dubfi.inversion.inversion_other¶

Inversion for arbitrary single-variant a priori probability distributions.

Mathematical construction¶

Consider an arbitrary probability density function (PDF) P(x) of a scalar variable x with mean 0 and variance 1. This can be used as a priori PDF for the state vector s with (positive definite, real- symmetric) error covariance matrix B using the transformation s = Q x + s0 where Q Q^T = B. The PDF of s is P’(s) = P(Q^-1 (s - s0)).

Similarly, we can consider a PDF P(x) with mean x0 and standard deviation xstd. Then one can define s = O x where B = O D O^T is the (orthogonal) eigenvalue decomposition of B and D is a diagonal matrix with entries xstd^2. The prior of the state vector is s0 = O x0.

This module implements two helper classes for using almost arbitrary P(x) to construct P’(s) for these two transformations.

Added in version 0.1.5.

Classes¶

`InvertorOptimizerTransformedPrior`	Abstract solver for flux inversion problem with prior cost computed in transformed coordinates.
`InvertorOptimizerTransformedPriorNonnormalized`	Abstract solver for flux inversion problem with prior cost computed in transformed coordinates.
`InvertorOptimizerLaplace`	Numerical solver for flux inversion problem with Laplacian prior PDF.
`InvertorOptimizerLorentz`	Numerical solver for flux inversion problem with Lorentz/Cauchy/Breit-Wigner prior PDF.
`InvertorOptimizerSparseReconstruction`	Numerical solver for flux inversion problem for sparse reconstruction with zero prior.

Module Contents¶

class dubfi.inversion.inversion_other.InvertorOptimizerTransformedPrior(y: dubfi.linalg.types.AbstractVector, prior: numpy.ndarray, h: dubfi.linalg.generic.ParametrizedVector, b: numpy.ndarray, r: dubfi.linalg.generic.ParametrizedOperator, norm_prefactor: float = 0.0)¶

Bases: dubfi.inversion.inversion.InvertorOptimizer

Abstract solver for flux inversion problem with prior cost computed in transformed coordinates.

Helper for arbitrary prior probability density functions P(x) that are applied to the uncertainty-normalized deviation from the prior. Implement cost_prior_transformed() as -log(P(x)) and its derivatives in cost_prior_grad_transformed() and cost_prior_hess_transformed().

Added in version 0.1.5.

Solve inverse problem using full ensemble-estimated model uncertainty.

Parameters:

y (AbstractVector) – vector of observations (usually minus model far-field or background)
prior (np.ndarray) – a priori estimate for scaling factors
h (ParametrizedVector) – model equivalents depending on scaling factors
b (np.ndarray) – error covariance matrix of prior scaling factors
r (ParametrizedOperator) – error covariance matrix of model-observation comparison, depends on scaling factors
norm_prefactor (float, default=0.0) –

prefactor of normalization term in cost function.
prefactor of normalization term in cost function.

property b: numpy.ndarray¶: Prior error covariance matrix of scaling factors.

_check_before_solving() → None¶: Overwrite in inheriting classes to assert properties before solving.

cost_prior(s: numpy.ndarray) → numpy.float64¶: Prior contribution to cost function.

cost_prior_grad(s: numpy.ndarray) → numpy.ndarray¶: Prior contribution to gradient of cost function.

cost_prior_hess(s: numpy.ndarray) → numpy.ndarray¶: Prior contribution to hessian of cost function.

abstractmethod cost_prior_transformed(x: numpy.ndarray) → numpy.ndarray¶: Prior contribution to cost function in transformed coordinates.

abstractmethod cost_prior_grad_transformed(x: numpy.ndarray) → numpy.ndarray¶: Gradient of prior cost function in transformed coordinates.

abstractmethod cost_prior_hess_transformed(x: numpy.ndarray) → numpy.ndarray¶: Diagonal of Hessian of prior cost function in transformed coordinates.

class dubfi.inversion.inversion_other.InvertorOptimizerTransformedPriorNonnormalized(y: dubfi.linalg.types.AbstractVector, prior: numpy.ndarray, h: dubfi.linalg.generic.ParametrizedVector, b: numpy.ndarray, r: dubfi.linalg.generic.ParametrizedOperator, norm_prefactor: float = 0.0)¶

Bases: dubfi.inversion.inversion.InvertorOptimizer

Abstract solver for flux inversion problem with prior cost computed in transformed coordinates.

Helper for arbitrary prior probability density functions P(x). In this case, x is the full prior and not normalized to uncertainty. Implement cost_prior_transformed() as -log(P(x)) and its derivatives in cost_prior_grad_transformed() and cost_prior_hess_transformed().

Added in version 0.1.5.

Solve inverse problem using full ensemble-estimated model uncertainty.

Parameters:

y (AbstractVector) – vector of observations (usually minus model far-field or background)
prior (np.ndarray) – a priori estimate for scaling factors
h (ParametrizedVector) – model equivalents depending on scaling factors
b (np.ndarray) – error covariance matrix of prior scaling factors
r (ParametrizedOperator) – error covariance matrix of model-observation comparison, depends on scaling factors
norm_prefactor (float, default=0.0) –

prefactor of normalization term in cost function.
prefactor of normalization term in cost function.

property b: numpy.ndarray¶: Prior error covariance matrix of scaling factors.

property x_stdev: numpy.ndarray¶: Element-wise standard deviation of x.

property x_prior: numpy.ndarray¶: Prior of x.

