TransportMaps.Distributions.Examples.PoissonProblem.PoissonDistributions

Module Contents

Classes

Covariance

OrnsteinUhlenbeckCovariance

SquaredExponentialCovariance

IdentityCovariance

GaussianFieldPoissonDistribution

Posterior distribution of the conductivity field of a Poisson problem.

GaussianFieldIndependentLikelihoodPoissonDistribution

Posterior distribution of the conductivity field of a Poisson problem.
class TransportMaps.Distributions.Examples.PoissonProblem.PoissonDistributions.Covariance[source]

Bases: object

__call__(coord)[source]
abstract _evaluate(dists)[source]
class TransportMaps.Distributions.Examples.PoissonProblem.PoissonDistributions.OrnsteinUhlenbeckCovariance(var, l)[source]

Bases: Covariance

_evaluate(dists)[source]
class TransportMaps.Distributions.Examples.PoissonProblem.PoissonDistributions.SquaredExponentialCovariance(var, l)[source]

Bases: Covariance

_evaluate(dists)[source]
class TransportMaps.Distributions.Examples.PoissonProblem.PoissonDistributions.IdentityCovariance(var)[source]

Bases: Covariance

_evaluate(dists)[source]
class TransportMaps.Distributions.Examples.PoissonProblem.PoissonDistributions.GaussianFieldPoissonDistribution(solver, sens_pos_list, sens_geo_std, prior_mean_field_vals=None, prior_cov=OrnsteinUhlenbeckCovariance(1.0, 1.0), lkl_cov=IdentityCovariance(1.0), field_vals=None, real_observations=None)[source]

Bases: TransportMaps.Distributions.Inference.BayesPosteriorDistribution

Posterior distribution of the conductivity field of a Poisson problem.

The system is observed through the operator \(\mathcal{O}[{\bf c}]\) defined by

\[\mathcal{O}[{\bf c}](u) = \int u({\bf x}) \frac{1}{2\pi\sigma_y^2} \exp\left(-\frac{\Vert{\bf x} - {\bf c}\Vert^2}{2\sigma_y^2} \right) d{\bf x} \;,\]

where \(u\) is the solution associated to an underlying field \(\kappa\).

Given the sensor locations \(\{{\bf c}_i\}_{i=1}^s\), the Bayesian problem is defined as follows:

\[\begin{split}{\bf y}_i = \mathcal{O}[{\bf c}_i](u) + \varepsilon \;, \qquad \varepsilon \sim \mathcal{GP}(0, \mathcal{C}_y({\bf x},{\bf x}^\prime)) \\ \kappa({\bf x}) \sim \log\mathcal{GP}(\mu_\kappa({\bf x}), \mathcal{C}_\kappa({\bf x},{\bf x}^\prime))\end{split}\]
Parameters:

  • solver (PoissonSolver) – Poisson solver for a particular problem setting

  • sens_pos_list (list of tuple) – list of sensor locations

  • sens_geo_std (float) – observation kernel width \(\sigma_y\)

  • real_observations (ndarray) – observations \(\{{\bf y}_i\}_{i=1}^s\)

  • field_vals (ndarray) – generating field (default is None, generating a synthetic field)

  • prior_mean_field_vals (ndarray) – values \(\mu_\kappa({\bf x})\) (default is None, which corresponds to \(\mu_\kappa({\bf x})=0\))

  • prior_cov (Covariance) – covariance function \(\mathcal{C}_\kappa({\bf x},{\bf x}^\prime)\)

  • lkl_cov (Covariance) – covariance function \(\mathcal{C}_y({\bf x},{\bf x}^\prime)\)

__getstate__()[source]
__setstate__(state)[source]
set_true_field()[source]
class TransportMaps.Distributions.Examples.PoissonProblem.PoissonDistributions.GaussianFieldIndependentLikelihoodPoissonDistribution(solver, sens_pos_list, sens_geo_std, prior_mean_field_vals=None, prior_cov=OrnsteinUhlenbeckCovariance(1.0, 1.0), lkl_std=1.0, field_vals=None, real_observations=None)[source]

Bases: TransportMaps.Distributions.Inference.BayesPosteriorDistribution

Posterior distribution of the conductivity field of a Poisson problem.

The system is observed through the operator \(\mathcal{O}[{\bf c}]\) defined by

\[\mathcal{O}[{\bf c}](u) = \int u({\bf x}) \frac{1}{2\pi\sigma_y^2} \exp\left(-\frac{\Vert{\bf x} - {\bf c}\Vert^2}{2\sigma_y^2} \right) d{\bf x} \;,\]

where \(u\) is the solution associated to an underlying field \(\kappa\).

Given the sensor locations \(\{{\bf c}_i\}_{i=1}^s\), the Bayesian problem is defined as follows:

\[\begin{split}{\bf y}_i = \mathcal{O}[{\bf c}_i](u) + \varepsilon \;, \qquad \varepsilon \sim \mathcal{N}(0, \sigma^2 {\bf I}) \\ \kappa({\bf x}) \sim \log\mathcal{GP}(\mu_\kappa({\bf x}), \mathcal{C}_\kappa({\bf x},{\bf x}^\prime))\end{split}\]
Parameters:

  • solver (PoissonSolver) – Poisson solver for a particular problem setting

  • sens_pos_list (list of tuple) – list of sensor locations

  • sens_geo_std (float) – observation kernel width \(\sigma_y\)

  • real_observations (ndarray) – observations \(\{{\bf y}_i\}_{i=1}^s\)

  • field_vals (ndarray) – generating field (default is None, generating a synthetic field)

  • prior_mean_field_vals (ndarray) – values \(\mu_\kappa({\bf x})\) (default is None, which corresponds to \(\mu_\kappa({\bf x})=0\))

  • prior_cov (Covariance) – covariance function \(\mathcal{C}_\kappa({\bf x},{\bf x}^\prime)\)

  • lkl_std (floag) – standard deviation \(\sigma\)

__getstate__()[source]
__setstate__(state)[source]
set_true_field()[source]