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. 2014 Sep;98:521–527. doi: 10.1016/j.neuroimage.2014.04.040

Table 1.

Comparison of the different gradient computation methods. The flow eqn. is either linear or non-linear, with P parameters and N state variables.

Finite differences Forward sensitivities Adjoint
Suitability Arbitrary N ≫ P P ≫ N
Cost (1 + P) flow eqns. P non-linear sensitivity eqns. + 1 flow eqn. 1 linear adjoint eqn. + 1 flow eqn.
Key steps
  • 1.

    Integrate flow eqn.

  • 2.

    Parametrically perturb flow P times
  • 1.

    Integrate the coupled flow and sensitivity eqns.
  • 1.

    Integrate flow eqn.

  • 2.

    Integrate adjoint eqn.