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\title{MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations\\
Lecture 1: Overture: Motivations, scope and structure of the course}
\author{Lecturer: Naoki Saito\\
        Scribe: Lawrence Austria/Allen Xue}
\date{March 29, 2007}

\begin{document}
\maketitle

\setcounter{section}{-1}
\section{Scope and Structure of This Course}
\subsection{Scope}
\begin{itemize}
\item Why Laplacian eigenvalues?
\begin{itemize}
\item Reflect geometric information of a domain $\Omega$
\item Can be used for shape analysis, graph theory, etc.
\end{itemize}
\item Why Laplacian eigenfunctions?
\begin{itemize}
\item Provide an orthonormal basis for $L^2(\Omega)$
\item Can be used for `spectral analysis' of data defined on $\Omega$ 
$\Longrightarrow$ generalization of Fourier analysis
\item Can be applied to data approximation, pattern analysis, and image analysis
\end{itemize}
\end{itemize}

\subsection{Structure of this course}
\begin{itemize}
\item Laplacian eigenvalue problems in the continuum, i.e., $\Omega\subset\Rf^d$
with $| \Omega | < \infty$
\begin{itemize}
\item 1D vibrations of a string
\item 2D/3D vibrations of a membrane of various shape, e.g., rectangle, disk,
sphere, etc.
\item Necessary functional analysis basics
\item Vibrations of a membrane and heat conduction in $\Omega\subset\Rf^d$
\item Application to data analysis, spectral geometry, and shape recognition
\end{itemize}
\item Laplacian eigenvalue problems in the discrete setting
\begin{itemize}
\item Basics of graph theory
\item Laplacian of a graph
\item Laplacian eigenvalues of a graph
\item Random walks and heat propagation on a graph
\item Diffusion maps
\item Application to clustering, image segmentation, statistical learning theory, etc.
\end{itemize}
\item Fast algorithms to compute Laplacian eigenvalues and eigenfunctions.
\end{itemize}


\section{Introduction}
Consider a domain $\Omega\subset\Rf^d$ of finite volume. A
\textbf{Laplacian Eigenvalue Problem} on $\Omega$ is an equation of the form:
\begin{equation}
\label{laplacian}
  -\varDelta u = \lambda u \quad \text{in $\Omega$},
\end{equation}
where ${\displaystyle \varDelta=\frac{\partial^2}{\partial
x_1^2}+\cdots+\frac{\partial^2}{\partial x_d^2}=\nabla\cdot\nabla}$,
with one of the following boundary conditions (BCs):

\begin{equation}
\left\{\begin{array}{lcl}
  u=0, & \text{on $\partial\Omega$} & \text{Dirichlet BC}\\[6pt]
  \normalderiv{u}=\nu\cdot\nabla u=0, & \text{on $\partial\Omega$} & \text{Neumann  BC}\\[9pt]
  \normalderiv{u}+au=0, a\in\Rf & \text{on $\partial\Omega$} & \text{Robin (Mixed) BC}
\end{array}\right.
\end{equation}

\begin{figure}[t]
\centering
\includegraphics[scale=0.7]{Figure1.eps}
\caption{$\Omega \subset \Rf^d$ with $\nu$ being a normal vector.}
\end{figure}

If $u\ne0$ in~$\Omega$ satisfies the Laplacian eigenvalue problem
with either of the above BCs, then we say $u$ is an \textbf{
eigenfunction} and the corresponding $\lambda$ is called the \textbf{
eigenvalue}.

Laplacian eigenvalues and eigenfunctions allow us to perform numerous
analysis with a given domain $\Omega$.  We will see that the
eigenvalues of \eqref{laplacian} reflect geometric information about
$\Omega$. Also, the eigenfunctions can be used for spectral analysis
of data defined (or living) on $\Omega$. Furthermore, Laplacian
eigenfunctions allow us to generalize Fourier analysis.

%For instance, a rectangular region on $\Rf^d$ gives us
%$\{e^{in \text{\bf x}}\}_{n\in \Z}$ as an orthonormal basis.
%By changing the shape of our domain, we can construct a new set
%basis.

\section{History of and Introduction to Laplacian Eigenvalues/Eigenfunctions}

\subsection{Vibrations of a one dimensional string}
Around the mid 18th century, d'Alembert, Euler and Daniel Bernoulli
examined and created the theory behind vibrations of a one-dimensional string.
Consider a perfectly elastic and flexible string resting over the
interval $[0,\ell]$ of the $x$-axis.  Let $\rho(x)$ be a mass density
for $x\in[0,\ell]$.  Assume that the magnitude of
the tension $\mathcal{T}$ of the string is constant.  If $u(x,t)$ is
the vertical displacement of the string at location $x\in[0,\ell]$ and
time $t\geq 0$, then the string vibrates according to the
one-dimensional wave equation:
\begin{equation}
\label{wave} \rho(x) u_{tt}=\mathcal{T}u_{xx}\ \ \ \text{or}\ \ \
u_{tt} = c^{2}u_{xx},\text{with $c^2=\frac{\mathcal{T}}{\rho}$}.
\end{equation}
See \cite[Appendix 1]{FOLLAND-FOURIER} for more about the derivation
of the above wave equation.
From now on, for simplicity, we assume the uniform string, i.e.,
$\rho(x)=\rho=$ constant.

