Numerical Algorithms
|
|
- Tobias Ford
- 6 years ago
- Views:
Transcription
1 Chapter 10 Slide 464 Numerical Algorithms
2 Slide 465 Numerical Algorithms In textbook do: Matrix multiplication Solving a system of linear equations
3 Slide 466 Matrices A Review An n m matrix Column a 0,0 a 0,1 a 0,m-2 a 0,m-1 a 1,0 a 1,1 a 1,m-2 a 1,m-1 Row a n-2,0 a n-1,0 a n-2,1 a n-1,1 a n-2,m-2 a n-2,m-1 a n-1,m-2 an-1,m-1
4 Slide 467 Matrix Addition Involves adding corresponding elements of each matrix to form the result matrix. Given the elements of A as a i,j and the elements of B as b i,j, each element of C is computed as c i,j = a i,j + b i,j (0 i < n, 0 j < m)
5 Slide 468 Matrix Multiplication Multiplication of two matrices, A and B, produces the matrix C whose elements, c i,j (0 i < n, 0 j < m), are computed as follows: l 1 c i, j = a i,k b k,j k = 0 where A is an n l matrix and B is an l m matrix.
6 Matrix multiplication, C = A B Slide 469 Multiply Column j Sum results Row i c i,j A B = C
7 Matrix-Vector Multiplication c = A b Slide 470 Matrix-vector multiplication follows directly from the definition of matrix-matrix multiplication by making B an n 1 matrix (vector). Result an n 1 matrix (vector). A b = c Row i sum c i
8 Slide 471 Relationship of Matrices to Linear Equations A system of linear equations can be written in matrix form: Ax = b Matrix A holds the a constants x is a vector of the unknowns b is a vector of the b constants.
9 Implementing Matrix Multiplication Slide 472 Sequential Code Assume throughout that the matrices are square (n n matrices). The sequential code to compute A B could simply be for (i = 0; i < n; i++) for (j = 0; j < n; j++) { c[i][j] = 0; for (k = 0; k < n; k++) c[i][j] = c[i][j] + a[i][k] * b[k][j]; } This algorithm requires n 3 multiplications and n 3 additions, leading to a sequential time complexity of Ο(n 3 ). Very easy to parallelize.
10 Parallel Code Slide 473 With n processors (and n n matrices), can obtain: Time complexity of O(n 2 ) with n processors Each instance of inner loop independent and can be done by a separate processor Time complexity of O(n) with n 2 processors One element of A and B assigned to each processor. Cost optimal since O(n 3 ) = n O(n 2 ) = n 2 O(n)]. Time complexity of O(log n) with n 3 processors By parallelizing the inner loop. Not cost-optimal since O(n 3 ) n 3 O(log n)). O(log n) lower bound for parallel matrix multiplication.
11 Partitioning into Submatrices Slide 474 Suppose matrix divided into s 2 submatrices. Each submatrix has n/ s n/s elements. Using notation A p,q as submatrix in submatrix row p and submatrix column q: for (p = 0; p < s; p++) for (q = 0; q < s; q++) { C p,q = 0; /* clear elements of submatrix */ for (r = 0; r < m; r++)/* submatrix multiplication &*/ C p,q = C p,q + A p,r * B r,q ;/*add to accum. submatrix*/ } The line C p,q = C p,q + A p,r * B r,q ; means multiply submatrix A p,r and B r,q using matrix multiplication and add to submatrix C p,q using matrix addition. Known as block matrix multiplication.
12 Slide 475 Block Matrix Multiplication Multiply q Sum results p = A B C
13 Submatrix multiplication a 0,0 a 0,1 a 0,2 a 0,3 a 1,0 a 1,1 a 1,2 a 1,3 b 0,0 b 0,1 b 0,2 b 0,3 b 1,0 b 1,1 b 1,2 b 1,3 Slide 476 (a) Matrices a 2,0 a 2,1 a 2,2 a 2,3 b 2,0 b 2,1 b 2,2 b 2,3 a 3,0 a 3,1 a 3,2 a 3,3 b 3,0 b 3,1 b 3,2 b 3,3 A 0,0 B 0,0 A 0,1 B 1,0 a 0,0 a 0,1 b 0,0 b 0,1 a 0,2 a 0,3 b 2,0 b 2,1 + (b) Multiplying A 0,0 B 0,0 to obtain C 0,0 a 1,0 a 1,1 b 1,0 b 1,1 a 0,0 b 0,0 +a 0,1 b 1,0 a 0,0 b 0,1 +a 0,1 b 1,1 a 1,2 a 1,3 b 3,0 b 3,1 a 0,2 b 2,0 +a 0,3 b 3,0 a 0,2 b 2,1 +a 0,3 b 3,1 = + a 1,0 b 0,0 +a 1,1 b 1,0 a 1,0 b 0,1 +a 1,1 b 1,1 a 1,2 b 2,0 +a 1,3 b 3,0 a 1,2 b 2,1 +a 1,3 b 3,1 a 0,0 b 0,0 +a 0,1 b 1,0 +a 0,2 b 2,0 +a 0,3 b 3,0 a 0,0 b 0,1 +a 0,1 b 1,1 +a 0,2 b 2,1 +a 0,3 b 3,1 = a 1,0 b 0,0 +a 1,1 b 1,0 +a 1,2 b 2,0 +a 1,3 b 3,0 a 1,0 b 0,1 +a 1,1 b 1,1 +a 1,2 b 2,1 +a 1,3 b 3,1 = C 0,0
14 Direct Implementation Slide 477 One processor to compute each element of C - n 2 processors would be needed. One row of elements of A and one column of elements of B needed. Some of same elements sent to more than one processor. Can use submatrices. Column j b[][j] Row i a[i][] Processor P i,j c[i][j]
15 Performance Improvement Using tree construction n numbers can be added in log n steps using n processors: Slide 478 a 0,0 b 0,0 a 0,1 b 1,0 a 0,2 b 2,0 a 0,3 b 3,0 P 0 P 1 P 2 P P 0 P 2 Computational time complexity of Ο(log n) using n 3 processors. P 0 + c 0,0
16 Slide 479 Recursive Implementation i j i P 0 P 1 P 2 P 3 P 0 + P 1 P 2 + P 3 j A pp A pq B pp B pq C pp C pq P 4 + P 5 P 6 + P 7 A qp A qq B qp B qq C qp C qq P 4 P 5 P 6 P 7 Apply same algorithm on each submatrix recursivly. Excellent algorithm for a shared memory systems because of locality of operations.
