apma4301_IntroToNonlinear08

apma4301_IntroToNonlinear08 - Lecture #11 Part B 12...

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12 November 2008 Lecture #11 Part B Wed aft. 4:10-6:40 S. W. Mudd Bldg. 535 Prof. David Keyes, instructor S. W. Mudd Bldg. 215 apam4301@gmail.com Yan Yan, teaching assistant yy2150@columbia.edu Applied Mathematics 4301: Numerical Methods for PDEs Iterative methods for Nonlinear Problems
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Background z Several times in the course we have deferred nonlinearity; now we pay the price z A first look at nonlinearity can be very simple: ± we linearize locally and solve a succession of linear problems (e.g., Newton) z It turns out that there are both difficulties (lack of robustness) and synergisms that make the problem much more interesting than this: ± reductions in complexity of work and storage (Newton-Krylov) ± new algorithmic combinations that interleave linearization and coarsening (nonlinear multigrid) ± new algorithmic combinations that interleave linearization and basis selection (nonlinear Krylov) ± generalization that include optimization (Lagrange-Newton)
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from APMA 4301 Lecture #2, 10 Sep 2008
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Nonuniqueness a crucial feature from APMA 4300, M. Heath text no solutions 2 solutions 4 solutions 1 solution
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Individual roots can be multiple, e.g., in 1D from APMA 4300, M. Heath text
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Residual and error different, as with linear! from APMA 4300, M. Heath text
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Convergence rate from APMA 4300, M. Heath text
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Newton’s method asymptotically quadratic from APMA 4300, M. Heath text
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Simple geometric interpretation in 1D from APMA 4300, M. Heath text Generalization to n -dimensional, f ( x ) = 0 : the row gradient of each component of this system of equations defines an n -dimensional hyperplane
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Example in 1D: quadratic convergence from APMA 4300, M. Heath text
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n -dimensional generalization from APMA 4300, M. Heath text
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from APMA 4300, M. Heath text Example in 2D: first Newton step
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Example in 2D: second Newton step from APMA 4300, M. Heath text
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Notation for balance of presentation z Given and iterate we wish to pick such that where z Neglecting higher-order terms, we get where is the Jacobian matrix, generally large, sparse, and ill-conditioned for PDEs z In practice, require z In practice, set where is selected to minimize n n F u F = : , 0 ) ( 0 u 1 + k u 0 ) ( ) ( ) ( ' 1 = + + k k k k u u F u F u F δ ,... 2 , 1 , 0 , 1 = = + k u u u k k k ) ( )] ( [ 1 k k k u F u J u = ) ( ' k u F J = ε < + || ) ( ) ( || k k k u u J u F k k k u u u λδ + = + 1 λ || ) ( || k k u u F +
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Newton’s method: pros and cons z Locally quadratically convergent (if Jacobian is nonsingular at the solution) ± number of significant digits doubles asymptotically at each step ± not globally convergent from arbitrary initial iterate z Requires Jacobian evaluation at each iteration ± may be nontrivial for user to supply derivatives ± may require large fraction of code size and execution time ± if exact derivative information is sacrificed, so is proof of quadratic convergence z Requires solution of linear system with Jacobian at each iteration ± bottleneck when ill-conditioned
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This note was uploaded on 08/30/2011 for the course APMA 4301 taught by Professor Keyes during the Fall '08 term at Columbia.

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apma4301_IntroToNonlinear08 - Lecture #11 Part B 12...

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