Graph coloring models of various kinds have proved to be powerful abstractions for
minimizing the execution time, memory, and storage space needed in computing sparse
derivative matrices using automatic differentiation. Depending on the structure of the
derivative matrix of interest---whether it is Jacobian (nonsymmetric) or Hessian
(symmetric)---and the specifics of the computational techniques employed, as many as five
variants of coloring problems exist. Graph coloring models are also useful in discovering
concurrency in parallel scientific computation; examples include iterative methods for
sparse linear systems, adaptive mesh refinement, preconditioning, and sparse tiling.
We have developed novel sequential as well as parallel algorithms for a
variety of coloring problems such as acyclic coloring, star coloring, and
distance-2 coloring.
ColPack, a software package consisting of implementations of
sequential algorithms for coloring and related problems in sparse derivative computation,
has been released. Implementations of the parallel algorithms are being developed now.

We are currently developing new algorithms for computing sparse Hessian matrices
via the Reverse mode of Algorithmic Differentiation (AD).
This approach is based on an Edge Pushing algorithm recently proposed by
Gower and Mello, that we extend by viewing the AD computation from the perspective of
Live Variables in data flow analysis.
The Live Variables approach provides a simpler derivation of the algorithm,
can be made more efficient by the use of graph transformations called Preaccumulation
in this context,
and also lends itself to tighter complexity analyses.
Our software LivarH, is available through GitHub. Further, we are extending this
approach to compute higher order derivatives, as well as implementing
the algorithm in parallel using a new MPI formalism.

**Representative Presentation:**

** Mu Wang and Alex Pothen, ** Reverse Hessian Algorithm with Preaccumulation

The 17th EuroAD Workshop, Argonne, IL, August 2015.

**Representative Publication:**

**Assefaw H. Gebremedhin, Duc Nguyen, Mostofa Ali Patwary, and Alex Pothen,**
* ColPack: Graph coloring software for derivative computation and beyond, *
ACM Transactions on Mathematical Software, 40 (1), 30 pp., 2013.

**Here is a page that shows **

the impact of ColPack and our work on Coloring.

**Matching in Scientific Computing**

Various kinds of matchings in graphs are needed in several scientific or high-performance
computing contexts. Examples include sparse linear systems (numerical preprocessing and block
triangular decomposition) and graph partitioning (coarsening phase of multilevel methods).
Traditional matching algorithms compute optimal solutions in superlinear time, but current
trends are toward algorithms that find approximate solutions of massive graphs in
near-linear time. We have developed
parallel algorithms based on approximation techniques for edge-weighted matching, and
have applied them to the Network Alignment problem. We have also developed parallel
algorithms for maximum cardinality matching in bipartite graphs on multithreaded
computers.

Furthermore, we have investigated vertex-weighted matchings, b-matchings,
and b-edge covers.
For the first of these problems, we have developed a 2/3-approximation algorithm for
bipartite graphs;
for the second, we have developed efficient 1/2-approximation algorithms
for shared memory parallel machines by proposing a new b-Suitor algorithm;
and for the minimum weight b-edge cover, we are developing a 3/2-approximation algorithm.

**Representative Presentation:**

Parallel Graph Algorithms on Emerging Architectures

SIAM Annual Meeting, 2013.

**Representative Publication:**

**Mahantesh Halappanavar, Alex Pothen, Fredrik Manne, Ariful Azad, Johannes Langguth and
Arif Khan, **
* Codesign Lessons Learned from Implementing Graph Matching Algorithms on Multithreaded
Architectures, * IEEE Computer, pp. 46-55, August 2015.

**Bioinformatics and Computational Proteomics **

Flow Cytometry is a platform for profiling the changes in the phenotype of cells due to
various stimuli at the single-cell level; the phenotype is determined by the
abundances of multiple protein complexes, and is measured by fluorescence.
This technology was developed in the 1960s,
and is routinely used in clinical settings to measure the response of the immune system to
leukemias, Alzheimer's disease, vaccines, etc.
FC technology has been changing in recent years to become high content and high-throughput;
several tens of proteins can be measured simultaneously, and with plates containing
several hundreds of samples can be processed rapidly.
Data analytics now lags behind the ability to generate large volumes of FC data.
We have worked on various aspects of the data analysis pipeline for FC data:
Spectral unmixing computes correct abundances of each protein by correcting
for the overlaps of signals from multiple proteins, and this requires
Nonnegative matrix factorizations, Bregman's divergence, and nonlinear optimization.
Variance Stablilization decouples the dependence of the variance of a cell population
from the magnitude of its abundance.
We have developed a combinatorial dissimilarity measure to compare FC samples,
which are collections of cell populations, represented as clusters in
a multi-dimensional space of protein abundances.
We have used this dissimilarity measure to register cell populations across
large numbers of samples under varying experimental or physilogical conditions.
We have also developed the concept of metaclusters to summarize
cell populations from similar samples, and the concept of templates to
summarize samples with similar cell populations.
We have developed dynamic classification algorithms based on templates to classify samples
into classes based on the protein expression patterns of the cell populations.
We are continuing to develop algorithms and software for the FC pipeline
that will scale to large numbers of samples.

