\item The circulant embedding approach is a tool to create efficiently and independently identical distributed samples.
\item The covariance matrix corresponding to (\ref{covariance_function}) has block symmetric Toeplitz structure, possibly with symmetric blocks (BST(S)TB).
\item This BST(S)TB matrix can be described with its first block rows and its first block columns.
\end{itemize}
\end{frame}
\begin{frame}{Circulant embedding}
\begin{itemize}
\item Goal: Try to decompose the covariance matrix in an efficient way for
\begin{equation*}
X = RZ, \quad\text{where } A = RR^T.
\end{equation*}
\item This is done by embedding the BST(S)TB matrix in an block circulant matrix with circulant blocks.%
\item For circulant matrices it holds
\begin{equation*}
B = W_{N,N}\Lambda W^*_{N,N},
\end{equation*}
where $W_{N,N}$ is the Fourier matrix for 2 dimensions and $\Lambda=\diag(\lambda)$ with $\lambda$ being the vector of eigenvalues. The eigenvalues can be computed with $\lambda= N W_{N,N} b^r$, where here $b^r$ is a $N \times N$ matrix containing the first block rows.
\end{itemize}
\end{frame}
\begin{frame}{Circulant embedding - Algorithm}
\begin{algorithm}[H]
\caption{Circulant embedding.}
\label{Circulant embedding algorithm}
\begin{algorithmic}[1]
{\footnotesize
\STATE Compute first rows and first columns of covariance matrix: $a^r$ and $a^c$\hfill$\mathcal{O}(N^2)$
\STATE Embed $a^r$ and $a^c$ to create a circulant matrix: $b^r \leftarrow\texttt{MOCE}(a^r, a^c)$
{\footnotesize For almost all $\omega\in\Omega$, the bilinear form $b_\omega(u,v)$ is bounded and coercive in $H_0^1(D)$ with respect to the norm $\vert\cdot\vert_{H^1(D)}$ with the constants $\amax(\omega)$ and $\amin(\omega)$, respectively. Moreover, there exists a unique solution $u(\omega, \cdot)\in H_0^1(D)$ to the variational problem (\ref{weak_formulation}) and
\begin{equation*}
\vert u(\omega, \cdot) \vert_{H^1(D)}\lesssim\frac{\Vert f \Vert_{H^{t-1}(D)}}{\amin(\omega)}.
\end{equation*}}
\end{lemma}
\begin{itemize}
\item Proof idea: Straight forward calculations with assumptions and Lax-Milgram.
\end{itemize}
\begin{theorem}[Existence]
{\footnotesize The weak solution $u$ of (\ref{eq:model_problem}-\ref{boundary_conditions2}) is unique and belongs to $L^p(\Omega, H_0^1(D))$ for all $p < p_*$.}
\end{theorem}
\begin{itemize}
\item Proof idea: Extend the Lemma above on Bochner spaces with Hölder's inequality and the assumptions.
{\footnotesize Suppose that there are positive constants $\alpha, \beta, \gamma, c_1, c_2, c_3 > 0$ such that $\alpha\geq\frac{1}{2}\min(\beta, \gamma)$ and
