Iterative Stencil Loops

ISLs perform a sequence of sweeps (called timesteps) through a given array.[2] Generally this is a 2- or 3-dimensional regular grid.[3] The elements of the arrays are often referred to as cells. In each timestep, all array elements are updated.[2] Using neighboring array elements in a fixed pattern (the stencil), each cell's new value is computed. In most cases boundary values are left unchanged, but in some cases (e.g. LBM codes) those need to be adjusted during the computation as well. Since the stencil is the same for each element, the pattern of data accesses is repeated.[4]

More formally, we may define ISLs as a 5-tuple ${\displaystyle (I,S,S_{0},s,T)}$ with the following meaning:[3]

• ${\displaystyle I=\prod _{i=1}^{k}[0,\ldots ,n_{i}]}$ is the index set. It defines the topology of the array.
• ${\displaystyle S}$ is the (not necessarily finite) set of states, one of which each cell may take on on any given timestep.
• ${\displaystyle S_{0}\colon \mathbb {Z} ^{k}\to S}$ defines the initial state of the system at time 0.
• ${\displaystyle s\in \prod _{i=1}^{l}\mathbb {Z} ^{k}}$ is the stencil itself and describes the actual shape of the neighborhood. There are ${\displaystyle l}$ elements in the stencil.
• ${\displaystyle T\colon S^{l}\to S}$ is the transition function which is used to determine a cell's new state, depending on its neighbors.

Since I is a k-dimensional integer interval, the array will always have the topology of a finite regular grid. The array is also called simulation space and individual cells are identified by their index ${\displaystyle c\in I}$. The stencil is an ordered set of ${\displaystyle l}$ relative coordinates. We can now obtain for each cell ${\displaystyle c}$ the tuple of its neighbors indices ${\displaystyle I_{c}}$

${\displaystyle I_{c}=\{j\mid \exists x\in s:j=c+x\}\,}$

Their states are given by mapping the tuple ${\displaystyle I_{c}}$ to the corresponding tuple of states ${\displaystyle N_{i}(c)}$, where ${\displaystyle N_{i}\colon I\to S^{l}}$ is defined as follows:

${\displaystyle N_{i}(c)=(s_{1},\ldots ,s_{l}){\text{ with }}s_{j}=S_{i}(I_{c}(j))\,}$

This is all we need to define the system's state for the following time steps ${\displaystyle S_{i+1}\colon \mathbb {Z} ^{k}\to S}$ with ${\displaystyle i\in \mathbb {N} }$:

${\displaystyle S_{i+1}(c)={\begin{cases}T(N_{i}(c)),&c\in I\\S_{i}(c),&c\in \mathbb {Z} ^{k}\setminus I\end{cases}}}$

Note that ${\displaystyle S_{i}}$ is defined on ${\displaystyle \mathbb {Z} ^{k}}$ and not just on ${\displaystyle I}$ since the boundary conditions need to be set, too. Sometimes the elements of ${\displaystyle I_{c}}$ may be defined by a vector addition modulo the simulation space's dimension to realize toroidal topologies:

${\displaystyle I_{c}=\{j\mid \exists x\in s:j=((c+x)\mod (n_{1},\ldots ,n_{k}))\}}$

This may be useful for implementing periodic boundary conditions, which simplifies certain physical models.

Example: 2D Jacobi iteration

Data dependencies of a selected cell in the 2D array.

To illustrate the formal definition, we'll have a look at how a two dimensional Jacobi iteration can be defined. The update function computes the arithmetic mean of a cell's four neighbors. In this case we set off with an initial solution of 0. The left and right boundary are fixed at 1, while the upper and lower boundaries are set to 0. After a sufficient number of iterations, the system converges against a saddle-shape.

${\displaystyle {\begin{aligned}I&=[0,\ldots ,99]^{2}\\S&=\mathbb {R} \\S_{0}&:\mathbb {Z} ^{2}\to \mathbb {R} \\S_{0}((x,y))&={\begin{cases}1,&x<0\\0,&0\leq x<100\\1,&x\geq 100\end{cases}}\\s&=((0,-1),(-1,0),(1,0),(0,1))\\T&\colon \mathbb {R} ^{4}\to \mathbb {R} \\T((x_{1},x_{2},x_{3},x_{4}))&=0.25\cdot (x_{1}+x_{2}+x_{3}+x_{4})\end{aligned}}}$

${\displaystyle S_{0}}$

${\displaystyle S_{200}}$

${\displaystyle S_{400}}$

${\displaystyle S_{600}}$

${\displaystyle S_{800}}$

${\displaystyle S_{1000}}$

2D Jacobi Iteration on a ${\displaystyle 100^{2}}$ Array

The shape of the neighborhood used during the updates depends on the application itself. The most common stencils are the 2D or 3D versions of the von Neumann neighborhood and Moore neighborhood. The example above uses a 2D von Neumann stencil while LBM codes generally use its 3D variant. Conway's Game of Life uses the 2D Moore neighborhood. That said, other stencils such as a 25-point stencil for seismic wave propagation[5] can be found, too.

9-point 2D stencil

5-point 2D stencil

7-point 3D stencil

25-point 3D stencil

A selection of stencils used in various scientific applications.

Many simulation codes may be formulated naturally as ISLs. Since computing time and memory consumption grow linearly with the number of array elements, parallel implementations of ISLs are of paramount importance to research.[6] This is challenging since the computations are tightly coupled (because of the cell updates depending on neighboring cells) and most ISLs are memory bound (i.e. the ratio of memory accesses to calculations is high).[7] Virtually all current parallel architectures have been explored for executing ISLs efficiently;[8] at the moment GPGPUs have proven to be most efficient.[9]

Due to both the importance of ISLs to computer simulations and their high computational requirements, there are a number of efforts which aim at creating reusable libraries to support scientists in performing stencil-based computations. The libraries are mostly concerned with the parallelization, but may also tackle other challenges, such as IO, steering and checkpointing. They may be classified by their API.

• Advanced Simulation Library
• Finite difference method
• Computer simulation
• Five-point stencil
• Stencil jumping
• Stencil (numerical analysis)

• Physis
• LibGeoDecomp
• waLBerla