17 Recursive Algorithm mat_mult(a pp, B pp, s) { if (s == 1) /* if submatrix has one element */ C = A * B; /* multiply elements */ else { /* continue to make recursive calls */ s = s/2; /* no of elements in each row/column */ P0 = mat_mult(a pp, B pp, s); P1 = mat_mult(a pq, B qp, s); P2 = mat_mult(a pp, B pq, s); P3 = mat_mult(a pq, B qq, s); P4 = mat_mult(a qp, B pp, s); P5 = mat_mult(a qq, B qp, s); P6 = mat_mult(a qp, B pq, s); P7 = mat_mult(a qq, B qq, s); C pp = P0 + P1; /* add submatrix products to */ C pq = P2 + P3; /* form submatrices of final matrix */ C qp = P4 + P5; C qq = P6 + P7; } return (C); /* return final matrix */ } Slide 480
18 Slide 481 Mesh Implementations Cannon s algorithm Fox s algorithm (not in textbook but similar complexity) Systolic array All involve using processor arranged a mesh and shifting elements of the arrays through the mesh. Accumulate the partial sums at each processor.
19 Mesh Implementations Cannon s Algorithm Slide 482 Uses a mesh of processors with wraparound connections (a torus) to shift the A elements (or submatrices) left and the B elements (or submatrices) up. 1.Initially processor P i,j has elements a i,j and b i,j (0 i < n, 0 k < n). 2. Elements are moved from their initial position to an aligned position. The complete ith row of A is shifted i places left and the complete jth column of B is shifted j places upward. This has the effect of placing the element a i,j+i and the element b i+j,j in processor P i,j,. These elements are a pair of those required in the accumulation of c i,j. 3.Each processor, P i,j, multiplies its elements. 4. The ith row of A is shifted one place right, and the jth column of B is shifted one place upward. This has the effect of bringing together the adjacent elements of A and B, which will also be required in the accumulation. 5. Each processor, P i,j, multiplies the elements brought to it and adds the result to the accumulating sum. 6. Step 4 and 5 are repeated until the final result is obtained (n - 1 shifts with n rows and n columns of elements).
20 Slide 483 Movement of A and B elements j i A P i,j B
21 Slide 484 Step 2 Alignment of elements of A and B i j B A j places i places b i+j,j a i,j+i
22 Step 4 One-place shift of elements of A and B Slide 485 i j B A P i,j
23 Systolic array Pumping action b 3,0 b 2,0 b 1,0 b 0,0 b 3,1 b 2,1 b 1,1 b 0,1 b 3,2 b 2,2 b 1,2 b 0,2 b 3,3 b 2,3 b 1,3 b 0,3 Slide 486 a 0,3 a 0,2 a 0,1 a 0,0 c 0,0 c 0,1 c 0,2 c 0,3 One cycle delay a 1,3 a 1,2 a 1,1 a 1,0 c 1,0 c 1,1 c 1,2 c 1,3 a 2,3 a 2,2 a 2,1 a 2,0 c 2,0 c 2,1 c 2,2 c 2,3 a 3,3 a 3,2 a 3,1 a 3,0 c 3,0 c 3,1 c 3,2 c 3,3
24 Matrix-Vector Multiplication Pumping action b 3 b 2 b 1 b 0 Slide 487 a 0,3 a 0,2 a 0,1 a 0,0 c 0 a 1,3 a 1,2 a 1,1 a 1,0 c 1 a 2,3 a 2,2 a 2,1 a 2,0 c 2 a 3,3 a 3,2 a 3,1 a 3,0 c 3
25 Solving a System of Linear Equations Slide 488 a n 1,0 x 0 + a n 1,1 x 1 + a n 1,2 x 2 + a n 1,n 1 x n 1 = b n 1... a 2,0 x 0 + a 2,1 x 1 + a 2,2 x 2 + a 2,n 1 x n 1 = b 2 a 1,0 x 0 + a 1,1 x 1 + a 1,2 x 2 + a 1,n 1 x n 1 = b 1 a 0,0 x 0 + a 0,1 x 1 + a 0,2 x 2 + a 0,n 1 x n 1 = b 0 which, in matrix form, is Ax = b Objective is to find values for the unknowns, x 0, x 1,, x n 1, given values for a 0,0, a 0,1,, a n 1,n 1, and b 0,, b n.
26 Solving a System of Linear Equations Slide 489 Dense matrices Gaussian Elimination - parallel time complexity O(n 2 ) Sparse matrices By iteration - depends upon iteration method and number of iterations but typically O(log n) Jacobi iteration Gauss-Seidel relaxation (not good for parallelization) Red-Black ordering Multigrid
27 Slide 490 Gaussian Elimination Convert general system of linear equations into triangular system of equations. Then be solved by Back Substitution. Uses characteristic of linear equations that any row can be replaced by that row added to another row multiplied by a constant. Starts at the first row and works toward the bottom row. At the ith row, each row j below the ith row is replaced by row j + (row i) ( a j,i / a i,i ). The constant used for row j is a j,i /a i,i. Has the effect of making all the elements in the ith column below the ith row zero because a j, i a j, i = a j, i + a i, i = 0 a i, i
28 Gaussian elimination Slide 491 Column Row Row i Row j Already cleared to zero Column i a ji Cleared to zero Step through
29 Slide 492 Partial Pivoting If a i,i is zero or close to zero, we will not be able to compute the quantity a j,i /a i,i. Procedure must be modified into so-called partial pivoting by swapping the ith row with the row below it that has the largest absolute element in the ith column of any of the rows below the ith row if there is one. (Reordering equations will not affect the system.) In the following, we will not consider partial pivoting.
