4.2.4 Machine model parallelism

Codes operating under the WRAPPER target an abstract machine that is assumed to consist of one or more logical processors that can compute concurrently. Computational work is divided among the logical processors by allocating ``ownership'' to each processor of a certain set (or sets) of calculations. Each set of calculations owned by a particular processor is associated with a specific region of the physical space that is being simulated, only one processor will be associated with each such region (domain decomposition).

In a strict sense the logical processors over which work is divided do not need to correspond to physical processors. It is perfectly possible to execute a configuration decomposed for multiple logical processors on a single physical processor. This helps ensure that numerical code that is written to fit within the WRAPPER will parallelize with no additional effort and is also useful when debugging codes. Generally, however, the computational domain will be subdivided over multiple logical processors in order to then bind those logical processors to physical processor resources that can compute in parallel.

4.2.4.1 Tiles

Computationally, associated with each region of physical space allocated to a particular logical processor, there will be data structures (arrays, scalar variables etc...) that hold the simulated state of that region. We refer to these data structures as being owned by the processor to which their associated region of physical space has been allocated. Individual regions that are allocated to processors are called tiles. A processor can own more than one tile. Figure 4.3 shows a physical domain being mapped to a set of logical processors, with each processors owning a single region of the domain (a single tile). Except for periods of communication and coordination, each processor computes autonomously, working only with data from the tile (or tiles) that the processor owns. When multiple tiles are alloted to a single processor, each tile is computed on independently of the other tiles, in a sequential fashion.

**Figure 4.3:** The WRAPPER provides support for one and two dimensional decompositions of grid-point domains. The figure shows a hypothetical domain of total size $N_{x}N_{y}N_{z}$ . This hypothetical domain is decomposed in two-dimensions along the $N_{x}$ and $N_{y}$ directions. The resulting **tiles** are **owned** by different processors. The **owning** processors perform the arithmetic operations associated with a **tile**. Although not illustrated here, a single processor can **own** several **tiles**. Whenever a processor wishes to transfer data between tiles or communicate with other processors it calls a WRAPPER supplied function.
$\resizebox{5in}{!}{ \includegraphics{part4/domain_decomp.eps} }$

4.2.4.2 Tile layout

Tiles consist of an interior region and an overlap region. The overlap region of a tile corresponds to the interior region of an adjacent tile. In figure 4.4 each tile would own the region within the black square and hold duplicate information for overlap regions extending into the tiles to the north, south, east and west. During computational phases a processor will reference data in an overlap region whenever it requires values that outside the domain it owns. Periodically processors will make calls to WRAPPER functions to communicate data between tiles, in order to keep the overlap regions up to date (see section 4.2.8). The WRAPPER functions can use a variety of different mechanisms to communicate data between tiles.

**Figure 4.4:** A global grid subdivided into tiles. Tiles contain a interior region and an overlap region. Overlap regions are periodically updated from neighboring tiles.
$\resizebox{5in}{!}{ \includegraphics{part4/tiled-world.eps} }$