1. Notes on Assignment 3 OpenMP Stencil Pattern
ITCS 4/5145 Parallel computing, UNC-Charlotte, B. Wilkinson Feb 25, Assign3Notes.ppt 1

2. Stencil Pattern Examples
A stencil describes a 2- or 3- dimensional layout of processes, with each process able to communicate only with its neighbors.
Appears in simulating many real-life situations.
Examples
Solving partial differential equations using discretized methods, which may be for:
Modeling engineering structures
Weather forecasting, see intro to course slides1a-1
Particle dynamics simulations
Modeling chemical and biological structures 2

3. Stencil pattern On each iteration, each node communicates
with neighbors to get stored
computed values
Two-way connection
Compute node
Source/sink 3

4. (Iterative synchronous) stencil pattern
Often globally synchronous and iterative:
Processes compute and communicate only with their neighbors, exchanging results
Check termination condition
Repeat
Stop 4

5. Finite Difference Method
Example application
Finite Difference Method
Find values of f(x,y) in this area, given boundary conditions.
Divide space into an array of solution points

6. Example (Static) heat distribution
Static heat equation (Laplace’s equation):

7. Value of f(x,y) given by average of four neighboring point
(left, right, up, down).

8. Sequential Code
Assume N x N array of points, h[i][j], including fixed boundary points. Using a fixed number of iterations
for (iteration = 0; iteration < limit; iteration++) {
for (i = 1; i < n-1; i++)
for (j = 1; j < n-1; j++)
g[i][j] = 0.25*(h[i-1][j]+h[i+1][j]+h[i][j-1]+h[i][j+1]);
for (i = 1; i < n; i++) /* update points */
for (j = 1; j < n; j++)
h[i][j] = g[i][j]; }
1 to n-2, skip boundary edges (values 0 and n-1)
Newly computed values are placed in a new array.
Then copied back to the original.
Called a Jacobi iteration when next iteration use only last iteration values
Multiplying by 0.25 faster than dividing by 4
6b.8

9. Improved Jacobi Iteration
int main() { int i, j, current, next; double A[2][N][M]; // A[0] is initialized with data somehow and duplicated into A[1] current = 0; next = (current + 1) % 2; for (time = 0; time < MAX_ITERATION; time++) { for (i = 1; i < N-1; i++) for (j = 1; j < M-1; j++) A[next][i][j] = (A[current][i-1][j] + A[current][i+1][j] + A[current][i][j-1] + A[current][i][j+1]) * 0.25; current = next; } // Final result is in A[current] ...
Add another dimension of size 2 to A.
A[0] is h and A[1] is g, alternating.
We toggle between copies of the array
Avoids copying values back into original array.

10. Application Example
A room has four walls and a fireplace. Temperature of wall is 20°C, and temperature of fireplace is 100°C. Write a parallel program using Jacobi iteration to compute the temperature inside the room and plot (preferably in color) temperature contours at 10°C intervals. 10

11. Sample student output 11

12. Adding heat sources
One can add heat sources inside the room such as humans (stationary).
Just make these points of fixed values in array
Similar problems -- heat distribution on a printed circuit board.

13. Extra credit
Dynamic Heat Equation

15. Possible code for (int t=t0; t<t1; t++) for (int x=1; x<M; x++)
for (int y=1; y<L; y++)
F(t+1,x,y) = F(t, x, y)
+ CX * (F(t,x-1,y) + F(t,x+1,y) -2*F(t,x,y))
+ CY * (F(t,x,y-1) + F(t,x,y+1) -2*F(t,x,y));
where:CX =α∆t/∆x2, and CY = α∆t/∆y2
See

16. Questions