30 Slide 493 Without partial pivoting: Sequential Code for (i = 0; i < n-1; i++) /* for each row, except last */ for (j = i+1; j < n; j++) {/*step thro subsequent rows */ m = a[j][i]/a[i][i]; /* Compute multiplier */ for (k = i; k < n; k++)/*last n-i-1 elements of row j*/ a[j][k] = a[j][k] - a[i][k] * m; b[j] = b[j] - b[i] * m;/* modify right side */ } The time complexity is O(n 3 ).
31 Parallel Implementation Slide 494 Column Row n i +1 elements (including b[i]) Row i Broadcast ith row Already cleared to zero
32 Analysis Communication Slide 495 n 1 broadcasts performed sequentially. ith broadcast contains n i + 1 elements. Time complexity of Ο(n 2 ) (see textbook) Computation After row broadcast, each processor P j beyond broadcast processor P i will compute its multiplier, and operate upon n j + 2 elements of its row. Ignoring the computation of the multiplier, there are n j + 2 multiplications and n j + 2 subtractions. Time complexity of Ο(n 2 ) (see textbook). Efficiency will be relatively low because all the processors before the processor holding row i do not participate in the computation again.
33 Pipeline implementation of Gaussian elimination Slide 496 P 0 P 1 P 2 P n-1 Row Broadcast rows
34 Strip Partitioning Row 0 Slide 497 P 0 n/p P 1 2n/p P 2 3n/p P 3 Poor processor allocation! Processors do not participate in computation after their last row is processed.
35 Cyclic-Striped Partitioning Slide 498 An alternative which equalizes the processor workload: Row 0 n/p P 0 2n/p P1 3n/p
36 Iterative Methods Slide 499 Time complexity of direct method at Ο(N 2 ) with N processors, is significant. Time complexity of iteration method depends upon: the type of iteration, number of iterations number of unknowns, and required accuracy but can be less than the direct method especially for a few unknowns i.e a sparse system of linear equations.
37 Slide 500 Jacobi Iteration Iteration formula - ith equation rearranged to have ith unknown on left side: x k i = a b i i, i j i k 1 a i, j x j Superscript indicates iteration: x k k 1 i is kth iteration of x i, x j is (k 1)th iteration of x j.
38 Example of a Sparse System of Linear Equations Laplace s Equation 2 f x 2 2 f = 0 y 2 Slide 501 Solve for f over the two-dimensional x-y space. For a computer solution, finite difference methods are appropriate Two-dimensional solution space is discretized into a large number of solution points.
39 Finite Difference Method Slide 502 Solution space f(x, y) y x
40 If distance between points,, made small enough: 2 f x [ f ( x +, y ) 2 f ( x, y ) + f ( x, y ) ] 2 2 f y [ f ( x, y + ) 2 f ( x, y ) + f ( x, y ) ] 2 Substituting into Laplace s equation, we get [ f ( x +, y ) + f ( x, y ) + f ( x, y + ) + f ( x, y ) 4 f ( x, y ) ] = 0 2 Rearranging, we get [ f ( x, y ) + f ( x, y ) + f ( x +, y ) + f ( x, y + ) ] f ( x, y ) = Rewritten as an iterative formula: f k ( x, y ) [ f k 1 ( x, y ) + f k 1 ( x, y ) + f k 1 ( x +, y ) + f k 1 ( x, y + ) ] = f k (x, y) - kth iteration, f k 1 (x, y) - (k 1)th iteration. Slide 503
41 Natural Order Boundary points (see text) Slide 504 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 11 x 12 x 13 x 14 x 15 x 16 x 17 x 18 x 19 x 21 x 22 x 23 x 24 x 25 x 26 x 27 x 28 x 29 x 31 x 32 x 33 x 34 x 35 x 36 x 37 x 38 x 39 x 41 x 42 x 43 x 44 x 45 x 46 x 47 x 48 x 49 x 51 x 52 x 53 x 54 x 55 x 56 x 57 x 58 x 59 x 61 x 62 x 63 x 64 x 65 x 66 x 67 x 68 x 69 x 71 x 72 x 73 x 74 x 75 x 76 x 77 x 78 x 79 x 81 x 82 x 83 x 84 x 85 x 86 x 87 x 88 x 89 x 10 x 20 x 30 x 40 x 50 x 60 x 70 x 80 x 90 x 91 x 92 x 93 x 94 x 95 x 96 x 97 x 98 x 99 x 100
42 Relationship with a General System of Linear Equations Slide 505 Using natural ordering, ith point computed from ith equation: or x x i n + x i 1 + x i x i = i + n x i n + x i 1 4x i + x i+1 + x i+n = 0 which is a linear equation with five unknowns (except those with boundary points). In general form, the ith equation becomes: a i,i n x i n + a i,i 1 x i 1 + a i,i x i + a i,i+1 x i+1 + a i,i+n x i+n = 0 where a i,i = 4, and a i,i n = a i,i 1 = a i,i+1 = a i,i+n = 1.
