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

Lab.tex

@@ -97,6 +97,9 @@
 \label{sec:curve_fitting}
 \label{sec:curve_fitting}
 \input{curve.tex}
 \input{curve.tex}
 
 
+\section{Spectral Methods for ODEs and PDEs}
+\input{PDE1.tex}
+
 \section{Statistical inference}
 \section{Statistical inference}
 \input{statinf.tex}
 \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}
 \subsubsection{Problems}
 \begin{enumerate}
 \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 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}
 \end{enumerate}
\ No newline at end of file