We know \eqref{wave} has infinitely many solutions. We would need
to specify a boundary condition and an initial condition on
\eqref{wave} to obtain the desired solution. For instance, say both
ends of the string are held fixed for all time $t$.  This
corresponds to the Dirichlet boundary condition $u(0,t)=u(\ell,t)=0$.
If $f(x)$ is the initial vertical displacement at $x$ and $g(x)$ is
its initial velocity, then we would have the initial condition (IC)
$u(x,0)=f(x)$ and $u_t(x,0)=g(x)$ for all $x\in[0,\ell]$.  What we have
then is:
\begin{equation}\label{1dstringDBC}
  \left\{ \begin{array}{ll}
    u_{tt} = c^{2} u_{xx} & \text{for $x\in(0,\ell)$  and  $t > 0$}; \\
    u(0,t)=u(\ell,t)=0 & \text{for $t \geq 0$}; \\
    u(x,0) = f(x),\ \ u_{t}(x,0) = g(x) & \text{for $x\in[0,\ell]$}.
\end{array} \right.
\end{equation}

%To solve this boundary value problem, we start by finding the
%general solution of the partial differential equation and make
%adjustments to fit the boundary and initial conditions.

\subsection{The behavior of the string $u(x,t)$}
First we use the method of separation of variables to seek a nontrivial solution
of the form $u(x,t) = X(x)T(t)$.  Plugging $X(x)T(t)$ into the
differential equation \eqref{1dstringDBC}, we get:
\begin{equation*}
  XT'' = c^2 X''T
  \Rightarrow \frac{X''}{X} = \frac{T}{c^2 T} = k
\end{equation*}
where $k$ must be constant. This relation yields the pair of ordinary differential equations:
\begin{equation}
\label{XODE}
X'' - kX = 0 \quad \text{with $X(0) = X(\ell) = 0$},
\end{equation}
\begin{equation}
\label{TODE}
T'' - c^2 kT = 0
\end{equation}

\emph{\textbf{Case I:}} $k>0$

By solving the characteristic equation $r^2-k=0$, we get
$r=\pm\sqrt{k}$ and
\[ X(x)=A\e^{\sqrt{k}x} + B\e^{-\sqrt{k}x}\ \ \text{or}\ \ A\cosh(\sqrt{k}x)+B\sinh(\sqrt{k}x). \]
Applying the BC $X(0)=X(\ell)=0$ yields $A=B=0$, thus the case
of $k>0$ is not feasible.

\ \\
\emph{\textbf{Case II:}} $k=0$
\[ X''=0 \Longrightarrow X(x) = Ax+B, \] 

which again leads to $X(x)\equiv 0$.

\ \\
\emph{\textbf{Case III:}} $k<0$

Set $k = -\nu^2$ and $\nu >0$. We then have the characteristic equation
$r^2+\nu^2=0$, i.e., $r=\pm \im\nu$. Therefore we get

\[X(x)=A\cos(\nu x)+B\sin(\nu x)\]

By the BC $X(0)=X(\ell)=0$, we get:
\begin{equation*}
\left\{\begin{array}{lcl}

X(0)=0 & \Longrightarrow & A=0 \\
X(\ell)=B\sin(\nu \ell)=0 & \Longrightarrow & \nu=\frac{n\pi}{\ell},\ \
\forall n\in\N
\end{array}\right.
\end{equation*}
(Note $n=0$ leads to $X(x) \equiv 0$ in this case, so it should not be included.)
Therefore we have $X(x)=B\sin(\frac{n\pi}{\ell}x)$ and for convenience,
we let $B=\sqrt{2/\ell}$ and define:
\begin{equation*}
X_{n}(x) = \phi_n(x) \triangleq \sqrt{\frac{2}{\ell}} \sin\(\frac{n\pi}{\ell}x\),
\end{equation*}
so that $\|\phi_n \|_{L^{2}[0,\ell]}=1$.  Note that $\{ \phi_n \}_{n \in \N}$
form an orthonormal basis for $L^2[0,\ell]$.

Similarly, by $T''=-\nu^2c^2T$ we obtain the family of solutions
\[ T_n(t)=a_n \cos\(\frac{n \pi c}{\ell}t\) + b_n \sin\(\frac{n \pi c}{\ell}t\). \]

Now, for each $n \in \N$, the function
\begin{equation*}
u_n(x,t)= \left\{a_n \cos\(\frac{n \pi c}{\ell}t\) + b_n \sin\(\frac{n \pi
c}{\ell}t\)\right\}\sqrt{\frac{2}{\ell}}\sin\(\frac{n\pi}{\ell}x\)
\end{equation*}
satisfies the equation \eqref{1dstringDBC}, and by the Superposition
Principle,
\begin{equation}
\label{gensol}
  u(x,t)= \sum_{n=1}^{\infty} u_n(x,t) = \sum_{n=1}^{\infty} \left\{ a_n
  \cos\(\frac{n \pi c}{\ell}t\) + b_n \sin\(\frac{n \pi
  c}{\ell}t\) \right\}\phi_n(x)
\end{equation}
is a general solution with yet undetermined coefficients $a_n$ and $b_n$.