43 Slide 506 A Those equations with a boundary point on diagonal unnecessary for solution To include boundary values and some zero entries (see text) x x 1 x ith equation a i,i n a i,i 1 a i,i a i,i a i,i+n 1 1 = x N-1 x N 0 0
44 Gauss-Seidel Relaxation Uses some newly computed values to compute other values in that iteration. Slide 507 Sequential order of computation Basic form not suitable for parallelization Point computed Point to be computed
45 Gauss-Seidel Iteration Formula Slide 508 x k i 1 1 i a b i a i, j x k N = j a i, j x k j 1 i, i j = 1 j = i + 1 where the superscript indicates the iteration. With natural ordering of unknowns, formula reduces to x k i = ( 1/a i,i )[a i,i n x k i n + a i,i 1 x k i 1 + a i,i+1 x k 1 i+1 + a i,i+n x k 1 i+n ] At the kth iteration, two of the four values (before the ith element) taken from the kth iteration and two values (after the ith element) taken from the (k 1)th iteration. We have: f k ( x, y) [ f k ( x, y) + f k ( x, y ) + f k 1 ( x +, y) + f k 1 ( x, y + ) ] =
46 Red-Black Ordering First, black points computed. Next, red points computed. Black points computed simultaneously, and red points computed simultaneously. Slide 509 Red Black
47 Slide 510 Red-Black Parallel Code forall (i = 1; i < n; i++) forall (j = 1; j < n; j++) if ((i + j) % 2 == 0) /* compute red points */ f[i][j] = 0.25*(f[i-1][j] + f[i][j-1] + f[i+1][j] + f[i][j+1]); forall (i = 1; i < n; i++) forall (j = 1; j < n; j++) if ((i + j) % 2!= 0) /* compute black points */ f[i][j] = 0.25*(f[i-1][j] + f[i][j-1] + f[i+1][j] + f[i][j+1]);
48 Slide 511 Higher-Order Difference Methods More distant points could be used in the computation. The following update formula: f k ( x, y) = f k 1 ( x, y) + 16 f k 1 ( x, y ) + 16 f k 1 ( x +, y) + 16 f k 1 ( x, y + ) f k 1 ( x 2, y) f k 1 ( x, y 2 ) f k 1 ( x + 2, y) f k 1 ( x, y + 2 )
49 Slide 512 Nine-point stencil
50 Overrelaxation Slide 513 Improved convergence obtained by adding factor (1 ω)x i to Jacobi or Gauss-Seidel formulae. Factor ω is the overrelaxation parameter. Jacobi overrelaxation formula x i k where 0 < ω < 1. = ω b k 1 k 1 i a ij x i + ( 1 ω)x i a ii j i k x i = Gauss-Seidel successive overrelaxation i 1 N ω b k k 1 k 1 i a ij x i a ij x i + ( 1 ω)x i a ii j = 1 j = i + 1 where 0 < ω 2. If ω = 1, we obtain the Gauss-Seidel method.
51 Slide 514 Multigrid Method First, a coarse grid of points used. With these points, iteration process will start to converge quickly. At some stage, number of points increased to include points of the coarse grid and extra points between the points of the coarse grid. Initial values of extra points found by interpolation. Computation continues with this finer grid. Grid can be made finer and finer as computation proceeds, or computation can alternate between fine and coarse grids. Coarser grids take into account distant effects more quickly and provide a good starting point for the next finer grid.
52 Multigrid processor allocation Slide 515 Coarsest grid points Finer grid points Processor
53 Slide 516 (Semi) Asynchronous Iteration As noted early, synchronizing on every iteration will cause significant overhead - best if one can is to synchronize after a number of iterations.
Matrix Multiplication
Matrix Multiplication Material based on Chapter 10, Numerical Algorithms, of B. Wilkinson et al., PARALLEL PROGRAMMING. Techniques and Applications Using Networked Workstations and Parallel Computers c
More informationDense Matrix Algorithms
Dense Matrix Algorithms Ananth Grama, Anshul Gupta, George Karypis, and Vipin Kumar To accompany the text Introduction to Parallel Computing, Addison Wesley, 2003. Topic Overview Matrix-Vector Multiplication
More informationChapter 8 Dense Matrix Algorithms
Chapter 8 Dense Matrix Algorithms (Selected slides & additional slides) A. Grama, A. Gupta, G. Karypis, and V. Kumar To accompany the text Introduction to arallel Computing, Addison Wesley, 23. Topic Overview
More informationLecture 17: Array Algorithms
Lecture 17: Array Algorithms CS178: Programming Parallel and Distributed Systems April 4, 2001 Steven P. Reiss I. Overview A. We talking about constructing parallel programs 1. Last time we discussed sorting
More informationParallelizing The Matrix Multiplication. 6/10/2013 LONI Parallel Programming Workshop
Parallelizing The Matrix Multiplication 6/10/2013 LONI Parallel Programming Workshop 2013 1 Serial version 6/10/2013 LONI Parallel Programming Workshop 2013 2 X = A md x B dn = C mn d c i,j = a i,k b k,j
More informationMatrix multiplication
Matrix multiplication Standard serial algorithm: procedure MAT_VECT (A, x, y) begin for i := 0 to n - 1 do begin y[i] := 0 for j := 0 to n - 1 do y[i] := y[i] + A[i, j] * x [j] end end MAT_VECT Complexity:
More informationParallel Implementations of Gaussian Elimination
s of Western Michigan University vasilije.perovic@wmich.edu January 27, 2012 CS 6260: in Parallel Linear systems of equations General form of a linear system of equations is given by a 11 x 1 + + a 1n
More informationContents. F10: Parallel Sparse Matrix Computations. Parallel algorithms for sparse systems Ax = b. Discretized domain a metal sheet
Contents 2 F10: Parallel Sparse Matrix Computations Figures mainly from Kumar et. al. Introduction to Parallel Computing, 1st ed Chap. 11 Bo Kågström et al (RG, EE, MR) 2011-05-10 Sparse matrices and storage
More informationLecture 12 (Last): Parallel Algorithms for Solving a System of Linear Equations. Reference: Introduction to Parallel Computing Chapter 8.
CZ4102 High Performance Computing Lecture 12 (Last): Parallel Algorithms for Solving a System of Linear Equations - Dr Tay Seng Chuan Reference: Introduction to Parallel Computing Chapter 8. 1 Topic Overview
More informationHigh Performance Computing Programming Paradigms and Scalability Part 6: Examples of Parallel Algorithms
High Performance Computing Programming Paradigms and Scalability Part 6: Examples of Parallel Algorithms PD Dr. rer. nat. habil. Ralf-Peter Mundani Computation in Engineering (CiE) Scientific Computing
More informationWhat is Multigrid? They have been extended to solve a wide variety of other problems, linear and nonlinear.
