Detailed-description-of-employed-methods-and-their-adaptations.md

@@ -16,17 +16,36 @@ with $`r_0`$ being the residual $`r_0=b-Ax_0`$ for the $`k=0`$ iteration. Partic
 
 
 
-<details><summary>The Smoother</summary>
+<details><summary>Domain Decomposition</summary>
 
 
 Block decompositions are used for parallelization. Each process obtains a block of lattice
 sites from a block decomposition and performs the calculations associated with these lattice sites. 
 
-Each process owns a lattice-block Pi of a block decomposition and ghost cells on the boundaries which contain the recent values from neighboring processes. Ghost cells
+Each process owns a lattice-block $`P_i`$ of a block decomposition and ghost cells on the boundaries which contain the recent values from neighboring processes. Ghost cells
 have to be updated whenever a computation demands information from nearest
 neighbor lattice sites, e.g., when evaluating the Wilson-Dirac operator D on a
 vector $`\psi`$. Here, this updating process is represented by the arrows pointing to
 the ghost cells.
 
 <img align="center" src="uploads/c71599ce697e6921cc347ebd12f4b1d5/Domain_decomposition.png" alt="drawing" width="400"/>
+
+</details>
+
+
+
+<details><summary>The Smoother</summary>
+
+The smoother in DD-alphaAMG is the Schwarz Alternating procedure, it is
+a colored version of the multiplicative Schwarz method that allows for
+parallelization. 
+
+It relies on block solves of the local systems using $`k`$ steps of MINRES, with a restart after every iteration.
+
+In a 2 Color SAP the lattice is partitioned in red and black lattice blocks in a chessboard-like manner as shown:
+
+<img align="center" src="uploads/a7437d50bbb45de5c58f9102730a2d5a/2color_bloc_decomposition.png" alt="drawing" width="400"/>
+
+
+
 </details>