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\begin{document}


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\boldmath

\LARGE\bf


%-----------------------------------------------------------
% Page 1

%\begin{landscape}

\begin{center}

\resizebox{7.5in}{!}{\includegraphics*{MIT.jpg}}


{\huge\bf Plamen Koev
\\
\medskip
\LARGE\bf
Department of Mathematics
\\
\medskip

Massachusetts Institute of Technology

}

\end{center}




% -------------------------------------------------------------

\slide{Outline}

\begin{itemize}
\item My Recent Background

\item Research
\begin{itemize}
\item Numerical linear algebra
\item (Numerical) Stochastic Eigenanalysis
\end{itemize}

\item Industrial and Academic Collaborations on Practical Problems

\item Research Plans

\end{itemize}


% ------------------------------------------------------------

\slide{Ph.D, Applied Maths, University of California, Berkeley, May 2002}

\resizebox{!}{4.6in}{\includegraphics*{Berkeley.jpg}}
\resizebox{!}{4.6in}{\includegraphics*{EvansHall.jpg}}
% \resizebox{4in}{!}{\includegraphics*{BerkeleySeal.jpg}}


\bigskip
\bigskip

\centerline{Thesis:}


\centerline{\tb{``Accurate and Efficient Computations with Structured Matrices''}}



% ------------------------------------------------------------

\slide{Ph.D Advisor: James Demmel }

\centerline{\resizebox{!}{5.5in}{\includegraphics*{JimDemmel.jpg}}}

\begin{itemize}
\item Professor of Mathematics and Computer Science
\item Chief Scientist, CITRIS
% \item Member of the National Academy of Engineering
% \item Leading the development of LAPACK
\end{itemize}



% ------------------------------------------------------------

\slide{Currently: Massachusetts Institute of Technology }

\begin{itemize}


\item Instructor of Applied Mathematics

\item Working with Alan Edelman

\bigskip

\centerline{\resizebox{!}{4in}{\includegraphics*{AlanEdelman.jpg}}}

\bigskip

\item Random matrices, Numerical Stochastic Eigenanalysis

\end{itemize}

% ------------------------------------------------------------

\slide{Current Research }

\begin{itemize}


\vskip2in

\item Numerical Linear Algebra
\begin{itemize}
\item Accurate and efficient computations with structured matrices
\end{itemize}

\item (Numerical) Stochastic Eigenanalysis



\vskip2in


\centerline{ Supported by NSF Grant 2003--2006: \$126,000 }

\end{itemize}

%-----------------------------------------------------------

\slide{Accurate and Efficient Computations with Structured Matrices}

\begin{itemize}

\item Conventional algorithms for an $n\times n$ matrix $A$:
   \begin{itemize}
   \item Ignore structure
   \item $O(n^3)$ time
   \item High \tr{absolute} accuracy:
   $$
   |\lambda_i-\hat \lambda_i|\le O(\varepsilon) \tr{\|A\|}
   $$
   \item Tiny eigenvalues, singular values, etc.\ lost
   \end{itemize}

\end{itemize}

%-----------------------------------------------------------

\slide{Accurate and Efficient Computations with Structured Matrices}

\begin{itemize}

\item Conventional algorithms for an $n\times n$ matrix $A$:
   \begin{itemize}
   \item Ignore structure
   \item $O(n^3)$ time
   \item High \tr{absolute} accuracy:
   $$
   |\lambda_i-\hat \lambda_i|\le O(\varepsilon) \tr{\|A\|}
   $$
   \item Tiny eigenvalues, singular values, etc.\ lost
   \end{itemize}

\item Contrast: New \tb{structured} algorithms
\begin{itemize}
   \item $\le O(n^3)$ time
   \item High \tb{relative} accuracy
   $$
   |\lambda_i-\hat \lambda_i|\le O(\varepsilon) \tb{\lambda_i}
   $$