AMSC 600/CMSC 760 Fall 2007 Solution of Sparse Linear Systems Multigrid, Part 1 Dianne P. O Leary c 2006, 2007 What is Multigrid? Originally, multigrid algorithms were proposed as an iterative method to
More informationMultigrid Pattern. I. Problem. II. Driving Forces. III. Solution
Multigrid Pattern I. Problem Problem domain is decomposed into a set of geometric grids, where each element participates in a local computation followed by data exchanges with adjacent neighbors. The grids
More informationParallel Numerical Algorithms
Parallel Numerical Algorithms Chapter 5 Vector and Matrix Products Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign CS 554 / CSE 512 Michael T. Heath Parallel
More informationf xx + f yy = F (x, y)
Application of the 2D finite element method to Laplace (Poisson) equation; f xx + f yy = F (x, y) M. R. Hadizadeh Computer Club, Department of Physics and Astronomy, Ohio University 4 Nov. 2013 Domain
More information1 2 (3 + x 3) x 2 = 1 3 (3 + x 1 2x 3 ) 1. 3 ( 1 x 2) (3 + x(0) 3 ) = 1 2 (3 + 0) = 3. 2 (3 + x(0) 1 2x (0) ( ) = 1 ( 1 x(0) 2 ) = 1 3 ) = 1 3
6 Iterative Solvers Lab Objective: Many real-world problems of the form Ax = b have tens of thousands of parameters Solving such systems with Gaussian elimination or matrix factorizations could require
More informationCSCE 5160 Parallel Processing. CSCE 5160 Parallel Processing
HW #9 10., 10.3, 10.7 Due April 17 { } Review Completing Graph Algorithms Maximal Independent Set Johnson s shortest path algorithm using adjacency lists Q= V; for all v in Q l[v] = infinity; l[s] = 0;
More informationIntroduction to Multigrid and its Parallelization
Introduction to Multigrid and its Parallelization! Thomas D. Economon Lecture 14a May 28, 2014 Announcements 2 HW 1 & 2 have been returned. Any questions? Final projects are due June 11, 5 pm. If you are
More informationLinear Equation Systems Iterative Methods
Linear Equation Systems Iterative Methods Content Iterative Methods Jacobi Iterative Method Gauss Seidel Iterative Method Iterative Methods Iterative methods are those that produce a sequence of successive
More informationHypercubes. (Chapter Nine)
Hypercubes (Chapter Nine) Mesh Shortcomings: Due to its simplicity and regular structure, the mesh is attractive, both theoretically and practically. A problem with the mesh is that movement of data is
More informationMatrix Multiplication
Matrix Multiplication CPS343 Parallel and High Performance Computing Spring 2013 CPS343 (Parallel and HPC) Matrix Multiplication Spring 2013 1 / 32 Outline 1 Matrix operations Importance Dense and sparse
More informationMatrix Multiplication
Matrix Multiplication CPS343 Parallel and High Performance Computing Spring 2018 CPS343 (Parallel and HPC) Matrix Multiplication Spring 2018 1 / 32 Outline 1 Matrix operations Importance Dense and sparse
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 20: Sparse Linear Systems; Direct Methods vs. Iterative Methods Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 26
More informationAn Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (2000/2001)
An Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (000/001) Summary The objectives of this project were as follows: 1) Investigate iterative
More informationParallelizing LU Factorization
Parallelizing LU Factorization Scott Ricketts December 3, 2006 Abstract Systems of linear equations can be represented by matrix equations of the form A x = b LU Factorization is a method for solving systems
More informationLinear Arrays. Chapter 7
Linear Arrays Chapter 7 1. Basics for the linear array computational model. a. A diagram for this model is P 1 P 2 P 3... P k b. It is the simplest of all models that allow some form of communication between
More informationWork Allocation. Mark Greenstreet. CpSc 418 Oct. 25, Mark Greenstreet Work Allocation CpSc 418 Oct. 25, / 13
Work Allocation Mark Greenstreet CpSc 418 Oct. 25, 2012 Mark Greenstreet Work Allocation CpSc 418 Oct. 25, 2012 0 / 13 Lecture Outline Work Allocation Static Allocation (matrices and other arrays) Stripes
More informationParallel Computing. Slides credit: M. Quinn book (chapter 3 slides), A Grama book (chapter 3 slides)
Parallel Computing 2012 Slides credit: M. Quinn book (chapter 3 slides), A Grama book (chapter 3 slides) Parallel Algorithm Design Outline Computational Model Design Methodology Partitioning Communication
More informationMatrix Multiplication
Matrix Multiplication Nur Dean PhD Program in Computer Science The Graduate Center, CUNY 05/01/2017 Nur Dean (The Graduate Center) Matrix Multiplication 05/01/2017 1 / 36 Today, I will talk about matrix
More information(Sparse) Linear Solvers
(Sparse) Linear Solvers Ax = B Why? Many geometry processing applications boil down to: solve one or more linear systems Parameterization Editing Reconstruction Fairing Morphing 2 Don t you just invert
More informationComparison of different solvers for two-dimensional steady heat conduction equation ME 412 Project 2
Comparison of different solvers for two-dimensional steady heat conduction equation ME 412 Project 2 Jingwei Zhu March 19, 2014 Instructor: Surya Pratap Vanka 1 Project Description The purpose of this
More informationLecture 4: Principles of Parallel Algorithm Design (part 4)
Lecture 4: Principles of Parallel Algorithm Design (part 4) 1 Mapping Technique for Load Balancing Minimize execution time Reduce overheads of execution Sources of overheads: Inter-process interaction
More informationLecture 6: Parallel Matrix Algorithms (part 3)
Lecture 6: Parallel Matrix Algorithms (part 3) 1 A Simple Parallel Dense Matrix-Matrix Multiplication Let A = [a ij ] n n and B = [b ij ] n n be n n matrices. Compute C = AB Computational complexity of
More informationCOSC6365. Introduction to HPC. Lecture 21. Lennart Johnsson Department of Computer Science
Introduction to HPC Lecture 21 Department of Computer Science Most slides from UC Berkeley CS 267 Spring 2011, Lecture 12, Dense Linear Algebra (part 2), Parallel Gaussian Elimination. Jim Demmel Dense
More informationPARALLEL METHODS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS. Ioana Chiorean
5 Kragujevac J. Math. 25 (2003) 5 18. PARALLEL METHODS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS Ioana Chiorean Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania (Received May 28,
More information10/26/ Solving Systems of Linear Equations Using Matrices. Objectives. Matrices
6.1 Solving Systems of Linear Equations Using Matrices Objectives Write the augmented matrix for a linear system. Perform matrix row operations. Use matrices and Gaussian elimination to solve systems.
