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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 [email protected] Yan Yan, teaching assistant [email protected] 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! It’s 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, δ 4,6 , or δ 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 ={ δ 1 , δ 2 ,…, δ 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
 Linear Algebra, Numerical Analysis, Gradient descent, Iterative method, conjugate gradient, conjugate gradients

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