Stokhos: Stokhos::KL::ExponentialRandomField< value

Class representing a KL expansion of an exponential random field. More...

#include <Stokhos_KL_ExponentialRandomField.hpp>

Public Member Functions
	ExponentialRandomField (Teuchos::ParameterList &solverParams)
	Constructor.
	~ExponentialRandomField ()
	Destructor.
int	spatialDimension () const
	Return spatial dimension of the field.
int	stochasticDimension () const
	Return stochastic dimension of the field.
template<typename rvar_type >
Teuchos::PromotionTraits < rvar_type, value_type > ::promote	evaluate (const Teuchos::Array< value_type > &point, const Teuchos::Array< rvar_type > &random_variables) const
	Evaluate random field at a point.
value_type	evaluate_mean (const Teuchos::Array< value_type > &point) const
	Evaluate mean of random field at a point.
value_type	evaluate_standard_deviation (const Teuchos::Array< value_type > &point) const
	Evaluate standard deviation of random field at a point.
value_type	evaluate_eigenfunction (const Teuchos::Array< value_type > &point, int i) const
	Evaluate given eigenfunction at a point.
const Teuchos::Array < ProductEigenPair< value_type > > &	getEigenPairs () const
	Get eigenpairs.
void	print (std::ostream &os) const
	Print KL expansion.
Protected Attributes
int	num_KL
	Number of KL terms.
int	dim
	Dimension of expansion.
Teuchos::Array< value_type >	domain_upper_bound
	Domain upper bounds.
Teuchos::Array< value_type >	domain_lower_bound
	Domeain lower bounds.
Teuchos::Array< value_type >	correlation_length
	Correlation lengths.
value_type	mean
	Mean of random field.
value_type	std_dev
	Standard deviation of random field.
Teuchos::Array < ProductEigenPair< value_type > >	product_eig_pairs
	Product eigenfunctions.

Detailed Description

template<typename value_type>
class Stokhos::KL::ExponentialRandomField< value_type >

Class representing a KL expansion of an exponential random field.

This class provides a means for evaluating a random field $a(x,\theta)$ , $x\in D$ , $\theta\in\Omega$ through the KL expansion

$a(x,\theta) \approx a_0 + \sigma\sum_{k=1}^M \sqrt{\lambda_k}b_k(x)\xi_k(\theta)$

where $D$ is a $d$ -dimensional hyper-rectangle, for the case when the covariance function of $a$ is exponential:

$\mbox{cov}(x,x') = \sigma\exp(-|x_1-x_1'|/L_1-...-|x_d-x_d'|/L_d).$

In this case, the covariance function and domain factor into a product 1-dimensional covariance functions over 1-dimensional domains, and thus the eigenfunctions $b_k$ and eigenvalues $\lambda_k$ factor into a product of corresponding 1-dimensional eigenfunctions and values. These are computed by the OneDExponentialCovarianceFunction class For a given value of $M$ , the code works by computing the $M$ eigenfunctions for each direction using this class. Then all possible tensor products of these one-dimensional eigenfunctions are formed, with corresponding eigenvalue given by the product of the one-dimensional eigenvalues. These eigenvalues are then sorted in increasing order, and the first $M$ are chosen as the $M$ KL eigenpairs. Then given values for the random variables $\xi_k$ , the class provides a routine for evaluating a realization of the random field.

The idea for this approach was provided by Chris Miller.

All data is passed into this class through a Teuchos::ParameterList, which accepts the following parameters:

"Number of KL Terms" -- [int] (Required) Number of KL terms
"Domain Upper Bounds" -- [Teuchos::Array<value_type>] (Required) Domain upper bounds for each dimension
"Domain Lower Bounds" -- [Teuchos::Array<value_type>] (Required) Domain lower bounds for each dimension
"Correlation Lengths" -- [Teuchos::Array<value_type>[ (Required) Correlation length for each dimension
"Mean" -- [value_type] (Required) Mean of the random field
"Standard Deviation" -- [value_type] (Required) Standard devation $\sigma$ of the random field

Additionally parameters for the OneDExponentialCovarianceFunction are also accepted.

The documentation for this class was generated from the following files:

Stokhos_KL_ExponentialRandomField.hpp
Stokhos_KL_ExponentialRandomFieldImp.hpp

Public Member Functions

Protected Attributes

Detailed Description

template<typename value_type> class Stokhos::KL::ExponentialRandomField< value_type >

template<typename value_type>
class Stokhos::KL::ExponentialRandomField< value_type >