_check_before_solving() → None¶: Overwrite in inheriting classes to assert properties before solving.

cost_prior(s: numpy.ndarray) → numpy.float64¶: Prior contribution to cost function.

cost_prior_grad(s: numpy.ndarray) → numpy.ndarray¶: Prior contribution to gradient of cost function.

cost_prior_hess(s: numpy.ndarray) → numpy.ndarray¶: Prior contribution to hessian of cost function.

abstractmethod cost_prior_transformed(x: numpy.ndarray) → numpy.ndarray¶: Prior contribution to cost function in transformed coordinates.

abstractmethod cost_prior_grad_transformed(x: numpy.ndarray) → numpy.ndarray¶: Gradient of prior cost function in transformed coordinates.

abstractmethod cost_prior_hess_transformed(x: numpy.ndarray) → numpy.ndarray¶: Diagonal of Hessian of prior cost function in transformed coordinates.

class dubfi.inversion.inversion_other.InvertorOptimizerLaplace(y: dubfi.linalg.types.AbstractVector, prior: numpy.ndarray, h: dubfi.linalg.generic.ParametrizedVector, b: numpy.ndarray, r: dubfi.linalg.generic.ParametrizedOperator, norm_prefactor: float = 0.0)¶

Bases: InvertorOptimizerTransformedPrior

Numerical solver for flux inversion problem with Laplacian prior PDF.

Added in version 0.1.5.

Solve inverse problem using full ensemble-estimated model uncertainty.

Parameters:

y (AbstractVector) – vector of observations (usually minus model far-field or background)
prior (np.ndarray) – a priori estimate for scaling factors
h (ParametrizedVector) – model equivalents depending on scaling factors
b (np.ndarray) – error covariance matrix of prior scaling factors
r (ParametrizedOperator) – error covariance matrix of model-observation comparison, depends on scaling factors
norm_prefactor (float, default=0.0) –

prefactor of normalization term in cost function.
prefactor of normalization term in cost function.

cost_prior_transformed(x: numpy.ndarray) → numpy.ndarray¶: Prior contribution to cost function in transformed coordinates.

cost_prior_grad_transformed(x: numpy.ndarray) → numpy.ndarray¶: Gradient of prior cost function in transformed coordinates.

cost_prior_hess_transformed(x: numpy.ndarray) → numpy.ndarray¶: Diagonal of Hessian of prior cost function in transformed coordinates.

class dubfi.inversion.inversion_other.InvertorOptimizerLorentz(y: dubfi.linalg.types.AbstractVector, prior: numpy.ndarray, h: dubfi.linalg.generic.ParametrizedVector, b: numpy.ndarray, r: dubfi.linalg.generic.ParametrizedOperator, norm_prefactor: float = 0.0)¶

Bases: InvertorOptimizerTransformedPrior

Numerical solver for flux inversion problem with Lorentz/Cauchy/Breit-Wigner prior PDF.

Added in version 0.1.5.

Solve inverse problem using full ensemble-estimated model uncertainty.

Parameters:

y (AbstractVector) – vector of observations (usually minus model far-field or background)
prior (np.ndarray) – a priori estimate for scaling factors
h (ParametrizedVector) – model equivalents depending on scaling factors
b (np.ndarray) – error covariance matrix of prior scaling factors
r (ParametrizedOperator) – error covariance matrix of model-observation comparison, depends on scaling factors
norm_prefactor (float, default=0.0) –

prefactor of normalization term in cost function.
prefactor of normalization term in cost function.

cost_prior_transformed(x: numpy.ndarray) → numpy.ndarray¶: Prior contribution to cost function in transformed coordinates.

cost_prior_grad_transformed(x: numpy.ndarray) → numpy.ndarray¶: Gradient of prior cost function in transformed coordinates.

cost_prior_hess_transformed(x: numpy.ndarray) → numpy.ndarray¶: Diagonal of Hessian of prior cost function in transformed coordinates.

class dubfi.inversion.inversion_other.InvertorOptimizerSparseReconstruction(y: dubfi.linalg.types.AbstractVector, prior: numpy.ndarray, h: dubfi.linalg.generic.ParametrizedVector, b: numpy.ndarray, r: dubfi.linalg.generic.ParametrizedOperator, norm_prefactor: float = 0.0)¶

Bases: InvertorOptimizerTransformedPriorNonnormalized

Numerical solver for flux inversion problem for sparse reconstruction with zero prior.

Added in version 0.1.5.

Solve inverse problem using full ensemble-estimated model uncertainty.

Parameters:

y (AbstractVector) – vector of observations (usually minus model far-field or background)
prior (np.ndarray) – a priori estimate for scaling factors
h (ParametrizedVector) – model equivalents depending on scaling factors
b (np.ndarray) – error covariance matrix of prior scaling factors
r (ParametrizedOperator) – error covariance matrix of model-observation comparison, depends on scaling factors
norm_prefactor (float, default=0.0) –

prefactor of normalization term in cost function.
prefactor of normalization term in cost function.

cost_prior_transformed(x: numpy.ndarray) → numpy.ndarray¶: Prior contribution to cost function in transformed coordinates.

cost_prior_grad_transformed(x: numpy.ndarray) → numpy.ndarray¶: Gradient of prior cost function in transformed coordinates.

cost_prior_hess_transformed(x: numpy.ndarray) → numpy.ndarray¶: Diagonal of Hessian of prior cost function in transformed coordinates.