More information15. The Software System ParaLab for Learning and Investigations of Parallel Methods
15. The Software System ParaLab for Learning and Investigations of Parallel Methods 15. The Software System ParaLab for Learning and Investigations of Parallel Methods... 1 15.1. Introduction...1 15.2.
More informationParallel Computing: Parallel Algorithm Design Examples Jin, Hai
Parallel Computing: Parallel Algorithm Design Examples Jin, Hai School of Computer Science and Technology Huazhong University of Science and Technology ! Given associative operator!! a 0! a 1! a 2!! a
More informationContents. I The Basic Framework for Stationary Problems 1
page v Preface xiii I The Basic Framework for Stationary Problems 1 1 Some model PDEs 3 1.1 Laplace s equation; elliptic BVPs... 3 1.1.1 Physical experiments modeled by Laplace s equation... 5 1.2 Other
More informationDense LU Factorization
Dense LU Factorization Dr.N.Sairam & Dr.R.Seethalakshmi School of Computing, SASTRA Univeristy, Thanjavur-613401. Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 6 Contents 1. Dense LU Factorization...
More informationChapter Introduction
Chapter 4.1 Introduction After reading this chapter, you should be able to 1. define what a matrix is. 2. identify special types of matrices, and 3. identify when two matrices are equal. What does a matrix
More informationDistributed-memory Algorithms for Dense Matrices, Vectors, and Arrays
Distributed-memory Algorithms for Dense Matrices, Vectors, and Arrays John Mellor-Crummey Department of Computer Science Rice University johnmc@rice.edu COMP 422/534 Lecture 19 25 October 2018 Topics for
More informationNumerical Method (2068 Third Batch)
1. Define the types of error in numerical calculation. Derive the formula for secant method and illustrate the method by figure. There are different types of error in numerical calculation. Some of them
More informationSolving Sparse Linear Systems. Forward and backward substitution for solving lower or upper triangular systems
AMSC 6 /CMSC 76 Advanced Linear Numerical Analysis Fall 7 Direct Solution of Sparse Linear Systems and Eigenproblems Dianne P. O Leary c 7 Solving Sparse Linear Systems Assumed background: Gauss elimination
More informationChapter 11 Image Processing
Chapter Image Processing Low-level Image Processing Operates directly on a stored image to improve or enhance it. Stored image consists of a two-dimensional array of pixels (picture elements): Origin (0,
More informationSorting Algorithms. - rearranging a list of numbers into increasing (or decreasing) order. Potential Speedup
Sorting Algorithms - rearranging a list of numbers into increasing (or decreasing) order. Potential Speedup The worst-case time complexity of mergesort and the average time complexity of quicksort are
More informationIterative Methods for Linear Systems
Iterative Methods for Linear Systems 1 the method of Jacobi derivation of the formulas cost and convergence of the algorithm a Julia function 2 Gauss-Seidel Relaxation an iterative method for solving linear
More informationLecture 5: Matrices. Dheeraj Kumar Singh 07CS1004 Teacher: Prof. Niloy Ganguly Department of Computer Science and Engineering IIT Kharagpur
Lecture 5: Matrices Dheeraj Kumar Singh 07CS1004 Teacher: Prof. Niloy Ganguly Department of Computer Science and Engineering IIT Kharagpur 29 th July, 2008 Types of Matrices Matrix Addition and Multiplication
More informationImage Processing. Application area chosen because it has very good parallelism and interesting output.
Chapter 11 Slide 517 Image Processing Application area chosen because it has very good parallelism and interesting output. Low-level Image Processing Operates directly on stored image to improve/enhance
More informationHomework # 2 Due: October 6. Programming Multiprocessors: Parallelism, Communication, and Synchronization
ECE669: Parallel Computer Architecture Fall 2 Handout #2 Homework # 2 Due: October 6 Programming Multiprocessors: Parallelism, Communication, and Synchronization 1 Introduction When developing multiprocessor
More informationSparse matrices, graphs, and tree elimination
Logistics Week 6: Friday, Oct 2 1. I will be out of town next Tuesday, October 6, and so will not have office hours on that day. I will be around on Monday, except during the SCAN seminar (1:25-2:15);
More informationMatrix Inverse 2 ( 2) 1 = 2 1 2
Name: Matrix Inverse For Scalars, we have what is called a multiplicative identity. This means that if we have a scalar number, call it r, then r multiplied by the multiplicative identity equals r. Without
More informationThe Matrix-Tree Theorem and Its Applications to Complete and Complete Bipartite Graphs
The Matrix-Tree Theorem and Its Applications to Complete and Complete Bipartite Graphs Frankie Smith Nebraska Wesleyan University fsmith@nebrwesleyan.edu May 11, 2015 Abstract We will look at how to represent
More informationOutline. Parallel Algorithms for Linear Algebra. Number of Processors and Problem Size. Speedup and Efficiency
1 2 Parallel Algorithms for Linear Algebra Richard P. Brent Computer Sciences Laboratory Australian National University Outline Basic concepts Parallel architectures Practical design issues Programming
More informationParallel Numerical Algorithms
Parallel Numerical Algorithms Chapter 3 Dense Linear Systems Section 3.3 Triangular Linear Systems Michael T. Heath and Edgar Solomonik Department of Computer Science University of Illinois at Urbana-Champaign
More informationChallenges and Advances in Parallel Sparse Matrix-Matrix Multiplication
Challenges and Advances in Parallel Sparse Matrix-Matrix Multiplication Aydin Buluc John R. Gilbert University of California, Santa Barbara ICPP 2008 September 11, 2008 1 Support: DOE Office of Science,
More informationNumerical Linear Algebra
Numerical Linear Algebra Probably the simplest kind of problem. Occurs in many contexts, often as part of larger problem. Symbolic manipulation packages can do linear algebra "analytically" (e.g. Mathematica,
More informationCS 140: Sparse Matrix-Vector Multiplication and Graph Partitioning
CS 140: Sparse Matrix-Vector Multiplication and Graph Partitioning Parallel sparse matrix-vector product Lay out matrix and vectors by rows y(i) = sum(a(i,j)*x(j)) Only compute terms with A(i,j) 0 P0 P1
More informationPipelined Computations
Pipelined Computations In the pipeline technique, the problem is divided into a series of tasks that have to be completed one after the other. In fact, this is the basis of sequential programming. Each
More informationFor example, the system. 22 may be represented by the augmented matrix
Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural of matrix) may be
More informationCHAPTER 5 Pipelined Computations
CHAPTER 5 Pipelined Computations In the pipeline technique, the problem is divided into a series of tasks that have to be completed one after the other. In fact, this is the basis of sequential programming.
