Yi%20Heng

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Second Order Differentiation

• Yi Heng

Bommerholz 14.08.2006 Summer School 2006
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Outline
Background What are derivatives? Where do we need
derivatives? How to compute derivatives? Basics
of Automatic Differentiation Introduction Forward
mode strategy Reverse mode strategy Second-Order
Automatic Differentiation Module Introduction Forw
ard mode strategy Taylor Series strategy Hessian
Performance An Application in Optimal Control
Problems Summary
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Background
What are derivatives?
Jacobian Matrix
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Background
What are derivatives?
Hessian Matrix
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Background
Where do we need derivatives?

• Linear approximation
• Bending and Acceleration (Second derivatives)
• Solve algebraic and differential equations
• Curve fitting
• Optimization Problems
• Sensitivity analysis
• Inverse Problem (data assimilation)
• Parameter identification

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Background
How to compute derivatives?
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Background
How to compute derivatives?
Automatic differentiation
To be continued ...
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Basics of Automatic Differentiation
Introduction
Automatic differentiation ...

• Is also known as computational differentiation,
  algorithmic differentiation, and differentiation
  of algorithms
• Is a systematic application of the familar rules
  of calculus to computer programs, yielding
  programs for the propagation of numerical values
  of first, second, or higher order derivatives
• Traverses the code list (or computational graph)
  in the forward mode, the reverse mode, or a
  combination of the two
• Typically is implemented by using either source
  code transformation or operator overloading
• Is a process for evaluating derivatives which
  depends only on an algorithmic specification of
  the function to be differentiated.

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Basics of Automatic Differentiation
Introduction
Rules of arithmetic operations for gradient vector
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Basics of Automatic Differentiation
Forward mode Reverse mode
An example
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Basics of Automatic Differentiation
Forward mode Reverse mode - Forward mode
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Basics of Automatic Differentiation
Forward mode Reverse mode - Reverse mode
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Second-Order AD Module
Introduction
Divided differences
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Second-Order AD Module
Introduction
Rules of arithmetic operations for Hessian
matrices
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Second-Order AD Module
Forward Mode Stategy
An example
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Second-Order AD Module
Forward Mode Stategy
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Second-Order AD Module
Forward Mode Stategy
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Second-Order AD Module
Forward Mode Stategy
Hessian Type Cost
H(f) O(n2)
H(f)V O(nnv)
VTH(f)V O(nv2)
VTH(f)W O(nvnw)
H(f) n by n matrix
V n by nv matrix
W n by nw matrix
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Second-Order AD Module
Taylor Series Strategy
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Second-Order AD Module
Taylor Series Strategy
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Second-Order AD Module
Hessian Performance

• Twice ADIFOR, first produces a gradient code with
  ADIFOR 2.0, and then runs the gradient code
  through ADIFOR again.
• Forward, implements the forward mode.
• Adaptive Forward, uses the forward mode, with
  preaccumulation at a statement level where deemed
  appropriate.
• Sparse Taylor Series, uses the Taylor series mode
  to compute the needed entries.

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An Application in OCPs
Problem Definition and Theoretical Analysis
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An Application in OCPs
The first order sensitivity equations
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An Application in OCPs
The second order sensitivity equations
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An Application in OCPs
The second order sensitivity equations
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An Application in OCPs
Optimal control problem
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An Application in OCPs
Truncated Newton method for the solution of the
NLP
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An Application in OCPs
Implementation Details
Step 1

• Automatic derivation of the first and second
  order sensitivity equations to construct a full
  augmented IVP.
• Creates corresponding program subroutines in a
  format suitable to a standard IVP solver.

Step 2

• Numerical solution of the outer NLP using a
  truncated Newton method which solves
  bound-constrained problems.

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An Application in OCPs
Two approaches with TN method
TN algorithm with finite difference scheme

• Gradient evaluation requires the solution of the
  first order sensitivity system
• Gradient information is used to approximate the
  Hessian vector product with a finite difference
  scheme

TN algorithm with the exact Hessian vector
product calculation

• It uses the second order sensitivity equations
  defined in Eq. (5a) to obtain the exact Hessian
  vector product. (Earlier methods of the CVP type
  were based on first order sensitivities only,
  i.e. Gradient based algorithms mostly).
• This approach has been shown more robust and
  reliable due to the use of exact second order
  information.

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Summary

• Basics of derivatives
• Definition of derivatives
• Application of derivatives
• Methods to compute derivatives
• Basics of AD
• Compute first order derivatives with forward mode
• Compute first order derivatives with reverse mode
• Second Order Differentiation
• Compute second order derivatives with forward
  mode strategy
• Compute second order derivatives with Taylor
  Series strategy
• Hessian Performance
• An Application in Optimal Control Problems
• First order and second order sensitivity
  equations of DAE
• Solve optimal control problem with CVP method
• Solve nonlinear programming problems with
  truncated Newton method
• Truncated Newton method with exact Hessian vector
  product calculation

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References

• Abate, Bischof, Roh,Carle "Algorithms and Design
  for a Second-Order Automatic Differentiation
  Module
• Eva Balsa Canto, Julio R. Banga, Antonio A.
  Alonso, Vassilios S. Vassiliadis "Restricted
  second order information for the solution of
  optimal control problems using control vector
  parameterization
• Louis B. Rall, George F. Corliss An Introduction
  to Automatic Differentiation
• Andreas Griewank Evaluating Derivatives
  Principles and Techniques of Algorithmic
  Differentiation
• Stephen G. Nash A Survey of Truncated-Newton
  Methods