aMerge branch 'master' of gitlab.icts.res.in:ajith/NM2024

Lab.tex

@@ -97,6 +97,9 @@
 \label{sec:curve_fitting}
 \input{curve.tex}
 
+\section{Spectral Methods for ODEs and PDEs}
+\input{PDE1.tex}
+
 \section{Statistical inference}
 \input{statinf.tex}
 

PDE1.tex

+\subsection{Elliptic equations}
+
+We consider the example discussed in class, 
+
+\begin{equation}
+    \dfrac{d^2 u}{dx^2} - x^6(3+x^2)u = 0,
+\end{equation}
+
+on the interval $ [-1,1] $ with boundary conditions given by $ u(\pm 1) = 1 $.
+
+\subsubsection{Problems}
+
+\begin{enumerate}
+
+    \item Discretize this equation using Chebyshev polynomials. This will result in a set of equations for N unknowns, where two of the equations correspond to boundary conditions. Use direct matrix-inversion to find the solution.
+    \item Provide plots of the solution.
+    \item Solve for various different resolutions $N$, and provide plots that demonstrate how accurate your solution is, and that your numerical solution converges with $N$ (i.e. error goes to 0 as N increases).
+    \item How many modes are needed for an accuracy of $10^{-5}$? How many for $ 10^{-12} $?
+\end{enumerate}
+
+\subsection{Hyperbolic equations [Optional]}
+
+Consider the wave equation in 1+1 dimensions, as discussed in class, over the interval $ [0,2\pi] $,
+
+\begin{equation}
+    \Box g = -\partial_t ^2 g + \partial_x ^2 g = 0.
+\end{equation}
+
+This can be written as a set of first order differential equations,
+
+\begin{align*}
+    \dot{g} &= -\Pi, \\
+    \dot{\Pi} &= -\partial_x \Phi, \\
+    \dot{\Phi} &= -\partial_x \Pi.
+\end{align*}
+    
+\subsubsection{Problems}
+
+Solve the above equations using Fourier polynomial basis. Choose an initial condition which looks like a localized pulse. Enforce the scalar field to be 0 at both boundaries, i.e. set
+
+\begin{equation}
+    g(0) = \Pi(0) = \Phi(0) = g(2\pi) = \Pi(2\pi) = \Phi(2\pi) = 0.
+\end{equation}
+
+\begin{enumerate}
+    \item What do you observe if you evolve for times longer than $2\pi$?
+    \item Increase the domain size without changing the initial pulse size. Comment on the behaviour of the scalar in this case. 
+\end{enumerate}
\ No newline at end of file

statinf.tex

@@ -14,9 +14,9 @@ is the evidence of the model $\mathcal{M}$. Bayesian model selection involves co
 
 \subsubsection{Problems}
 \begin{enumerate}
-\item Compute the posterior distribution $p(H_0 | d, \mathcal{M}_1)$ of the Hubble constant $H_0$ using the Supernova Cosmology project data $z \leq 0.1$. Assume the Hubble's law [Eq.\eqref{eq:Hubble_law}] as the model $\mathcal{M}_1$. Assume uniform prior for $H_0$ in the interval $(10, 100)$ km/s/Mpc. 
+\item Compute the posterior distribution $p(H_0 | d, \mathcal{M}_1)$ of the Hubble constant $H_0$ using the Supernova Cosmology project data $z \leq 0.1$. Assume the Hubble's law [Eq.~\eqref{eq:Hubble_law}] as the model $\mathcal{M}_1$. Assume uniform prior for $H_0$ in the interval $(10, 100)$ km/s/Mpc. 
 \item Repeat the analysis using the full data set. What are the differences that you see in the posterior?
-\item Compute the posterior distribution  $p(H_0, \Omega_M | d, \mathcal{M}_2)$ of the Hubble constant $H_0$ and matter density $\Omega_M$ using the $\Lambda$CDM model $\mathcal{M}_2$ [Eq.\eqref{eq:lcdm}]. You can compute the posterior on a 2-dimensional grid. Assume uniform priors for $H_0$ in the interval $(10, 100)$ km/s/Mpc and for $\Omega_M$ in the interval (0, 1). 
-\item Compute the likelihood ratio (Bayes factor) between the Hubble's law and $\Lambda$CDM model by computing the evidences [Eq.\ref{eq:evidence}] of the two models using a  numerical integration method that we learned in Sec.~\ref{sec:integr}. 
+\item Compute the posterior distribution  $p(H_0, \Omega_M | d, \mathcal{M}_2)$ of the Hubble constant $H_0$ and matter density $\Omega_M$ using the $\Lambda$CDM model $\mathcal{M}_2$ [Eq.~\eqref{eq:lcdm}]. You can compute the posterior on a 2-dimensional grid. Assume uniform priors for $H_0$ in the interval $(10, 100)$ km/s/Mpc and for $\Omega_M$ in the interval (0, 1). 
+\item Compute the likelihood ratio (Bayes factor) between the Hubble's law and $\Lambda$CDM model by computing the evidences [Eq.~\eqref{eq:evidence}] of the two models using a  numerical integration method that we learned in Sec.~\ref{sec:integr}. 
 
 \end{enumerate}
\ No newline at end of file