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 36
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 36 In last class, we have derived element equations for two d elasticity problems
More information1. (a) O(log n) algorithm for finding the logical AND of n bits with n processors
1. (a) O(log n) algorithm for finding the logical AND of n bits with n processors on an EREW PRAM: See solution for the next problem. Omit the step where each processor sequentially computes the AND of
More informationSmoothers. < interactive example > Partial Differential Equations Numerical Methods for PDEs Sparse Linear Systems
Smoothers Partial Differential Equations Disappointing convergence rates observed for stationary iterative methods are asymptotic Much better progress may be made initially before eventually settling into
More informationIntroduction to Parallel. Programming
University of Nizhni Novgorod Faculty of Computational Mathematics & Cybernetics Introduction to Parallel Section 9. Programming Parallel Methods for Solving Linear Systems Gergel V.P., Professor, D.Sc.,
More information(Sparse) Linear Solvers
(Sparse) Linear Solvers Ax = B Why? Many geometry processing applications boil down to: solve one or more linear systems Parameterization Editing Reconstruction Fairing Morphing 1 Don t you just invert
More informationMatrix Multiplication on an Experimental Parallel System With Hybrid Architecture
Matrix Multiplication on an Experimental Parallel System With Hybrid Architecture SOTIRIOS G. ZIAVRAS and CONSTANTINE N. MANIKOPOULOS Department of Electrical and Computer Engineering New Jersey Institute
More informationLinear systems of equations
Linear systems of equations Michael Quinn Parallel Programming with MPI and OpenMP material do autor Terminology Back substitution Gaussian elimination Outline Problem System of linear equations Solve
More informationHomework # 1 Due: Feb 23. Multicore Programming: An Introduction
C O N D I T I O N S C O N D I T I O N S Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.86: Parallel Computing Spring 21, Agarwal Handout #5 Homework #
More informationLecture 15: More Iterative Ideas
Lecture 15: More Iterative Ideas David Bindel 15 Mar 2010 Logistics HW 2 due! Some notes on HW 2. Where we are / where we re going More iterative ideas. Intro to HW 3. More HW 2 notes See solution code!
More informationPROGRAMMING OF MULTIGRID METHODS
PROGRAMMING OF MULTIGRID METHODS LONG CHEN In this note, we explain the implementation detail of multigrid methods. We will use the approach by space decomposition and subspace correction method; see Chapter:
More informationReducing Communication Costs Associated with Parallel Algebraic Multigrid
Reducing Communication Costs Associated with Parallel Algebraic Multigrid Amanda Bienz, Luke Olson (Advisor) University of Illinois at Urbana-Champaign Urbana, IL 11 I. PROBLEM AND MOTIVATION Algebraic
More informationMPI Programming. Henrik R. Nagel Scientific Computing IT Division
1 MPI Programming Henrik R. Nagel Scientific Computing IT Division 2 Outline Introduction Finite Difference Method Finite Element Method LU Factorization SOR Method Monte Carlo Method Molecular Dynamics
More informationParallelization of an Example Program
Parallelization of an Example Program [ 2.3] In this lecture, we will consider a parallelization of the kernel of the Ocean application. Goals: Illustrate parallel programming in a low-level parallel language.
More informationSimulating ocean currents
Simulating ocean currents We will study a parallel application that simulates ocean currents. Goal: Simulate the motion of water currents in the ocean. Important to climate modeling. Motion depends on
More informationSynchronous Computations
Chapter 6 slides6-1 Synchronous Computations Synchronous Computations slides6-2 In a (fully) synchronous application, all the processes synchronized at regular points. Barrier A basic mechanism for synchronizing
More informationCSE 590: Special Topics Course ( Supercomputing ) Lecture 6 ( Analyzing Distributed Memory Algorithms )
CSE 590: Special Topics Course ( Supercomputing ) Lecture 6 ( Analyzing Distributed Memory Algorithms ) Rezaul A. Chowdhury Department of Computer Science SUNY Stony Brook Spring 2012 2D Heat Diffusion
More informationAMath 483/583 Lecture 21 May 13, 2011
AMath 483/583 Lecture 21 May 13, 2011 Today: OpenMP and MPI versions of Jacobi iteration Gauss-Seidel and SOR iterative methods Next week: More MPI Debugging and totalview GPU computing Read: Class notes
More informationIterative Algorithms I: Elementary Iterative Methods and the Conjugate Gradient Algorithms
Iterative Algorithms I: Elementary Iterative Methods and the Conjugate Gradient Algorithms By:- Nitin Kamra Indian Institute of Technology, Delhi Advisor:- Prof. Ulrich Reude 1. Introduction to Linear
More information0_PreCNotes17 18.notebook May 16, Chapter 12
Chapter 12 Notes BASIC MATRIX OPERATIONS Matrix (plural: Matrices) an n x m array of elements element a ij Example 1 a 21 = a 13 = Multiply Matrix by a Scalar Distribute scalar to all elements Addition
More informationModule 16: Data Flow Analysis in Presence of Procedure Calls Lecture 32: Iteration. The Lecture Contains: Iteration Space.
