lec10 (1)

lec10 (1) - Introduction to Simulation - Lecture 10...

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Introduction to Simulation - Lecture 10 Modified Newton Methods Jacob White Thanks to Deepak Ramaswamy, Jaime Peraire, Michal Rewienski, and Karen Veroy
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SMA-HPC ©2003 MIT Outline • Damped Newton Schemes – Globally Convergent if Jacobian is Nonsingular – Difficulty with Singular Jacobians • Introduce Continuation Schemes – Problem with Source/Load stepping – More General Continuation Scheme • Improving Continuation Efficiency – Better first guess for each continuation step • Arc Length Continuation
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SMA-HPC ©2003 MIT Newton Algorithm Multidimensional Newton Method ( ) Newton Algorithm for Solving 0 Fx = 0 Initial Guess, = 0 xk = Repeat { () ( ) Compute , kk F F xJx ( ) ( ) 11 Solve for k k k F J xx x F x x + + −= 1 =+ } Until ( ) 1 1 , small en ug oh k xx + +
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SMA-HPC ©2003 MIT Multidimensional Convergence Theorem Multidimensional Newton Method Theorem Statement Main Theorem If () ( ) 1 ) Inverse is bounded k F aJx β () () ( ) ) Derivative is Lipschitz Cont FF bJxJy xy −≤ A Then Newton’s method converges given a sufficiently close initial guess
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SMA-HPC ©2003 MIT Multidimensional Convergence Theorem Multidimensional Newton Method Implications If a function’s first derivative never goes to zero, and its second derivative is never too large… Then Newton’s method can be used to find the zero of the function provided you all ready know the answer . Need a way to develop Newton methods which converge regardless of initial guess!
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SMA-HPC ©2003 MIT Non-converging Case f(x) X 0 x 1 x 1-D Picture Limiting the changes in X might improve convergence
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SMA-HPC ©2003 MIT Newton Algorithm Newton Method with Limiting ( ) Newton Algorithm for Solving 0 Fx = 0 Initial Guess, = 0 xk = Repeat { () ( ) Compute , kk F F xJx ( ) 11 Solve for k k F J xx F x x + + ∆= ( ) limited k x ++ =+ } Until ( ) 1 1 , small enough k k x + + 1
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SMA-HPC ©2003 MIT Damped Newton Scheme Newton Method with Limiting General Damping Scheme () ( ) 11 Solve for kk k k F J xx F x x + + ∆= k k x α ++ =+∆ Key Idea: Line Search 2 1 2 Pick to minimize k k Fx x αα + +∆ 2 1 2 T k k k x x x + Method Performs a one-dimensional search in Newton Direction
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SMA-HPC ©2003 MIT Damped Newton Newton Method with Limiting Convergence Theorem If () ( ) 1 ) Inverse is bounded k F aJx β () () ( ) ) Derivative is Lipschitz Cont FF bJxJy xy −≤ A Then ( ] There exists a set of ' 0,1 such that k s α ( ) ( ) 11 with <1 kk k k k Fx x γγ ++ =+ ∆< Every Step reduces F-- Global Convergence!
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