inversion_ideas.conjugate_gradient

inversion_ideas.conjugate_gradient(objective, initial_model, preconditioner=None, **kwargs)

Minimize objective function with a Conjugate Gradient method.

Important

This minimizer should be used only for linear objective functions.

Parameters:

objectiveObjective

Objective function to be minimized.

initial_model(n_params) array

Initial model used to start the minimization.

preconditioner(n_params, n_params) array, sparse array or LinearOperator, optional

Matrix used as preconditioner in the conjugate gradient algorithm. If None, no preconditioner will be used. If the preconditioner implements an initialize method, the preconditioner will be initialized before being used in the conjugate gradient algorithm.

kwargsdict

Extra arguments that will be passed to the scipy.sparse.linalg.cg() function.

Returns:

inverted_model(n_params) array

Inverted model obtained after minimization.

Notes

Minimize the objective function \(\phi(\mathbf{m})\) by solving the system:

\[\bar{\bar{\nabla}} \phi \mathbf{m}^{*} = - \bar{\nabla} \phi\]

through a Conjugate Gradient algorithm, where \(\bar{\bar{\nabla}} \phi\) and \(\bar{\nabla} \phi\) are the the Hessian and the gradient of the objective function, respectively.