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\begin{document}

\title{{\color{red} Avenues for Future Research}}
\author{}
\date{}

\maketitle

\thispagestyle{empty}

%Slide
\stitle{Avenues for Ongoing/Future Research}

\begin{itemize}
\item
Continuing efforts on linear solvers:
\begin{itemize}
\item
	Broaden algorithmic and architectural scope of performance studies.
\item
	Autotuning and composition of linear solvers.
\item
	Application studies (structures, fluids, nanoscale modeling)
\end{itemize}
\item
Linear and non-linear Eigenvalue Problems
\item
Examine role of emerging language concepts (atomics, transactions) in
linear algebra kernels.
\end{itemize}


\stitle{Eigenvalue Problems}

\begin{itemize}
\item
Along with linear system solvers, eigenvalue problems are among the most
frequently used HPC kernels.
\item
They are essential to diverse applications in structures, fluids,
electromagnetics, nanoelectronics, etc.
\end{itemize}

\stitle{Eigenvalue Problems: Example}

\begin{itemize}
\item
Electronic structure calculation requires minimization of the
Kohn-Sham (KS) total energy functional associated with an
atomistic system with respect to its electron orbitals.
\item
The total energy can be expressed as:
\begin{equation}
E_{tot}(X) = E_{kin}(X) + E_{ion}(X) + E_H(X) + E_{XC}(X)
\end{equation}
Here $X = (x_1, \ldots, x_k)$ represents the electronic wavefunctions
corresponding to $k$ occupied states, and $E$ corresponds to various
energy terms.
\end{itemize}


\stitle{Eigenvalue Problems: Example}

\begin{itemize}
\item
The total energy must be minimized under the orthonormality constraint
$X^TX = I$.
\item
The Karush-Kuhn-Tucker (KKT) condition yields the following nonlinear
eigenvalue problem:
\begin{equation}
H(X)X = X\Lambda_k
\end{equation}
\item
$H(X)$ is the Kohn-Sham Hamiltonian that depends on $X$.
\end{itemize}

\stitle{Eigenvalue Problems: Example}

\begin{itemize}
\item
The most commonly used algorithm is the SCF (self consistent field)
iteration (fixed point of the KKT condition).
\item
While the parallelization characteristics of SCF are identical to the
linear eigenvalue problem, its convergence is often suspect.
\item
Other techniques such as direct inversion of iterative subspaces (DIIS)
have been proposed to improve robustness.
\item
While there is some work on parallelization of these schemes, their
scaling properties are ill-understood. 
\item
Scaling solutions to this eigenvalue problem to petascale platforms would
represent the most significant development in materials modeling.
\end{itemize}

\end{document}