\end{itemize}
\item Motivation: Determined accurately and needed accurately

\item Example
\end{itemize}



% ----------------------------------------------------------

\slide{The Pascal Matrix}


\vskip1in

$$
\bmat{rrrrr}
1 & 1 & 1  & 1  & 1 \\
1 & 2 & 3  & 4  & 5 \\
1 & 3 & 6  & 10 & 15 \\
1 & 4 & 10 & 20 & 35 \\
1 & 5 & 15 & 35 & 70
\emat
$$

\vskip1in

\begin{itemize}

\item Notoriously ill-conditioned, totally positive matrix
\item $\lambda_i=\lambda_{n-i}^{-1}$

\end{itemize}

% ----------------------------------------------------------

\slide{ Eigenvalues of PASCAL(40) }

\centerline{\resizebox{8.5in}{!}{ \includegraphics*{pascal40.pdf} }}





% ---------------------------------------------------------

\slide{ Accurate and Efficient Computations with Structured Matrices }

\begin{itemize}

\item Developed algorithms, techniques, perturbation theory
\item Publicly available MATLAB software
\item A large number of matrix classes
\begin{itemize}
\item Polynomial Vandermonde
\item Generalized Vandermonde
\item Green's matrices
\item Sturm--Liouville ODE
\item M-matrices
\item Totally Positive matrices\tb{$^*$}
\begin{itemize}
\item Vibrations of mechanical systems
\item Stochastic processes
\item CAD
\end{itemize}
\end{itemize}

\end{itemize}

\vskip1in

\tb{$^*$first accurate algorithm for non-symmetric matrices}



% ----------------------------------------------------------

\slide{ Numerical Stochastic Eigenanalysis }

\begin{itemize}

\item $=$ computing distributions of random covariance matrices

\item Appear in situations with multiple signal/noise sources

\begin{itemize}

\item Cellular communications

\item Military

\end{itemize}

\end{itemize}



%-----------------------------------------------------------

\slide{ Random Covariance Matrices }

\begin{eqnarray*}
B&=&\bmat{ccccccc}
N(0,1) & &N(0,1) & &\ldots & & N(0,1) \\
N(0,1) & &N(0,1) & &\ldots & &N(0,1) \\
\vdots && \vdots & & \ldots && \vdots \\
N(0,1) & &N(0,1) & &\ldots & &N(0,1)
\emat
\\
\\
\\
\Sigma&=&
\bmat{cccc}
s_1\\ & s_2 \\ && \ddots  \\ &&& s_n
\emat
\\
\\
\\
\tb{A}& \tb\equiv & \tb{B^T\cdot \Sigma \cdot B}
\end{eqnarray*}

\vskip1in

Need distributions of $\lambda_i$, trace, det, moments





%
%-----------------------------------------------------------
% Page 1


\newpage
\begin{center}
{\LARGE \bf Explicit Formulas Exist---James 1964 }
\end{center}
\hrule
\bigskip


{\LARGE \bf

\vskip1in
$$P(\lambda_{\max}(A)<x)
=\tb{\frac{\Gamma_n\left({\textstyle\frac{n+1}{2}}\right)}
     {\Gamma_n\left({\textstyle\frac{l+n+1}{2}}\right)}
     \cdot
     \det\left({\textstyle\frac{1}{2}}x\Sigma^{-1}\right)^{l/2}
     \cdot\;
     }
     \tr{
     _1F_1\left(
     {\textstyle\frac{l}{2}};{\textstyle\frac{n+l+1}{2}};
     -{\textstyle\frac{1}{2}} x\Sigma^{-1}\right)}
$$


\vskip0.5in
\centerline{(\tb{Blue} $=$ easy; \tr{Red} $=$ hard)}

\vskip0.5in
\begin{itemize}
\item 40-year-old problem, impossible until now
\begin{itemize}
\item Previous best algorithm ($n=5$): \hskip0.5in\tr{8 weeks}

{\large (Guti{\'e}rrez, Rodriguez, S{\'a}ez, 2000)}

\end{itemize}
\end{itemize}


}

%
%-----------------------------------------------------------
% Page 1


\newpage
\begin{center}
{\LARGE \bf Explicit Formulas Exist---James 1964 }
\end{center}
\hrule
\bigskip


