Wrote initial version of admm implementation details

latex/thesis/chapters/lp_dec_using_admm.tex

@@ -724,6 +724,70 @@ The method chosen here is the one presented in \cite{lautern}.
 \section{Implementation Details}%
 \label{sec:lp:Implementation Details}
 
+The development process used to implement this decoding algorithm is the same
+as outlined in section
+\ref{sec:prox:Implementation Details} for proximal decoding.
+At first, an initial version was implemented in Python, before repeating the
+process using C++ to achieve higher performance.
+Again, the performance can be increased by reframing the operations in such
+a way that the computation can take place primarily with element-wise
+operations and matrix-vector multiplication, since these operations
+are highly optimized in the software libraries used for the implementation.
+
+In the summation operation in line 8 of algorithm \ref{alg:admm}, the
+components of each $\boldsymbol{z}_j$ and $\boldsymbol{u}_j$ relating to a
+given \ac{VN} $i$ have to be found.
+This operation can be streamlined by observing that the transfer matrices
+$\boldsymbol{T}_j,\hspace{1mm}j\in\mathcal{J}$ are able to perform the mapping
+they were devised for in both directions:
+with $\boldsymbol{T}_j \tilde{\boldsymbol{c}}$, the $d_j$ components of
+$\tilde{\boldsymbol{c}}$ required for parity check $i$ are selected;
+with $\boldsymbol{T}_j^\text{T} \boldsymbol{z}_j$, the $d_j$ components of
+$\boldsymbol{z}_j$ can be mapped onto a vector of length $n$, each component
+at the position corresponding to the \ac{VN} it relates to.
+Using this observation, the sum can be written as%
+%
+\begin{align*}
+    \sum_{j\in N_v\left( i \right) }\left( \boldsymbol{T}_j^\text{T} \left( \boldsymbol{z}_j
+        - \boldsymbol{u}_j \right) \right)_i
+.\end{align*}
+Further noticing that the vectors
+$\boldsymbol{T}_j^\text{T}\left( \boldsymbol{z}_j - \boldsymbol{u}_j \right),
+\hspace{1mm} j\in\mathcal{J} $
+unrelated to \ac{VN} $i$ have $0$ as the $i$th component, the set of indices
+the summation takes place over can be extended to $\mathcal{J}$, allowing the
+expression to be rewritten to%
+%
+\begin{align*}
+    \sum_{j\in \mathcal{J}}\left( \boldsymbol{T}_j^\text{T} \left( \boldsymbol{z}_j
+        - \boldsymbol{u}_j \right) \right)_i
+    = \left( \sum_{j\in\mathcal{J}} \boldsymbol{T}_j^\text{T}
+        \left( \boldsymbol{z}_j - \boldsymbol{u}_j \right) \right)_i
+.\end{align*}
+%
+Defining%
+%
+\begin{align*}
+    \boldsymbol{D} := \begin{bmatrix} 
+        d_1 \\
+        \vdots \\
+        d_m
+    \end{bmatrix}%
+    \hspace{5mm}%
+    \text{and}%
+    \hspace{5mm}%
+    \boldsymbol{M} := \sum_{j\in\mathcal{J}} \boldsymbol{T}_j^\text{T}
+        \left( \boldsymbol{z}_j - \boldsymbol{u}_j \right)
+\end{align*}%
+%
+the $\tilde{\boldsymbol{c}}$ update can then be rewritten as%
+%
+\begin{align*}
+    \tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ -1} \circ
+        \left( \boldsymbol{M} - \frac{1}{\mu}\boldsymbol{\gamma} \right)  
+.\end{align*}
+%
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results}%