tlm_adjoint.hessian
Module Contents
- class tlm_adjoint.hessian.Hessian
Represents a Hessian associated with a given forward.
- abstract action(M, dM, M0=None)
Compute (the conjugate of) a Hessian action on some \(\zeta\) using an adjoint of a tangent-linear. i.e. compute
\[\left( \frac{d}{dm} \left[ \frac{d \mathcal{J}}{d m} \zeta \right] \right)^{*,T}.\]- Parameters:
- Mvariable or Sequence[variable, …]
Defines the control.
- dMvariable or Sequence[variable, …]
Defines the direction \(\zeta\).
- M0variable or Sequence[variable, …]
Defines the value of the control. M is used if not supplied.
- Returns:
- JComplex
Value of the functional.
- dJvariable or tuple[variable, …]
Value of the directional derivative \(\left( d \mathcal{J} / d m \right) \zeta\).
- ddJvariable or tuple[variable, …]
The (conjugate of) the Hessian action on \(\zeta\).
- class tlm_adjoint.hessian.GeneralHessian(forward, *, manager=None)
Represents a Hessian associated with a given forward. Calls to
GeneralHessian.action()re-run the forward.- Parameters:
- forwardcallable
Accepts one or more variable arguments, and returns a scalar variable defining the functional.
- manager
EquationManager Used to create an internal manager via
EquationManager.new(). manager() is used if not supplied.
- action(M, dM, M0=None)
Compute (the conjugate of) a Hessian action on some \(\zeta\) using an adjoint of a tangent-linear. i.e. compute
\[\left( \frac{d}{dm} \left[ \frac{d \mathcal{J}}{d m} \zeta \right] \right)^{*,T}.\]- Parameters:
- Mvariable or Sequence[variable, …]
Defines the control.
- dMvariable or Sequence[variable, …]
Defines the direction \(\zeta\).
- M0variable or Sequence[variable, …]
Defines the value of the control. M is used if not supplied.
- Returns:
- JComplex
Value of the functional.
- dJvariable or tuple[variable, …]
Value of the directional derivative \(\left( d \mathcal{J} / d m \right) \zeta\).
- ddJvariable or tuple[variable, …]
The (conjugate of) the Hessian action on \(\zeta\).
- class tlm_adjoint.hessian.GaussNewton(R_inv_action, B_inv_action=None)
Represents a Gauss-Newton approximation for a Hessian.
This defines a Hessian approximation
\[H = J^T R_\text{obs}^{-1} J + B^{-1},\]where \(J\) is the forward Jacobian. In a variational assimilation approach \(R_\text{obs}^{-1}\) corresponds to the observational inverse covariance and \(B^{-1}\) corresponds to the background inverse covariance.
- Parameters:
- R_inv_actioncallable
Accepts one or more variables as arguments, defining a direction, and returns a variable or a
Sequenceof variables defining the conjugate of the action of \(R^{-1}\) on this direction. Arguments should not be modified.- B_inv_actioncallable
Accepts one or more variables as arguments, defining a direction, and returns a variable or a
Sequenceof variables defining the conjugate of the action of \(B^{-1}\) on this direction. Arguments should not be modified.
- action(M, dM, M0=None)
Compute (the conjugate of) a Hessian action on some \(\zeta\), using the Gauss-Newton approximation for the Hessian. i.e. compute
\[\left[ J^T R_\text{obs}^{-1} J \zeta + B^{-1} \zeta \right]^{*,T}.\]- Parameters:
- Mvariable or Sequence[variable, …]
Defines the control.
- dMvariable or Sequence[variable, …]
Defines the direction \(\zeta\).
- M0variable or Sequence[variable, …]
Defines the value of the control. M is used if not supplied.
- class tlm_adjoint.hessian.GeneralGaussNewton(forward, R_inv_action, B_inv_action=None, *, manager=None)
Represents a Gauss-Newton approximation to a Hessian associated with a given forward. Calls to
GaussNewton.action()re-run the forward.- Parameters:
- forwardcallable
Accepts one or more variable arguments, and returns a variable or
Sequenceof variables defining the state.- R_inv_actioncallable
See
GaussNewton.- B_inv_actioncallable
See
GaussNewton.- manager
EquationManager Used to create an internal manager via
EquationManager.new(). manager() is used if not supplied.