solutions.tex

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\usetheme{kalkulo}

\graphicspath{{./figures/}}


\title{Polynomial chaos expansions: Solutions}
\author{Jonathan Feinberg and Simen Tennøe}

 \lstset{
escapeinside={||},
basicstyle=\ttfamily\footnotesize, 
columns=fixed
}


\begin{document}


\begin{frame}
  \maketitle
\end{frame}

%  \begin{chaospy}{Traveling with constant velocity.}
%  \scriptsize
%  \begin{lstlisting}[language=Python]
% def s(t, v):
%     return v*t
% 
% v = cp.Normal(5,1)
% t = np.linspace(0,10,1000)
% 
% P = cp.orth_ttr(5, v)
% nodes, weights = cp.generate_quadrature(6, v, rule="G")
% samples_u = [s(t, n) for n in nodes.T]
% u_hat = cp.fit_quadrature(P, nodes, weights, samples_u)
% 
% plot(t, cp.E(u_hat, v))
% plot(t, cp.Var(u_hat, v))
% \end{lstlisting}
% \end{chaospy}
% 
% 
% \begin{frame}
%  \frametitle{Traveling with constant velocity.}
%   \begin{figure}
%   \includegraphics[width=0.85\textwidth]{solution1.png}
%  \end{figure}
% \end{frame}

 \begin{chaospy}{Traveling with constant acceleration, classical Monte Carlo integration.}
 \scriptsize
 \begin{lstlisting}[language=Python]
def s(t, v0, a):
    return v0*t + 0.5*a*t**2

N = 1000
v0 = cp.Uniform(1,2)
a = cp.Beta(2,2)
t = np.linspace(0,10,1000)

samples_v0 = v0.sample(N)
samples_a = a.sample(N)

distance = np.array([s(t,v0_,a_) for v0_,a_ \
                     in zip(samples_v0.T, samples_a.T)])
E = np.sum(distance,0)/N
Var = np.sum(distance**2,0)/N - E**2
plot(t, E)
plot(t, Var)
\end{lstlisting}
\end{chaospy}


\begin{frame}
 \frametitle{Traveling with constant acceleration, classical Monte Carlo integration.}
  \begin{figure}
  \includegraphics[width=0.85\textwidth]{solution2.png}
 \end{figure}
\end{frame}




 \begin{chaospy}{Traveling with constant acceleration, Quasi-Monte Carlo using \underline{S}obol sequence.}
 \scriptsize
 \begin{lstlisting}[language=Python]
def s(t, v0, a):
    return v0*t + 0.5*a*t**2

N = 1000
v0 = cp.Uniform(1,2)
a = cp.Beta(2,2)
t = np.linspace(0,10,1000)

samples_v0 = v0.sample(N, "S")
samples_a = a.sample(N, "S")

distance = np.array([s(t,v0_,a_) for v0_,a_ \
                     in zip(samples_v0.T, samples_a.T)])
E = np.sum(distance,0)/N
Var = np.sum(distance**2,0)/N - E**2
plot(t, E)
plot(t, Var)
\end{lstlisting}
\end{chaospy}



\begin{frame}
 \frametitle{Traveling with constant acceleration, Quasi-Monte Carlo using \underline{S}obol sequence.}
  \begin{figure}
  \includegraphics[width=0.85\textwidth]{solution3.png}
 \end{figure}
\end{frame}




 \begin{chaospy}{Traveling with constant acceleration, pseudo-spectral projection with full tensor grid \underline{G}aussian quadrature.}
 \scriptsize
 \begin{lstlisting}[language=Python]
def s(t, v0, a):
    return v0*t + 0.5*a*t**2

v0 = cp.Uniform(1,2)
a = cp.Beta(2,2)
dist = cp.J(v0,a)

t = np.linspace(0,10,1000)

P = cp.orth_ttr(5, dist)
nodes, weights = cp.generate_quadrature(6, dist, rule="G")
samples_u = [s(t, *n) for n in nodes.T]
u_hat = cp.fit_quadrature(P, nodes, weights, samples_u)

plot(t, cp.E(u_hat, dist))
plot(t, cp.Var(u_hat, dist))
\end{lstlisting}
\end{chaospy}



\begin{frame}
 \frametitle{Traveling with constant acceleration, pseudo-spectral projection with full tensor grid \underline{G}aussian quadrature.}
  \begin{figure}
  \includegraphics[width=0.85\textwidth]{solution4.png}
 \end{figure}
\end{frame}



 \begin{chaospy}{Traveling with constant acceleration, pseudo-spectral projection with \underline{C}lenshaw-Curtis and Smolyak sparse grid.}
 \scriptsize
 \begin{lstlisting}[language=Python]
def s(t, v0, a):
    return v0*t + 0.5*a*t**2

v0 = cp.Uniform(1,2)
a = cp.Beta(2,2)
dist = cp.J(v0,a)

t = np.linspace(0,10,1000)

