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

#include <Stokhos_KL_ExponentialRandomField.hpp>

Public Types
typedef ExponentialOneDEigenFunction< value_type >	one_d_eigen_func_type

typedef OneDEigenPair< one_d_eigen_func_type >	one_d_eigen_pair_type

typedef ProductEigenPair< one_d_eigen_func_type, execution_space >	product_eigen_pair_type

typedef Kokkos::View< one_d_eigen_func_type **, execution_space >	eigen_func_array_type

typedef Kokkos::View< value_type *, execution_space >	eigen_value_array_type

Public Member Functions
	ExponentialRandomField ()
	Default constructor.

	ExponentialRandomField (Teuchos::ParameterList &solverParams)
	Constructor.

KOKKOS_INLINE_FUNCTION	~ExponentialRandomField ()
	Destructor.

KOKKOS_INLINE_FUNCTION int	spatialDimension () const
	Return spatial dimension of the field.

KOKKOS_INLINE_FUNCTION int	stochasticDimension () const
	Return stochastic dimension of the field.

template<typename point_type , typename rv_type >
KOKKOS_INLINE_FUNCTION Teuchos::PromotionTraits< typenamerv_type::value_type, value_type >::promote	evaluate (const point_type &point, const rv_type &random_variables) const
	Evaluate random field at a point.

template<typename point_type >
KOKKOS_INLINE_FUNCTION value_type	evaluate_mean (const point_type &point) const
	Evaluate mean of random field at a point.

template<typename point_type >
KOKKOS_INLINE_FUNCTION Teuchos::PromotionTraits< typenamepoint_type::value_type, value_type >::promote	evaluate_standard_deviation (const point_type &point) const
	Evaluate standard deviation of random field at a point.

template<typename point_type >
KOKKOS_INLINE_FUNCTION Teuchos::PromotionTraits< typenamepoint_type::value_type, value_type >::promote	evaluate_eigenfunction (const point_type &point, int i) const
	Evaluate given eigenfunction at a point.

value_type KOKKOS_INLINE_FUNCTION	eigenvalue (int i) const
	Return eigenvalue.

void	print (std::ostream &os) const
	Print KL expansion.

Protected Attributes
int	num_KL
	Number of KL terms.

int	dim
	Dimension of expansion.

value_type	mean
	Mean of random field.

value_type	std_dev
	Standard deviation of random field.

eigen_func_array_type	product_eigen_funcs
	Product eigenfunctions.

eigen_value_array_type	product_eigen_values
	Product eigenvalues.

Detailed Description

template<typename value_type, typename execution_space = Kokkos::DefaultExecutionSpace>
class Stokhos::KL::ExponentialRandomField< value_type, execution_space >

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 Types

Public Member Functions

Protected Attributes

Detailed Description