SimPEG.directives.BetaEstimate_ByEig#

class SimPEG.directives.BetaEstimate_ByEig(beta0_ratio=1.0, n_pw_iter=4, seed=None, **kwargs)[source]#

Bases: BaseBetaEstimator

Estimate initial trade-off parameter (beta) by power iteration.

The initial trade-off parameter (beta) is estimated by scaling the ratio between the largest eigenvalue in the second derivative of the data misfit and the model objective function. The largest eigenvalues are estimated using the power iteration method; see SimPEG.utils.eigenvalue_by_power_iteration(). The estimated trade-off parameter is used to update the beta property in the associated SimPEG.inverse_problem.BaseInvProblem object prior to running the inversion. Note that a separate directive is used for updating the trade-off parameter at successive beta iterations; see BetaSchedule.

Parameters:

beta0_ratio: float: Desired ratio between data misfit and model objective function at initial beta iteration.
n_pw_iterint: Number of power iterations used to estimate largest eigenvalues.
seedint, None: Seed used for random sampling.

Notes

Let \(\phi_d\) represent the data misfit, \(\phi_m\) represent the model objective function and \(\mathbf{m_0}\) represent the starting model. The first model update is obtained by minimizing the a global objective function of the form:

\[\phi (\mathbf{m_0}) = \phi_d (\mathbf{m_0}) + \beta_0 \phi_m (\mathbf{m_0})\]

where \(\beta_0\) represents the initial trade-off parameter (beta). Let \(\gamma\) define the desired ratio between the data misfit and model objective functions at the initial beta iteration (defined by the ‘beta0_ratio’ input argument). Using the power iteration approach, our initial trade-off parameter is given by:

\[\beta_0 = \gamma \frac{\lambda_d}{\lambda_m}\]

where \(\lambda_d\) as the largest eigenvalue of the Hessian of the data misfit, and \(\lambda_m\) as the largest eigenvalue of the Hessian of the model objective function. For each Hessian, the largest eigenvalue is computed using power iteration. The input parameter ‘n_pw_iter’ sets the number of power iterations used in the estimate.

For a description of the power iteration approach for estimating the larges eigenvalue, see SimPEG.utils.eigenvalue_by_power_iteration().

Attributes

`beta0_ratio`	The estimated ratio is multiplied by this to obtain beta.
`debug`	verbose.debug has been deprecated.
`dmisfit`	Data misfit associated with the directive.
`invProb`	Inverse problem associated with the directive.
`inversion`	Inversion object associated with the directive.
`n_pw_iter`	Number of power iterations for estimating largest eigenvalues.
`opt`	Optimization algorithm associated with the directive.
`reg`	Regularization associated with the directive.
`seed`	Random seed to initialize with.
`simulation`	Return simulation for all data misfits.
`survey`	Return survey for all data misfits
`verbose`	Whether or not to print debugging information.

Methods

`endIter`()	Update inversion parameter(s) according to directive at end of iteration.
`finish`()	Update inversion parameter(s) according to directive at end of inversion.
`initialize`()	Initialize inversion parameter(s) according to directive.
`validate`(directive_list)	Validate directive.