P = cp.orth_ttr(M, dist)
nodes, weights = cp.generate_quadrature(6, dist, rule="C" \
                                        , sparse=True)
samples_u = [s(t, *n) for n in nodes.T]
u_hat = cp.fit_quadrature(P, nodes, weights, samples_u)

plot(t, cp.E(u_hat, dist))
plot(t, cp.Var(u_hat, dist))
\end{lstlisting}
\end{chaospy}


\begin{frame}
 \frametitle{Traveling with constant acceleration, pseudo-spectral projection with \underline{C}lenshaw-Curtis and Smolyak sparse grid.}
  \begin{figure}
  \includegraphics[width=0.85\textwidth]{solution5.png}
 \end{figure}
\end{frame}



 \begin{chaospy}{Traveling with constant acceleration, point collocation with random samples and least squares minimization.}
 \scriptsize
 \begin{lstlisting}[language=Python]
def s(t, v0, a):
    return v0*t + 0.5*a*t**2

v0 = cp.Uniform(1,2)
a = cp.Beta(2,2)
dist = cp.J(v0,a)

t = np.linspace(0,10,1000)

P = cp.orth_ttr(5, dist)
nodes = dist.sample(2*len(P))
samples_u = [s(t, *n) for n in nodes.T]
u_hat = cp.fit_regression(P, nodes, samples_u, rule="LS")

plot(t, cp.E(u_hat, dist))
plot(t, cp.Var(u_hat, dist))
\end{lstlisting}
\end{chaospy}


\begin{frame}
 \frametitle{Traveling with constant acceleration, point collocation with random samples and least squares minimization.}
  \begin{figure}
  \includegraphics[width=0.85\textwidth]{solution6.png}
 \end{figure}
\end{frame}



 \begin{chaospy}{Traveling with constant acceleration, point collocation with Ha\underline{m}mersley samples and \underline{T}ikhonov regularization.}
 \scriptsize
 \begin{lstlisting}[language=Python]
def s(t, v0, a):
    return v0*t + 0.5*a*t**2

v0 = cp.Uniform(1,2)
a = cp.Beta(2,2)
dist = cp.J(v0,a)

t = np.linspace(0,10,1000)

P = cp.orth_ttr(5, dist)
nodes = dist.sample(2*len(P), "M")
samples_u = [s(t, *n) for n in nodes.T]
u_hat = cp.fit_regression(P, nodes, samples_u, rule="T")

plot(t, cp.E(u_hat, dist))
plot(t, cp.Var(u_hat, dist))
\end{lstlisting}
\end{chaospy}


\begin{frame}
 \frametitle{Traveling with constant acceleration, point collocation with Ha\underline{m}mersley samples and \underline{T}ikhonov regularization.}
  \begin{figure}
  \includegraphics[width=0.85\textwidth]{solution7.png}
 \end{figure}
\end{frame}


\begin{chaospy}{A different differential equation}
 \scriptsize
 \begin{lstlisting}[language=Python]
a = cp.Normal(4,1)
I = cp.Uniform(2, 6)
dist_Q = cp.J(a, I)
dist_R = cp.J(cp.Normal(), cp.Uniform())

x = np.linspace(0,1,100)

P = cp.orth_ttr(2, dist_R)
nodes_R, weights_R = cp.generate_quadrature(3, dist_R)
nodes_Q = dist_Q.inv(dist_R.fwd(nodes_R))
weights_Q = \
   weights_R*dist_Q.pdf(nodes_Q)/dist_R.pdf(nodes_R)

samples_u = [u(x, *node) for node in nodes_Q.T]
u_hat = cp.fit_quadrature(P, nodes_R, weights_Q, samples_u)
\end{lstlisting}
\end{chaospy}

\begin{chaospy}{A different differential equation}
  \begin{figure}
  \includegraphics[width=0.85\textwidth]{solution9.png}
 \end{figure}
\end{chaospy}




 \begin{chaospy}{A different differential equation}
 \scriptsize
 \begin{lstlisting}[language=Python]
a = cp.Normal(4,1)
I = cp.Uniform(2, 6)
dist = cp.J(a, I)

x = np.linspace(0,1,100)

P, norm = cp.orth_ttr(5, dist, retall=True)

q0, q1 = cp.variable(2)

P_nk = cp.outer(P, P)
E_ak = cp.E(q0*P, dist)
E_ik = cp.E(q1*P, dist)
E_nk = cp.E(P_nk, dist)
\end{lstlisting}
\end{chaospy}

 \begin{chaospy}{A different differential equation}
 \scriptsize
 \begin{lstlisting}[language=Python]
def f(c_k,x):
    return (c_k + E_ak)*cp.sum(E_nk, -1)/norm
        
solver = odespy.RK4(f)
c_0 = E_ik/norm
solver.set_initial_condition(c_0)
c_n, x_ = solver.solve(x)
u_hat = cp.sum(P*c_n,-1)

E = cp.E(u_hat, dist)
Var = cp.Var(u_hat, dist)
 
plot(x, cp.E(u_hat, dist))
plot(x, cp.Var(u_hat, dist))
\end{lstlisting}
\end{chaospy}

\begin{chaospy}{A different differential equation}
  \begin{figure}
  \includegraphics[width=0.85\textwidth]{solution8.png}
 \end{figure}
\end{chaospy}



\end{document}