Mathematical background: adjoints and their applications

Section author: Patrick E. Farrell <patrick.farrell@maths.ox.ac.uk>

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Sections

• Foreword
  • Why care about adjoints?
  • A note on the exposition
  • Notation
• PDE-constrained optimisation
  • Problem statement
  • Solution approaches
• Differentiating functionals
  • Finite differencing
  • The tangent linear approach
  • The adjoint approach
  • Summary
• Properties of the adjoint equations
  • The adjoint reverses the propagation of information
    • A simple advection example
    • A time-dependent example
  • The adjoint equation is linear
    • Summary
• Applications of adjoints
  • PDE-constrained optimisation
  • Sensitivity analysis
  • Data assimilation
  • Inverse problems
  • Generalised stability theory
  • Error estimation

Appendices

• Generalised stability theory
  • Introduction
  • The singular value decomposition of the propagator
  • Computing the propagator
  • Singular value computation