inversion_ideas.DataMisfit#

class inversion_ideas.DataMisfit(data, uncertainty, simulation, *, build_hessian=False, estimate_hessian_diagonal=False)#

L2 data misfit.

Parameters:

data(n_data) array: Array with observed data values.
uncertainty(n_data) array: Array with data uncertainty.
simulationSimulation: Instance of Simulation.
build_hessianbool, optional: If True, the hessian method will build the Hessian matrix and allocate it in memory. If False, the hessian method will return a linear operator that represents the Hessian matrix. Default to False.

Important

Hessian matrices are usually very large. Use build_hessian=True only if you need to build it.
estimate_hessian_diagonalbool, optional: If True, the hessian_diagonal method will estimate the diagonal of the Hessian, even if the Jacobian of the simulation is a LinearOperator. If False, an error will be raised when calling the hessian_diagonal method in case the Jacobian of the simulation is a LinearOperator.

Important

Estimating the diagonal of a LinearOperator require a high number of dot products between the operator and multiple unit vectors. This might lead to very expensive computations. Enable estimate_hessian_diagonal only if you really need to.

Attributes

`n_data`	Number of data values.
`n_params`	Number of model parameters.
`name`	Name of the objective function.
`weights`	Data weights: 1D array with the square of the inverse of the uncertainties.
`weights_matrix`	Diagonal matrix with the square root of the regularization weights.

Methods

`__call__`(model)	Evaluate the objective function for a given model.
`chi_factor`(model)	Compute chi factor.
`gradient`(model)	Gradient vector.
`hessian`(model)	Hessian matrix.
`hessian_diagonal`(model)	Get the main diagonal of the Hessian.
`info`()	Get information about the objective function.
`residual`(model)	Residual vector.
`set_name`(value)	Set name for the objective function.

Notes

The L2 data misfit objective function is defined as:

\[\phi_d(\mathbf{m}) = \sum\limits_{i=1}^N \frac{ \left\lvert f_i(\mathbf{m}) - d_i^\text{obs} \right\rvert^2 }{ \epsilon_i^2 }\]

where \(\mathbf{m}\) is the model vector, \(d_i^\text{obs}\) is the \(i\)-th observed datum, \(f_i(\mathbf{m})\) is the forward modelling function for the \(i\)-th datum, and \(\epsilon_i\) is the uncertainty of the \(i\)-th datum.

The data misfit term can be expressed in terms of weights \(w_i = 1 / \epsilon_i^2\):

\[\phi_d(\mathbf{m}) = \sum\limits_{i=1}^N w_i \left\lvert f_i(\mathbf{m}) - d_i^\text{obs} \right\rvert^2\]

And also in matrix form:

\[\phi_d(\mathbf{m}) = \left\lVert \mathbf{W} \left[ f(\mathbf{m}) - \mathbf{d}^\text{obs} \right] \right\rVert^2\]

where \(\mathbf{W}\) is a diagonal matrix with the square root of the weights, \(\mathbf{d}^\text{obs}\) is the vector of observed data, and \(f(\mathbf{m})\) is the forward modelling vector.

Attributes#

DataMisfit.n_data#: Number of data values.

DataMisfit.n_params#: Number of model parameters.

DataMisfit.name#: Name of the objective function.

DataMisfit.weights#: Data weights: 1D array with the square of the inverse of the uncertainties.

DataMisfit.weights_matrix#: Diagonal matrix with the square root of the regularization weights.

Methods#

Methods documentation

DataMisfit.__call__(model)#: Evaluate the objective function for a given model.

DataMisfit.chi_factor(model)#

Compute chi factor.

Parameters:

model(n_params) array: Array with model values.

DataMisfit.gradient(model)#: Gradient vector.

DataMisfit.hessian(model)#: Hessian matrix.

DataMisfit.hessian_diagonal(model)#

Get the main diagonal of the Hessian.

Important

If the Jacobian of the simulation is a LinearOperator, the diagonal of the Hessian will be estimated only if estimate_hessian_diagonal is True. Diagonal estimations can be expensive computational tasks for large problems. Make sure you to enable diagonal estimations only if you need it.

Parameters:

model(n_params) array: Array with model values.

Returns:

(n_params,) array: Array containing the diagonal of the Hessian.

DataMisfit.info()#: Get information about the objective function.

DataMisfit.residual(model)#

Residual vector.

Parameters:

model(n_params) array: Array with model values.

Returns:

(n_data) array: Array with residual vector.

Notes

Residual vector defined as:

\[\mathbf{r} = \mathcal{F}(\mathbf{m}) - \mathbf{d}\]

where \(\mathbf{d}\) is the vector with observed data, \(\mathcal{F}\) is the forward model, and \(\mathbf{m}\) is the model vector.

DataMisfit.set_name(value)#: Set name for the objective function.