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Computational Methods for Design Lecture 4 – Introduction to Sensitivities John A. Burns C enter for O ptimal D esign A nd C ontrol I nterdisciplinary C enter for A pplied M athematics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0531 A Short Course in Applied Mathematics 2 February 2004 – 7 February 2004 N∞M∞T Series Two Course Canisius College, Buffalo, NY
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A Falling Object “Newton’s Second Law”. y(t) { { AIR RESISTANCE
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System of Differential Equations
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Parameters IN REAL PROBLEMS THERE ARE PARAMETERS SOLUTIONS DEPEND ON THESE PARAMETERS WE WILL BE INTERESTED IN COMPUTING SENSITIVITIES WITH RESPECT TO THESE PARAMETERS
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Examples: n=m=1 CONTINUOUS EVERYWHERE UNIQUE SOLUTION CONTINUOUS WHEN UNIQUE SOLUTION
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Logistics Equation
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Computing Sensitivities HOW DO WE COMPUTE THE SENSITIVITIES …
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SEM Example 1 DIFFERENTIATE
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SEM Example 1
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SEM Method SOLVE THE SYSTEM (DE) – (SE) (DE) (SE) 1.WHY DO IT THIS WAY ? 2.WE DERIVED (SE) BY USING THE KNOWN SOLUTION … HOW DO WE FIND (SE) IN GENERAL? 3.HOW GENERAL IS THIS PROCESS?
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Derivation of SEN Eq DIFFERENTIATE THE EQUATION WITH RESPECT TO q INTERCHANGE THE ORDER OF DIFFERENTIATION
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Derivation of SEN Eq (DE) (SE)
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SEM Method (DE) (SE)
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Explicit Euler for SEQs t0t0 (x 0,s 0 ) R2R2
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Explicit Euler for SEQs
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SOLVE BOTH DE AND SE TOGETHER HOW DOES IT WORK?
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MATLAB Code for SEM Set q Set x 0 and s 0 Set h Time interval Set ICs Explicit Euler
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DE Solution x(t)
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SE Solution s(t)
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Special Structure of SE’s (DE) (SE) (DE) (SE) FIRST: SOLVE (DE) SECOND: SOLVE (SE)
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Logistics Equation
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SEQ for the Logistics Equation DIFFERENTIATE THE EQUATION WITH RESPECT TO q 1
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SEQ for the Logistics Equation
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INTERCHANGE THE ORDER OF DIFFERENTIATION
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SEQ for the Logistics Equation
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NEED SENSITIVITY WITH RESPECT TO q 2
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SEQ for the Logistics Equation 2 DIFFERENTIATE THE EQUATION WITH RESPECT TO q 2
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SEQ for the Logistics Equation 2
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INTERCHANGE THE ORDER OF DIFFERENTIATION
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SEQ for the Logistics Equation 2
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SEQ’s for the Logistics Equation FROM THE FIRST PARTIAL THE LOGISTICS EQUATION
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SEQ’s for the Logistics Equation FIRST: SOLVE (DE) SECOND: SOLVE (SEs)
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Model Problem #1 q q q SENSITIVITY The sensitivity equation for s(x, q ) = q w(x, q ) in the “ physical ” domain ( q ) = (0, q ) is given by Can be made “ rigorous ” by the method of mappings. MORE ABOUT THIS NEAR THE END q q q q,q, q q q
