Stencil Pattern. CS 472 Concurrent & Parallel Programming University of Evansville

Transcription

1 Stencil Pattern CS 472 Concurrent & Parallel Programming University of Evansville Selection of slides from CIS 41/51 Introduction to Parallel Computing Department of Computer and Information Science, University of Oregon

2 Recall: Partitioning 1D 2D Data is divided into non-overlapping regions (avoid write conflicts, race conditions) equal-sized regions (improve load balancing) 2

3 Recall: Partitioning 1D 2D Data is divided into non-overlapping regions (avoid write conflicts, race conditions) equal-sized regions (improve load balancing) 3D 3

4 Stencil Pattern A stencil pattern is a map where each output depends on a neighborhood of inputs These inputs are a set of fixed offsets relative to the output position A stencil output is a function of a neighborhood of elements in an input collection Applies the stencil to select the inputs Data access patterns of stencils are regular Stencil is the shape of neighborhood Stencil remains the same 4

5 Serial Stencil Example (part 1) 5

6 Serial Stencil Example (part 2) How would we parallelize this? 6

7 What is the stencil pattern? 7

8 What is the stencil pattern? Input array 8

9 What is the stencil pattern? Function 9

10 What is the stencil pattern? Output Array 1

11 What is the stencil pattern? neighborhood i i-1 i+1 This stencil has 3 elements in the neighborhood: i-1, i, i+1 11

12 What is the stencil pattern? neighborhood i i-1 i+1 Applies some function to them 12

13 What is the stencil pattern? And outputs to the i th position of the output array 13

14 Stencil Patterns Stencils can operate on one dimensional and multidimensional data Stencil neighborhoods can range from compact to sparse, square to cube, and anything else! It is the pattern of the stencil that determines how the stencil operates in an application 14

15 2-Dimensional Stencils 4-point stencil Center cell (P) is not used 5-point stencil Center cell (P) is used as well 9-point stencil Center cell (C) is used as well Source: 15

16 3-Dimensional Stencils 6-point stencil (7-point stencil) Source: 24-point stencil (25-point stencil) 16

17 Stencil Example A Here is our array, A

18 Stencil Example Here is our array A B is the output array Initialize to all Apply a stencil operation to the inner square of the form: B(i,j) = avg( A(i,j), A(i-1,j), A(i+1,j), A(i,j-1), A(i,j+1) ) A What is the stencil? 18

19 Stencil Pattern Procedure A 1) Average all blue squares

20 Stencil Pattern Procedure B 1) Average all blue squares 2) Store result in B 4.4 2

21 Stencil Pattern Procedure A 1) Average all blue squares 2) Store result in B 3) Repeat 1 and 2 for all green squares

22 Practice! A B

23 Stencil Pattern Practice A B

24 Stencil Pattern Practice A B

25 Stencil Pattern Practice A B

26 Stencil Pattern Practice A B

27 Serial Stencil Example (part 1) 27

28 Serial Stencil Example (part 2) How would we parallelize this? a[i] Updates occur in place!!! 28

29 Stencil Pattern with In Place Update 29

30 Stencil Pattern with In Place Update Input array 3

31 Stencil Pattern with In Place Update Function 31

32 Stencil Pattern with In Place Update Input Array!!! 32

33 Stencil Example A Here is our array, A

34 Stencil Example A Here is our array A Update A in place Apply a stencil operation to the inner square of the form: 9 7 A(i,j) = avg( A(i,j), A(i-1,j), A(i+1,j), A(i,j-1), A(i,j+1) ) 6 4 What is the stencil? 34

35 Stencil Pattern Procedure 1) Average all blue squares

36 Stencil Pattern Procedure 1) Average all blue squares 2) Store result in red square

37 Stencil Pattern Procedure 1) Average all blue squares 2) Store result in red square 3) Repeat 1 and 2 for all green squares

38 Practice! A B

39 Stencil Pattern Practice A

40 Stencil Pattern Practice A

41 What is the stencil pattern? A

42 What is the stencil pattern? A

43 What is the stencil pattern? A

44 What is the stencil pattern? A

45 What is the stencil pattern? A

46 What is the stencil pattern? A

47 Different Cases Input Output Separate output array Input Output Updates occur in place

48 Which is correct? Input Output Is this output incorrect? 48

49 Iterative Codes Iterative codes are ones that update their data in steps At each step, a new value of an element is computed using a formula based on other elements Once all elements are updated, the computation proceeds to the next step or completes Iterative codes are most commonly found in computer simulations of physical systems for scientific and engineering applications Computational fluid dynamics Electromagnetics modeling They are often applied to solve partial differential equations Jacobi iteration Gauss-Seidel iteration Successive over relaxation 49

50 Iterative Codes and Stencils Stencils essentially define which elements are used in the update formula Because the data is organized in a regular manner, stencils can be applied across the data uniformly 5

51 Simple 2D Example Consider the following code for k=1, 1 for i=1, N-2 for j = 1, N-2 a[i][j] =.25 * (a[i][j] } } } + a[i-1][j] + a[i+1][j] + a[i][j-1] + a[i][j+1]) 5-point stencil Do you see anything interesting? How would you parallelize? 51

52 2-Dimension Jacobi Iteration Initialize each array element to some value At each step, update each array element to the arithmetic mean of its N, S, E, W neighbors Iterate until array values converge Here we are using a 4-point stencil 4-point stencil It is different from before because we want to update all array elements simultaneously How? 52

53 2-Dimension Jacobi Iteration hot Consider a 2D array of elements Initialize each array element to some value At each step, update each array element to the arithmetic mean of its N, S, E, W neighbors Iterate until array values converge cold Heat equation simulation 4-point stencil Step Step 2 Step 4 Step 6 Step 8 Step 1 53