predecessors of v. At
bottom, is the symmetric graph for a mesh
smoothing application.
Modern microprocessors are highly sensitive to
the spatial and temporal locality of data.
Reordering the vertices and elements in a mesh
can have a significant impact on performance.
Researchers at Argonne have developed several
reordering algorithms that use the hypergraph
representation of a matrix. These algorithms can
improve the performance of a mesh smoothing
application by nearly 50. The left image shows
the nonzero pattern for the original Hessian
matrix, and the right image shows the reordered
version.
Sandia is a multiprogram laboratory operated by
Sandia Corporation, a LockheedMartin Company, for
the United States Department of Energys National
Nuclear Security Administration under contract
DE-AC04-94AL85000. The work at Argonne was
supported by the Mathematical, Information, and
Computational Sciences Division subprogram of the
Office of Advanced Scientific Computing Research,
U.S. Department of Energy under Contract
W-31-109-Eng-38.