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Unformatted text preview: Applied Mathematics 4301: Numerical Methods for PDEs Introduction to Krylov Methods Wed aft. 4:106:40 S. W. Mudd Bldg. 1024 Prof. David Keyes, instructor S. W. Mudd Bldg. 215 apam4301@gmail.com Yan Yan, teaching assistant yy2250@columbia.edu Lecture #10 (Part B) 5 November 2008 Main agenda http://www.cs.cmu.edu/~quakepapers/painlessconjugategradient.pdf J. R. Shewchuk (now professor at UC Berkeley) An Introduction to the Conjugate Gradient Method Without the Agonizing Pain Conjugate gradients: a Krylov iterative method for symmetric systems Why do we care? Krylov methods are key to contemporary large scale computation you must know them to have a hand in the future of computational science & engineering though their theory is involved, their implementation is simple They look for a solution to a linear system in a reduced basis, relative to problem dimension, in contrast to direct methods They adapt to the spectrum of the system matrix, in contrast to stationary iterative methods Krylov bases for sparse systems Examples: conjugate gradients (CG) for symmetric, positive definite systems, and generalized minimal residual (GMRES) for nonsymmetry or indefiniteness Krylov iteration is an algebraic projection method for converting a highdimensional linear system into a lowerdimensional linear system (like Galerkin) AV W H T = = b Ax = = b W g T = = g Hy = solve this using this! Its all about BASIS! key is this! = Vy x = m m v y v y x r L r r + + = 1 1 Krylov bases for sparse systems Krylov bases, cont. Cartesian bases This Krylov basis is an alternative to the standard Cartesian coordinate basis used in, e.g., Gaussian elimination A sample element of the Cartesian basis is shown to the right (this is the (4,6) element, 4,6 , or 49 ), out of the 81 possible unit vectors in the 9 9 array of nodes, including boundary nodes By choosing a basis adapted to the operator and the RHS, we may need many fewer vectors to accurately represent the solution x y Krylov vs. Cartesian bases Instead of the basis C ={ 1 , 2 ,, n } We choose the basis K ={ b , Ab ,, A m1 b }, or something with the same span as this basis that is easier to manipulate, K ={ v 1 , v 2 ,, v m } The key is that, for equivalent accuracy in solving the original linear system, Ax=b , we have m << n ! Easier may mean orthogonal (like the Cartesian basis): v i T v j = 0, i j or Aconjugate: v i T Av j = 0, i j Krylov bases, cont. Krylov bases, cont. History Aleksei N. Krylov (18631945) was a naval architect in St. Petersburg Gave formal definition for finding least squares solution to linear system within a subspace Method was not made practical in his lifetime Method of Conjugate Gradients (CG) invented in 1950 (published 1952) as a direct method by Hestenes and Stiefel Reinvented in 1971 by Reid as an iterative method ICCG popularized in 1977 1980s and 1990s full of discovery Key features...
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 Fall '08
 Keyes

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