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# AdjDirectSurrog_opt - Stanford University Derivatives and...

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Stanford University PE284 Derivatives and Surrogates D. Echeverría Ciaurri Optimization PE284 Stanford University

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Stanford University Derivatives and Surrogates Nov 5 2008 PE284 2 Outline Calculating Derivatives Derivative-Free Methods Surrogate-Based Optimization Summary
Stanford University Derivatives and Surrogates Nov 5 2008 PE284 3 Need for Derivatives They appear in many schemes They speed-up local convergence In practical applications their computation is not trivial Example: simulation-based design

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Stanford University Derivatives and Surrogates Nov 5 2008 PE284 4 Simulation-Based Design Objective function where x(p) is the solution of the system ( 29 ( 29 ( 29 dt p , p x f p J f 0 t t = ( 29 ( 29 = = 0 0 x t x 0 t , p , x , x g
Stanford University Derivatives and Surrogates Nov 5 2008 PE284 5 Simulation-Based Design Three strategies Finite Differences Forward Sensitivity Analysis (FSA) Adjoint Sensitivity Analysis (ASA)

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Stanford University Derivatives and Surrogates Nov 5 2008 PE284 6 Finite Differences Forward differences Central differences ( 29 ( 29 ( 29 i i i i dp p J dp e p J p p J - + 2245 ( 29 ( 29 ( 29 i i i i i i dp 2 dp e p J dp e p J p p J - - + 2245
Stanford University Derivatives and Surrogates Nov 5 2008 PE284 7 FSA It solves the perturbation size problem ( 29 ( 29 ( 29 dt p f p x x f dt p p , p x f p p J f 0 f 0 t t i t t i i + = =

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Stanford University Derivatives and Surrogates Nov 5 2008 PE284 8 FSA It solves the perturbation size problem ( 29 ( 29 ( 29 dt p f p x x f dt p p , p x f p p J f 0 f 0 t t i t t i i + = = ( 29 0 p g p x x g p x x g 0 t , p , x , x g i i i = + + =
Stanford University Derivatives and Surrogates Nov 5 2008 PE284 9 FSA It solves the perturbation size problem Not a black-box approach anymore A system for each derivative The systems appear to be similar and can be computed in parallel No more systems if more constraints

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Stanford University Derivatives and Surrogates Nov 5 2008 PE284 10 ASA Key: augmenting the cost function ( 29 ( 29 ( 29 ( 29 λ + = f 0 f 0 t t T t t dt t , p , x , x g dt p , p x f p J
Stanford University Derivatives and Surrogates Nov 5 2008 PE284 11 ASA Key: augmenting the cost function ( 29 ( 29 ( 29 ( 29 λ + = f 0 f 0 t t T t t dt t , p , x , x g dt p , p x f p J

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Stanford University Derivatives and Surrogates Nov 5 2008 PE284 12 ASA Key: augmenting the cost function ( 29 + + λ + + + = f 0 f 0 t t i i T t t i i dt p g p x x g p x x g dt p f p x x f p p J
Stanford University Derivatives and Surrogates Nov 5 2008 PE284 13 ASA Key: augmenting the cost function ( 29 λ + + λ + + λ + = f 0 f 0 f 0 t t i T t t i T t t T i dt p x

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