{\LARGE \bf

\vskip1in
$$
P(\lambda_{\max}(A)<x)
=\tb{\frac{\Gamma_n\left({\textstyle\frac{n+1}{2}}\right)}
     {\Gamma_n\left({\textstyle\frac{l+n+1}{2}}\right)}
     \cdot
     \det\left({\textstyle\frac{1}{2}}x\Sigma^{-1}\right)^{l/2}
     \cdot\;
     }
     \tr{
     _1F_1\left(
     {\textstyle\frac{l}{2}};{\textstyle\frac{n+l+1}{2}};
     -{\textstyle\frac{1}{2}} x\Sigma^{-1}\right)}
$$


\vskip0.5in
\centerline{(\tb{Blue} $=$ easy; \tr{Red} $=$ hard)}

\vskip0.5in
\begin{itemize}
\item 40-year-old problem, impossible until now
\begin{itemize}
\item Previous best algorithm ($n=5$): \hskip0.5in\tr{8 weeks}

{\large (Guti{\'e}rrez, Rodriguez, S{\'a}ez, 2000)}

\item New algorithm: \hskip2.8in\tb{$\frac{1}{100}$ second}

\end{itemize}
\end{itemize}


}

%
%-----------------------------------------------------------
% Page 2


\newpage
\begin{center}
{\LARGE \bf The Hypergeometric Function of a Matrix Argument $X$}
\end{center}
\hrule
\bigskip

{\LARGE \bf
\vskip0.5in
$$
\pFq(a_1,\ldots,a_p;
b_1,\ldots,b_q;X)
\equiv
\sum_{k=0}^{\tr\infty}
\sum_{\kappa\vdash k}
\frac{\tb{(a_1)_\kappa\cdots(a_p)_\kappa}}
{\tb{k!(b_1)_\kappa\cdots(b_q)_\kappa}} \cdot
\tr{C_\kappa(X)}
$$

\vskip0.5in
\begin{itemize}

\item Natural multivariate generalization of the classical HGF

\item The slow convergence is inherited (and exacerbated!)

\item $(a)_\kappa$ --- {\em Pochhammer symbol} (\tb{easy})

\tb{$\Rightarrow\; \pFq$ as hard as $_1F_1$}

\bigskip

\item $C_\kappa(X)$---{\em Zonal Polynomial} (Schur, Jack function)

\tr{$\Rightarrow$ Really hard: $O(n^m)$ each!}

\item \tb{Our Contribution: $O(n)$}

\item Trick: Connection with theory of symmetric functions and
       representation theory

\end{itemize}

}



%-----------------------------------------------------------

\slide{ Statistics We Can Now Compute }

\vskip1in

\begin{center}
\begin{tabular}{c||c|c}

Statistic & Wishart, \tb{\em any} $\Sigma$ & $\beta$-Laguerre, $\Sigma=I$ \\
\hline
\hline
$\lambda_{\min}$ &\chk  & \chk \\
$\lambda_{\max}$ &\chk  & \chk \\
trace            &\chk  & \chk \\
E(det$^k$)       &\chk  & \chk
\end{tabular}


\end{center}




% ---------------------------------------------------------
\slide{Academic and Industrial Collaborations/Applications}

\begin{itemize}

\item Automatic Target Classification
\\
(MITRE Coproration)

\item Electric Impedance Tomography  \\
(Prof.\ Yu Chen, New York University)

\item Using Neural Networks for solving Inverse Elliptic PDEs
\\
(Schlumberger Oilfield Research -- Boston)

\item Cosmic Microwave Background Radiation \\
      (with Julian Borrill, Lawrence Berkeley Lab)

\item Microprocessor Testing \\
      (Schlumberger Test Systems $\rightarrow$ NPTest $\rightarrow$
       Credence)
       \\
       San Jose, California