Next, we specify the coefficients $a_n$ and $b_n$ by matching \eqref{gensol}
with the ICs in \eqref{1dstringDBC}.  Thus we get
\[ u(x,0)=f(x)=\sum_{n=1}^{\infty} a_n
\sqrt{\frac{2}{\ell}}\sin\(\frac{n\pi}{\ell}x\)=\sum_{n=1}^{\infty}a_n\phi_n(x)\]
Then
\[ a_n=\inner{f}{\phi_n}=\sqrt{\frac{2}{\ell}}\int_{0}^{\ell}f(x)\sin\(\frac{n\pi}{\ell}x\)\dd{x}, \]
which is a Fourier sine series expansion of $f$.

Similarly,
\begin{equation*}
u_t(x,0)=g(x)=\sum_{n=1}^{\infty} \frac{n\pi
c}{\ell}b_n\sqrt{\frac{2}{\ell}}\sin\(\frac{n\pi}{\ell}x\).
\end{equation*}
We obtain
\begin{equation*}
  \frac{n\pi c}{\ell}b_n=\inner{g}{\phi_n}\ \ \Rightarrow\ \
  b_n=\frac{\ell}{n\pi c}\inner{g}{\phi_n}.
\end{equation*}

By construction, the particular solution:
\begin{equation*}
  u(x,t)= \sum_{n=1}^{\infty} \left\{\inner{f}{\phi_n}\cos\(\frac{n\pi c}{\ell}t\)+\frac{\ell}{n\pi c} \inner{g}{\phi_n}\sin\(\frac{n\pi c}{\ell}t\)\right\}\phi_n(x)
\end{equation*}

We need to check if our solution makes sense physically.  Notice
that
\begin{equation*}
c^2 = \frac{\mathcal{T}}{\rho} \Rightarrow \text{ the sound
frequency = } \frac{n\pi}{\ell} \sqrt{\frac{\mathcal{T}}{\rho}}.
\end{equation*}
For instance, if $\ell$ is short, $\mathcal{T}$ is high, and $\rho$ is
small (thin), then such a string generates a high frequency tone.  On the other
hand, if $\ell$ is long, $\mathcal{T}$ is low, and $\rho$ is large
(thick), then it generates a low frequency tone.

\ \\
\textbf{NOTE:} The Neumann boundary condition imposes
\[ u_x(0,t)=u_x(\ell,t)=0\ \ \ \forall t>0. \]
This leads to the Fourier \emph{cosine} series expansions of $f$ and $g$.
Note that the Neumann problem allows the solution
$u(x,t)=a_0=\text{const}$ and $\nu_0=0$.


\section{Remarks}
Through the process of finding a solution to the boundary value problem:
\begin{displaymath}
  \left\{ \begin{array}{ll}
    u_{tt} = c^{2} u_{xx} & \text{for $0 < x < \ell$ and $t > 0$} \\
    u(0,t)=u(\ell,t)=0 & \text{for $t \geq 0$}\\
    u(x,0) = f(x) & u_{t}(x,0) = g(x),
\end{array} \right.
\end{displaymath}
we arrive at the system
\begin{equation}
\label{1D}
  -X'' = \nu^2 X \quad \text{with $X(0) = X(\ell) = 0$}.
\end{equation}
Notice that \eqref{1D} is a one-dimensional version of the Dirichlet-Laplacian
eigenvalue problem with $\Omega = (0,\ell)$.  More importantly, we were
able to obtain two objects, namely:
\begin{itemize}
\item Eigenvalues:  ${\displaystyle \lambda_n = \nu_n^2 = \(\frac{n \pi}{\ell}\)^2}$
\item Eigenfunctions: ${\displaystyle \phi_n(x) =
  \sqrt{\frac{2}{\ell}}\sin\(\sqrt{\lambda_n}x\)}$
\end{itemize}
We see that $\{ \lambda_n \}_{n=1}^{\infty}$ contains geometric
information of $\Omega = (0,\ell)$. For instance, the size of the
first (or the smallest) eigenvalue tells us the volume of $\Omega$ 
(i.e., the length of $\Omega$ in this case).  If the first eigenvalue is small 
(under our assumption of constant tension and constant density), then length of 
the interval is long. 
Furthermore, the set $\{\phi_n\}_{n=1}^{\infty}$ forms an orthonormal basis for
$L^2(\Omega)$, so the eigenfunctions allows us to analyze functions living
on $\Omega$.

\begin{thebibliography}{10}
\bibitem{FOLLAND-FOURIER}
{\sc G.~B. Folland}, {\em Fourier Analysis and Its Applications},
Brooks/Cole Publishing Company, 1992.

\end{thebibliography}


\end{document}
