lec11a - Introduction to Simulation Lecture 11...

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Introduction to Simulation - Lecture 11 Newton-Method Case Study – Simulating An Image Smoother Jacob White Thanks to Deepak Ramaswamy, Andrew Lumsdaine, Jaime Peraire, Michal Rewienski, and Karen Veroy
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SMA-HPC ©2003 MIT Outline • Image Segmentation Example – Large nonlinear system of equations – Formulation? Continuation? Linear Solver? • Newton Iterative Methods – Accuracy Theorem – Matrix-free idea • Gershgorin Circle Theorem – Lends insight on iterative method convergence • Arc-Length Continuation
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SMA-HPC ©2003 MIT Circuit Diagram Simple Smoother Smoothed Output Image Input
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SMA-HPC ©2003 MIT Circuit Diagram Nonlinear Smoother
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SMA-HPC ©2003 MIT Nonlinear Resistor Constitutive Equation Nonlinear Smoother +, 2 1 v v iv e EJ D D 00 .
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SMA-HPC ©2003 MIT Nonlinear Resistor Constitutive Equation Nonlinear Smoother Current Voltage
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SMA-HPC ©2003 MIT Nonlinear Resistor Constitutive Equation Nonlinear Smoother Varying Beta
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SMA-HPC ©2003 MIT Questions • What Equation Formulation? – Node-Branch or Nodal? • What Newton Method? – Standard, Damped, or Continuation? – What kind of Continuation? • What Linear Solver? – Sparse Gaussian Elimination or Krylov? – Will Krylov converge rapidly? – How will formulation, Newton choices interact?
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SMA-HPC ©2003 MIT 0 Initial Guess, =0 xk Repeat { Solve (Using GCR) +, Compute , kk F F xJ x Basic Algorithm Newton-Iterative Method Nested Iteration 11 for k k F J xx F x x .. ' 0 ' k x . ' } Until 1 1 , small enough k k x Fx . . ' 1 .
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lec11a - Introduction to Simulation Lecture 11...

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