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Example: solving Poissons equation for temperature

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Title: Example: solving Poissons equation for temperature


1
Example solving Poissons equation for
temperature
  • For each i from 1 to n, except on the boundaries
  • t(i-k2) t(i-k) t(i-1) 6t(i) t(i1)
    t(ik) t(ik2) 0
  • n equations in n unknowns At b
  • Each row of A has at most 7 nonzeros.

2
Sparse matrix data structure (stored by rows)
  • Full
  • 2-dimensional array of real or complex numbers
  • (nrowsncols) memory
  • Sparse
  • compressed row storage
  • about (2nzs nrows) memory

3
Distributed row sparse matrix data structure
P0
P1
  • Each processor stores
  • of local nonzeros
  • range of local rows
  • nonzeros in CSR form

P2
Pn
4
Graphs and Sparse Matrices
  • Sparse matrix is a representation of a (sparse)
    graph

1 2 3 4 5 6
1 1 1 2 1 1
1 3
1 1 1 4 1
1 5 1 1
6 1 1
3
2
4
1
5
6
  • Matrix entries are edge weights
  • Diagonal contains self-edges (usually non-zero)
  • Number of nonzeros per row is the vertex degree

5
Link analysis of the web
  • Web page vertex
  • Link directed edge
  • Link matrix Aij 1 if page i links to page j

6
Web graph PageRank (Google) Brin,
Page
An important page is one that many important
pages point to.
  • Markov process follow a random link most of the
    time otherwise, go to any page at random.
  • Importance stationary distribution of Markov
    process.
  • Transition matrix is pA (1-p)ones(size(A)),
    scaled so each column sums to 1.
  • Importance of page i is the i-th entry in the
    principal eigenvector of the transition matrix.
  • But, the matrix is 8,000,000,000 by 8,000,000,000.

7
A Page Rank Matrix
  • Importance ranking of web pages
  • Stationary distribution of a Markov chain
  • Power method matvec and vector arithmetic
  • MatlabP page ranking demo (from SC03) on
    a web crawl of mit.edu (170,000 pages)

8
Conjugate gradient iteration
x0 0, r0 b, d0 r0 for k 1, 2,
3, . . . ak (rTk-1rk-1) / (dTk-1Adk-1)
step length xk xk-1 ak dk-1
approx solution rk rk-1 ak
Adk-1 residual ßk
(rTk rk) / (rTk-1rk-1) improvement dk
rk ßk dk-1
search direction
  • One matrix-vector multiplication per iteration
  • Two vector dot products per iteration
  • Four n-vectors of working storage

9
Matrix-vector product Parallel implementation
  • Lay out matrix and vectors by rows
  • y(i) sum(A(i,j)x(j))
  • Skip terms with A(i,j) 0
  • Algorithm
  • Each processor i
  • Broadcast x(i)
  • Compute y(i) A(i,)x
  • Optimizations reduce communication by
  • Only send as much of x as necessary to each proc
  • Reorder matrix for better locality by graph
    partitioning

10
Other memory layouts for matrix-vector product
  • A column layout of the matrix eliminates the
    broadcast
  • But adds a reduction to update the destination
    same total comm
  • A blocked layout uses a broadcast and reduction,
    both on a subset of sqrt(p) processors less
    total comm

P0 P1 P2 P3
P0 P1 P2 P3
P4 P5 P6 P7
P8 P9 P10 P11
P12 P13 P14 P15
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