The Lecture Contains: Iteration Space Iteration Vector Normalized Iteration Vector Dependence Distance Direction Vector Loop Carried Dependence Relations Dependence Level Iteration Vector - Triangular
More informationAMath 483/583 Lecture 24. Notes: Notes: Steady state diffusion. Notes: Finite difference method. Outline:
AMath 483/583 Lecture 24 Outline: Heat equation and discretization OpenMP and MPI for iterative methods Jacobi, Gauss-Seidel, SOR Notes and Sample codes: Class notes: Linear algebra software $UWHPSC/codes/openmp/jacobi1d_omp1.f90
More informationParallel Poisson Solver in Fortran
Parallel Poisson Solver in Fortran Nilas Mandrup Hansen, Ask Hjorth Larsen January 19, 1 1 Introduction In this assignment the D Poisson problem (Eq.1) is to be solved in either C/C++ or FORTRAN, first
More informationSynchronous Computations
Synchronous Computations Material based on B. Wilkinson et al., PARALLEL PROGRAMMING. Techniques and Applications Using Networked Workstations and Parallel Computers c 2002-2004 R. Leduc Introduction We
More informationMultigrid Algorithms for Three-Dimensional RANS Calculations - The SUmb Solver
Multigrid Algorithms for Three-Dimensional RANS Calculations - The SUmb Solver Juan J. Alonso Department of Aeronautics & Astronautics Stanford University CME342 Lecture 14 May 26, 2014 Outline Non-linear
More informationNumerical Modelling in Fortran: day 6. Paul Tackley, 2017
Numerical Modelling in Fortran: day 6 Paul Tackley, 2017 Today s Goals 1. Learn about pointers, generic procedures and operators 2. Learn about iterative solvers for boundary value problems, including
More informationThe Simplex Algorithm. Chapter 5. Decision Procedures. An Algorithmic Point of View. Revision 1.0
The Simplex Algorithm Chapter 5 Decision Procedures An Algorithmic Point of View D.Kroening O.Strichman Revision 1.0 Outline 1 Gaussian Elimination 2 Satisfiability with Simplex 3 General Simplex Form
More informationAMath 483/583 Lecture 24
AMath 483/583 Lecture 24 Outline: Heat equation and discretization OpenMP and MPI for iterative methods Jacobi, Gauss-Seidel, SOR Notes and Sample codes: Class notes: Linear algebra software $UWHPSC/codes/openmp/jacobi1d_omp1.f90
More informationChemical Engineering 541
Chemical Engineering 541 Computer Aided Design Methods Direct Solution of Linear Systems 1 Outline 2 Gauss Elimination Pivoting Scaling Cost LU Decomposition Thomas Algorithm (Iterative Improvement) Overview
More informationSELECTIVE ALGEBRAIC MULTIGRID IN FOAM-EXTEND
Student Submission for the 5 th OpenFOAM User Conference 2017, Wiesbaden - Germany: SELECTIVE ALGEBRAIC MULTIGRID IN FOAM-EXTEND TESSA UROIĆ Faculty of Mechanical Engineering and Naval Architecture, Ivana
More informationLesson 2 7 Graph Partitioning
Lesson 2 7 Graph Partitioning The Graph Partitioning Problem Look at the problem from a different angle: Let s multiply a sparse matrix A by a vector X. Recall the duality between matrices and graphs:
More informationSynchronous Shared Memory Parallel Examples. HPC Fall 2012 Prof. Robert van Engelen
Synchronous Shared Memory Parallel Examples HPC Fall 2012 Prof. Robert van Engelen Examples Data parallel prefix sum and OpenMP example Task parallel prefix sum and OpenMP example Simple heat distribution
More informationSpace Filling Curves and Hierarchical Basis. Klaus Speer
Space Filling Curves and Hierarchical Basis Klaus Speer Abstract Real world phenomena can be best described using differential equations. After linearisation we have to deal with huge linear systems of
More informationWhy Use the GPU? How to Exploit? New Hardware Features. Sparse Matrix Solvers on the GPU: Conjugate Gradients and Multigrid. Semiconductor trends
Imagine stream processor; Bill Dally, Stanford Connection Machine CM; Thinking Machines Sparse Matrix Solvers on the GPU: Conjugate Gradients and Multigrid Jeffrey Bolz Eitan Grinspun Caltech Ian Farmer
More informationSynchronous Computation Examples. HPC Fall 2008 Prof. Robert van Engelen
Synchronous Computation Examples HPC Fall 2008 Prof. Robert van Engelen Overview Data parallel prefix sum with OpenMP Simple heat distribution problem with OpenMP Iterative solver with OpenMP Simple heat
More informationFast Iterative Solvers for Markov Chains, with Application to Google's PageRank. Hans De Sterck
Fast Iterative Solvers for Markov Chains, with Application to Google's PageRank Hans De Sterck Department of Applied Mathematics University of Waterloo, Ontario, Canada joint work with Steve McCormick,
More informationParallel Programs. EECC756 - Shaaban. Parallel Random-Access Machine (PRAM) Example: Asynchronous Matrix Vector Product on a Ring
Parallel Programs Conditions of Parallelism: Data Dependence Control Dependence Resource Dependence Bernstein s Conditions Asymptotic Notations for Algorithm Analysis Parallel Random-Access Machine (PRAM)
More informationAlgorithms and Applications
Algorithms and Applications 1 Areas done in textbook: Sorting Algorithms Numerical Algorithms Image Processing Searching and Optimization 2 Chapter 10 Sorting Algorithms - rearranging a list of numbers
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 5: Sparse Linear Systems and Factorization Methods Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 18 Sparse
More informationHPC Algorithms and Applications
HPC Algorithms and Applications Dwarf #5 Structured Grids Michael Bader Winter 2012/2013 Dwarf #5 Structured Grids, Winter 2012/2013 1 Dwarf #5 Structured Grids 1. dense linear algebra 2. sparse linear
More informationExam Design and Analysis of Algorithms for Parallel Computer Systems 9 15 at ÖP3
UMEÅ UNIVERSITET Institutionen för datavetenskap Lars Karlsson, Bo Kågström och Mikael Rännar Design and Analysis of Algorithms for Parallel Computer Systems VT2009 June 2, 2009 Exam Design and Analysis
More information