CS 290H: Sparse Matrix Algorithms

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CS 290H Sparse Matrix Algorithms

• John R. Gilbert (gilbert_at_cs.ucsb.edu)
• www.cs.ucsb.edu/gilbert/cs290

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The (2-dimensional) model problem

• Graph is a regular square grid with n k2
  vertices.
• Corresponds to matrix for regular 2D finite
  difference mesh.
• Gives good intuition for behavior of sparse
  matrix algorithms on many 2-dimensional physical
  problems.
• Theres also a 3-dimensional model problem.

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Solving Poissons equation for temperature

• For each i from 1 to n, except on the boundaries
• x(i-k2) x(i-k) x(i-1) 6x(i) x(i1)
  x(ik) x(ik2) 0
• n equations in n unknowns Ax b
• Each row of A has at most 7 nonzeros.

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Link analysis of the web

• Web page vertex
• Link directed edge
• Link matrix Aij 1 if page i links to page j

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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 10,000,000,000 by
  10,000,000,000.

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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)

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Graphs and Sparse Matrices Cholesky
factorization
Fill new nonzeros in factor
Symmetric Gaussian elimination for j 1 to n
add edges between js higher-numbered
neighbors
G(A)chordal
G(A)
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The Landscape of Sparse Axb Solvers
D
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Complexity of linear solvers
Time to solve model problem (Poissons equation)
on regular mesh
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CS 290H Administrivia

• Course web site www.cs.ucsb.edu/gilbert/cs290
• Join the email (Google) discussion group!! (see
  web site)
• First homework is on the web site, due next
  Monday
• About 5 weekly homeworks, then a final project
  (implementation experiment, application, or
  survey paper)
• Assigned readings some online, some from Davis
  book.