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Typical Cost Function WHERE w( x, q ) USUALLY SATISFIES A DIFFERENTIAL EQUATION AND q IS A PARAMETER (OR VECTOR OF PARAMETERS) q q q q q q q q q q q q q q THE CHAIN RULE PRODUCES OR (Reality) USING NUMERICAL SOLUTIONS hh hh CONTINUOUS SENSITIVITY DISCRETE SENSITIVITY
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Computing Gradients (I) BY FINITE DIFFERENCES q q0q0 q0q0 q0q0 q h h h TYPICAL APPROACHES TO COMPUTE q q =q0q =q0 h (II) BY DISCRETE SENSITIVITIES q0q0 q0q0 q0q0 q0q0 q0q0 h h h h
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Computing Gradients FINITE DIFFERENCES REQUIRES 2 NON-LINEAR SOLVES IF SHAPE IS A DESIGN VARIABLE, FD REQUIRES 2 MESH GENERATIONS DISCRETE SENSITIVITIES REQUIRES THE EXISTENCE OF THE DISCRETE SENSITIVITY IF SHAPE IS A DESIGN VARIABLE, THE DISCRETE SENSITIVITY LEADS TO MESH DERIVATIVES COMPUTATIONS WHAT IS THE “ CONTINUOUS / HYBRID ” SENSITIVITY EQUATION METHOD? --- SEM q0q0 q0q0 q0q0 h h h, k APPROXIMATE
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A Sensitivity Equation Method FOR q > 1 AND h=q / (N+1) CONSIDER (FORMAL ) h h h h h NUMERICAL APPROXIMATION x=0 x=1 x=q x w(x) w h (x) = Finite Element Approximation q q,q, DISCRETE STATE EQUATION
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A Sensitivity Equation Method h h h h h h h q q,q, q q q,q, q h IMPORTANT OBSERVATIONS l The sensitivity equations are linear The sensitivity equation “ solver ” can be constructed independently of the forward solver -- SENSE™ When done correctly “ mesh gradients ” are not required
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A Sensitivity Equation Method FOR q > 1 AND k = q / (M+1) CONSIDER (FORMAL) 2 nd NUMERICAL APPROXIMATION x=0 x=1 x=q x s(x)= q w(x,q) s h,k (x) = Finite Element Approximation of h h,k q q q,q, q q q h h h
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Convergence Issues q q q h h h q k h,k THEOREM. The finite element scheme is asymptotically consistent. qq h h h k k a trust region method should (might?) converge. qq h h k When the erroris small, then IDEA: J. T. Borggaard and J. A. Burns, “ A PDE Sensitivity Equation Method for Optimal Aerodynamic Design ”, Journal of Computational Physics, Vol.136 (1997), 366-384. R. G. Carter, “ On the Global Convergence of Trust-Region Algorithms Using Inexact Gradient Information ”, SIAM J. Num. Anal., Vol 28 (1991), 251-265. J. T. Borggaard, “ The Sensitivity Equation Method for Optimal Design ”, Ph.D. Thesis, Virginia Tech, Blacksburg, VA, 1995.
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Convergence Issues N=16, M=32
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Convergence Issues qq h hh THE CASE k = h is often used, but may not be “ good enough ” NOT CONVERGENT
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Timing Issues THE CASE k = 2h offers flexibility and qq h h 2h2h convergence. But, what about timings? Approximately 96.6% of cpu time spent in function evaluations Approximately 02.4% of cpu time spent in gradient evaluations
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Mathematics Impacts “ Practically ” UNDERSTANDING THE PROPER MATHEMATICAL FRAMEWORK CAN BE EXPLOITED TO PRODUCE BETTER SCIENTIFIC COMPUTING TOOLS A REAL JET ENGINE WITH 20 DESIGN VARIABLES l PREVIOUS ENGINEERING DESIGN METHODOLOGY REQUIRED 8400 CPU HRS ~ 1 YEAR l USING A HYBRID SEM DEVELOPED AT VA TECH AS IMPLEMENTED BY AEROSOFT IN SENSE™ REDUCED THE DESIGN CYCLE TIME FROM... 8400 CPU HRS ~ 1 YEAR TO 480 CPU HRS ~ 3 WEEKS NEW MATHEMATICS WAS THE ENABLING TECHNOLOGY
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Special Structure of SE’s (DE) (SE) (DE) (SE) FIRST: SOLVE (DE) SECOND: SOLVE (SE)
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END
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