\end{itemize}


% ---------------------------------------------------------

\slide{Cosmic Microwave Background Radiation }

\centerline{We looked 15 billion years back in time to see this:}

\centerline{\resizebox{7in}{!}{ \includegraphics*{boom_map_orange.jpg} }}


% ----------------------------------------------------------

\slide{ Microprocessor Testing }

\centerline{Elaborate Model to Monitor and Control Heat Exchange}


\bigskip

\centerline{\resizebox{7in}{!}{ \includegraphics*{heatexch.jpg} }}

\bigskip

\centerline{(Paper in J. Electronic Packaging)}



% ------------------------------------------------------------

\slide{Heat must be controlled so people can look at this:}


\bigskip

\centerline{\resizebox{6in}{!}{ \includegraphics*{chip.jpg} }}



% -----------------------------------------------------------

\slide{Research Plans}

\begin{itemize}

\item Numerical Linear Algebra

\begin{itemize}

\item Nonsymmetric tridiagonal eigenvalue problem

\item Other structured matrix classes

\item Theoretical results in matrix theory

\end{itemize}

\item Stochastic Eigenanalysis
\begin{itemize}

\item Random Matrix Toolbox

\item Cooley--Tukey--like alg.\ for
      multivariate orth.\ polynomials
$$\mbox{New cost} \approx  O(\sqrt{\mbox{Current cost}})$$

\end{itemize}
\end{itemize}





%-----------------------------------------------------------------
\slide{List of collaborators}

{\large

\begin{itemize}
\item Numerical Linear Algebra

\begin{itemize}

\item James Demmel (Berkeley)
\item Beresford Parlett (Berkeley)
\item William Kahan (Berkeley)
\item Alan Edelman (M.I.T.)
\item Froil\'an  Dopico (Madrid, Carlos III)
\item Jose--Javier Martinez (Madrid, Alcala)
\item Juan Manuel Pe\~na (Zaragoza, Spain)
\item Ilse Ipsen (North Carolina)
\item Charlie Jonhnson (William and Mary)
\item Shaun Fallat (Regina, Canada)

\end{itemize}
\item Stochastic Eigenanalysis

\begin{itemize}
\item Alan Edelman (M.I.T.)
\item Raj Rao (M.I.T.)
\item Iain Johnstone (Stanford)
\item Ioana Dumitriu (Berkeley)
\item Vesselin Drensky (Bulgarian Academy of Sciences)
\end{itemize}

\item Industrial collaborations
\begin{itemize}
\item MITRE Corporation
\item Schlumberger Oilfield Research -- Boston
\item Credence, Inc. -- San Jose, California
\end{itemize}

\end{itemize}

}


% -----------------------------------------------------------

\slide{Teaching}

\begin{itemize}

\item Berkeley:

\begin{itemize}
\item TA for 6 undergrad and grad courses
\item Instructor for Numerical Analysis
\item Instructor for Preliminary Exam workshop (3 times)
\end{itemize}

\item \tb{Outstanding Graduate Student Instructor Award -- 1999}

\item M.I.T.:

\begin{itemize}

\item PDE
\item Numerical Analysis
\item Numerical Linear Algebra
\item Numerical PDEs

(last 2 pubished on OpenCourseWare $=\$4000$)

\end{itemize}



\item At the University Carlos III in Madrid:
\begin{itemize}
\item A Ph.D course on ``Advanced Topics in Matrix Analysis''

\end{itemize}


\item Excellent teaching evaluations from all three schools



\end{itemize}

% ------------------------------------------------------------
\slide{ Courses I can contribute to the department }

\begin{itemize}
\item Undergraduate / lower division
\begin{itemize}
\item Calculus
\item Linear Algebra
\item $\ldots$
\end{itemize}
\item Upper division / Graduate
\begin{itemize}
\item Numerical Analysis
\item Differential Equations
\item Numerical Linear Algebra
\item Numerical ODEs, PDEs
\item Computer Arithmetic
\item Matrix Analysis
\end{itemize}



\end{itemize}



\end{document}
