tlm_adjoint.hessian

Module Contents

class tlm_adjoint.hessian.Hessian

Represents a Hessian associated with a given forward. Abstract base class.

abstract compute_gradient(M, M0=None)

Compute the (conjugate of the) derivative of a functional with respect to a control using an adjoint.

Parameters:

  • M – A variable or a Sequence of variables defining the control.

  • M0 – A variable or a Sequence of variables defining the control value. M is used if not supplied.

Returns:

The (conjugate of the) derivative. A variable or Sequence of variables, depending on the type of M.

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. considering derivatives to be row vectors, compute

\[\left( \frac{d}{dm} \left[ \frac{d \mathcal{J}}{d m} \zeta \right] \right)^{*,T}.\]
Parameters:

  • M – A variable or a Sequence of variables defining the control.

  • dM – A variable or a Sequence of variables defining \(\zeta\). The (conjugate of the) Hessian action on \(\zeta\) is computed.

  • M0 – A variable or a Sequence of variables defining the control value. M is used if not supplied.

Returns:

A tuple (J, dJ, ddJ). J is the value of the functional. dJ is the value of \(\left( d \mathcal{J} / d m \right) \zeta\). ddJ stores the (conjugate of the) result of the Hessian action on \(\zeta\), and is a variable or a Sequence of variables depending on the type of M.

action_fn(m, m0=None)

Return a callable which can be used to compute Hessian actions.

Parameters:

  • m – A variable defining the control.

  • m0 – A variable defining the control value. m is used if not supplied.

Returns:

A callable which accepts a single variable argument, and returns the result of the Hessian action on that argument as a variable. Note that the result is not the conjugate of the Hessian action on the input argument.

class tlm_adjoint.hessian.GeneralHessian(forward, *, manager=None)

Represents a Hessian associated with a given forward. Calls to GeneralHessian.compute_gradient() or GeneralHessian.action() re-run the forward.

Parameters:

  • forward – A callable which accepts one or more variable arguments, and which returns a variable defining the forward functional.

  • manager – An EquationManager used to create an internal manager via EquationManager.new(). manager() is used if not supplied.

compute_gradient(M, M0=None)

Compute the (conjugate of the) derivative of a functional with respect to a control using an adjoint.

Parameters:

  • M – A variable or a Sequence of variables defining the control.

  • M0 – A variable or a Sequence of variables defining the control value. M is used if not supplied.

Returns:

The (conjugate of the) derivative. A variable or Sequence of variables, depending on the type of M.

action(M, dM, M0=None)

Compute (the conjugate of) a Hessian action on some \(\zeta\) using an adjoint of a tangent-linear. i.e. considering derivatives to be row vectors, compute

\[\left( \frac{d}{dm} \left[ \frac{d \mathcal{J}}{d m} \zeta \right] \right)^{*,T}.\]
Parameters:

  • M – A variable or a Sequence of variables defining the control.

  • dM – A variable or a Sequence of variables defining \(\zeta\). The (conjugate of the) Hessian action on \(\zeta\) is computed.

  • M0 – A variable or a Sequence of variables defining the control value. M is used if not supplied.

Returns:

A tuple (J, dJ, ddJ). J is the value of the functional. dJ is the value of \(\left( d \mathcal{J} / d m \right) \zeta\). ddJ stores the (conjugate of the) result of the Hessian action on \(\zeta\), and is a variable or a Sequence of variables depending on the type of M.

class tlm_adjoint.hessian.GaussNewton(R_inv_action, B_inv_action=None)

Represents a Gauss-Newton approximation for a Hessian. Abstract base class.

In terms of matrices 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_action – A callable which accepts one or more variables, and returns the conjugate of the action of the operator corresponding to \(R_\text{obs}^{-1}\) on those variables, returning the result as a variable or a Sequence of variables.

  • B_inv_action – A callable which accepts one or more variables, and returns the conjugate of the action of the operator corresponding to \(B^{-1}\) on those variables, returning the result as a variable or a Sequence of variables.

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

\[\left( H \zeta \right)^{*,T}.\]
Parameters:

  • M – A variable or a Sequence of variables defining the control.

  • dM – A variable or a Sequence of variables defining \(\zeta\). The (conjugate of the) approximated Hessian action on \(\zeta\) is computed.

  • M0 – A variable or a Sequence of variables defining the control value. M is used if not supplied.

Returns:

The (conjugate of the) result of the approximated Hessian action on \(\zeta\). A variable or a Sequence of variables depending on the type of M.

action_fn(m, m0=None)

Return a callable which can be used to compute Hessian actions using the Gauss-Newton approximation.

Parameters:

  • m – A variable defining the control.

  • m0 – A variable defining the control value. m is used if not supplied.

Returns:

A callable which accepts a single variable argument, and returns the result of the approximated Hessian action on that argument as a variable. Note that the result is not the conjugate of the approximated Hessian action on the input argument.

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:

  • forward – A callable which accepts one or more variable arguments, and which returns a variable or Sequence of variables defining the state.

  • R_inv_action – See GaussNewton.

  • B_inv_action – See GaussNewton.

  • manager – An EquationManager used to create an internal manager via EquationManager.new(). manager() is used if not supplied.