simpeg.directives.BetaEstimateMaxDerivative

class simpeg.directives.BetaEstimateMaxDerivative(beta0_ratio=1.0, random_seed=None, **kwargs)

Bases: BaseBetaEstimator

Estimate initial trade-off parameter (beta) using largest derivatives.

The initial trade-off parameter (beta) is estimated by scaling the ratio between the largest derivatives in the gradient of the data misfit and model objective function. 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. 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.

random_seedNone or RandomSeed, optional

Random seed used for random sampling. It can either be an int, a predefined Numpy random number generator, or any valid input to numpy.random.default_rng.

Attributes

beta0_ratio	The estimated ratio is multiplied by this to obtain beta.
dmisfit	Data misfit associated with the directive.
invProb	Inverse problem associated with the directive.
inversion	Inversion object associated with the directive.
opt	Optimization algorithm associated with the directive.
random_seed	Random seed to initialize with.
reg	Regularization associated with the directive.
seed	random_seed.seed has been deprecated.
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.

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).

We define \(\gamma\) as the desired ratio between the data misfit and model objective functions at the initial beta iteration (defined by the ‘beta0_ratio’ input argument). Here, the initial trade-off parameter is computed according to:

\[\beta_0 = \gamma \frac{| \nabla_m \phi_d (\mathbf{m_0}) |_{max}}{| \nabla_m \phi_m (\mathbf{m_0 + \delta m}) |_{max}}\]

where

\[\delta \mathbf{m} = \frac{m_{max}}{\mu_{max}} \boldsymbol{\mu}\]

and \(\boldsymbol{\mu}\) is a set of independent samples from the continuous uniform distribution between 0 and 1.