**Representative Presentation:**

**Ariful Azad, Bartek Rajwa, and Alex Pothen,**
*Classifying immunophenotypes for Acute Myeloid Leukemia via templates, *
CYTO 2014 (Conference of the International Society for the Advancement of Cytometry)

** Representative Publication:**

**Ariful Azad, Arif Khan, Bartek Rajwa, Saumyadipta Pyne and Alex Pothen,**
*Classifying immunophenotypes with templates from Flow Cytometry,*
Proceedings of the ACM Conference on Bioinformatics and Computational Biology (BCB), 10 pages, 2013

**Augmented Matrices, Computational Surgery, and Contingency Analysis of the Power Grid **

Computational surgery requires interactive visualization of solid finite element models of organs and their
deformations as they are being cut. An example is astigmatism surgery of the eye, with the cornea cut to
restore the normal curvature of the eye in orthogonal directions.
The computational challenge here is to provide 10-100 updates of a system of equations of involving
the stiffness matrix of a large mesh consisting of hundreds of thousands of nodes and elements
to enable the visualization.

A similar problem arises in contingency analysis of the electric Power Grid.
The operators of the Grid are required by law to compute what happens to the Grid when a single generator or
transmission line goes down, so that they know the corrective action to take.
The problem arises when we want to compute what happens to the Grid when some larger number k of the
generators or lines go down. Now the number of scenarios to consider increases even for a grid consisting
of a few thousand nodes.

Both these problems can be modeled by an initial state of the mesh or the Grid, described by a large mesh
or network, which is then changed by removing or adding a few nodes or edges.
This change happens at each cutting step as we remesh around the cut, or in the Power Grid as elements
go down. All of the changes can be modeled by adding rows and columns to a 2 by 2 block matrix
in such a way that the (1,1) block remains unchanged.
Since the changes affect only a relatively small number of mesh elements or network nodes,
we solve the Augmented system of equations implicitly (i.e., the matrix is not formed,
but the solution is computed using only matrix-vector operations.)
Our work shows that the Augmented matrix approach is capable of surgical simulations that could be used
in a haptic or graphic simulator.
We also reduce the time to compute a solution to the contingency analysis problem by two or more orders
of magnitude. Our work pays careful attention to sparsity in the matrices, the solution vectors,
and the right-hand-side vectors.

**Representative Presentation:**

**Alex Pothen and Yu-Hong Yeung**
*Augmented Systems for Fast Updates: Computational Surgery and Power Grids, *
Sparse Days in St. Girons III Conference, France July 2015

** Representative Publication:**

**Alex Pothen, Jessica Crouch, Yu-Hong Yeung**
*Interactively Cutting and Constraining Meshes Using Augmented Matrices,*
ACM Transactions on Computer Graphics (ACM TOG), 16 pages, 2015. (Accepted subject to revisions)

**Combinatorial Scientific Computing and Petascale Simulations Institute**

The CSCAPES Institute was a multi-institutional project funded by Advanced Scientific
Computing Research (ASCR) Office of the U.S. Department of Energy from 2006-2012.
The major goals of the project were to develop parallel algorithms and software for
problems in computational science and engineering (CSE) that could be modeled by graphs
and hypergraphs. We developed graph and hypergraph partitioning tools to
enable load-balancing; parallel algorithms to compute sparse Jacobians and Hessians in
Automatic Differentiation via graph coloring; approximation algorithms to
compute matchings on parallel computers; and ordering algorithms to
improve memory-system performance of sparse graph and matrix computations.
Our software tools were used by research teams in Accelerator Physics, Computational
Fluid Dynamics and Systems Biology.
Thirty researchers from Purdue, Sandia National Labs, Argonne National Lab,
Ohio State, Colorado State and Old Dominion University participated in this work.

**Project Website:**

CSCAPES Institute