Class representing an exponential covariance function and its KL eigevalues/eigenfunctions. More...

#include <Stokhos_KL_OneDExponentialCovarianceFunction.hpp>

Classes
struct	EigFuncCos
	Nonlinear function whose roots define eigenvalues for cos() eigenfunction. More...

struct	EigFuncSin
	Nonlinear function whose roots define eigenvalues for sin() eigenfunction. More...

Public Types
typedef ExponentialOneDEigenFunction < value_type >	eigen_function_type

typedef OneDEigenPair < eigen_function_type >	eigen_pair_type

Public Member Functions
	OneDExponentialCovarianceFunction (int M, const value_type &a, const value_type &b, const value_type &L, const int dim_name, Teuchos::ParameterList &solverParams)
	Constructor. More...

	~OneDExponentialCovarianceFunction ()
	Destructor. More...

value_type	evaluateCovariance (const value_type &x, const value_type &xp) const
	Evaluate covariance. More...

const Teuchos::Array < eigen_pair_type > &	getEigenPairs () const
	Get eigenpairs. More...

Protected Types
typedef Teuchos::ScalarTraits < value_type >::magnitudeType	magnitude_type

Protected Member Functions
template<class Func >
value_type	newton (const Func &func, const value_type &a, const value_type &b, magnitude_type tol, int max_num_its)
	A basic root finder based on Newton's method. More...

template<class Func >
value_type	bisection (const Func &func, const value_type &a, const value_type &b, magnitude_type tol, int max_num_its)
	A basic root finder based on bisection. More...

Protected Attributes
value_type	L
	Correlation length. More...

Teuchos::Array< eigen_pair_type >	eig_pair
	Eigenpairs. More...

Private Member Functions
	OneDExponentialCovarianceFunction (const OneDExponentialCovarianceFunction &)
	Prohibit copying. More...

OneDExponentialCovarianceFunction &	operator= (const OneDExponentialCovarianceFunction &)
	Prohibit copying. More...

Detailed Description

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

Class representing an exponential covariance function and its KL eigevalues/eigenfunctions.

This class provides the exponential covariance function

$\mbox{cov}(x,x') = \exp(-|x-x'|/L).$

The corresponding eigenfunctions can be shown to be $A_n\cos(\omega_n(x-\beta))$ and $B_n\sin(\omega^\ast_n(x-\beta))$ for $x\in[a,b]$ where $\omega_n$ and $\omega^\ast_n$ satisfy

$1 - L\omega_n\tan(\omega_n\alpha) = 0$

and

$L\omega^\ast_n + \tan(\omega^\ast_n\alpha) = 0$

respectively, where $\alpha=(b-a)/2$ and $\beta=(b+a)/2$ . Then

$A_n = \frac{1}{\left(\int_a^b\cos^2(\omega_n(x-\beta)) dx\right)^{1/2}} = \frac{1}{\sqrt{\alpha + \frac{\sin(2\omega_n\alpha)}{2\omega_n}}},$

$B_n = \frac{1}{\left(\int_a^b\sin^2(\omega^\ast_n(x-\beta)) dx\right)^{1/2}} = \frac{1}{\sqrt{\alpha - \frac{\sin(2\omega_n^\ast\alpha)}{2\omega^\ast_n}}}$

and the corresponding eigenvalues are given by

$\lambda_n = \frac{2L}{(L\omega_n)^2 + 1}$

and

$\lambda^\ast_n = \frac{2L}{(L\omega^\ast_n)^2 + 1}.$

It is straightforward to show that for each $n$ , $\omega^\ast_n < \omega_n$ , and thus $\lambda_n < \lambda^\ast_n$ . Hence when sorted on decreasing eigenvalue, the eigenfunctions alternate between cosine and sine. See "Stochastic Finite Elements" by Ghanem and Spanos for a complete description of how to derive these relationships.

For a given value of $M$ , the code works by computing the $M$ eigenfunctions using a bisection root solver to compute the frequencies $\omega_n$ and $\omega^\ast_n$ .

Data for the root solver is passed through a Teuchos::ParameterList, which accepts the following parameters:

"Bound Peturbation Size" – value_type Perturbation away from $i\pi/2$ for bounding the frequencies $\omega$ in the bisection algorithm
"Nonlinear Solver Tolerance" – value_type Tolerance for bisection nonlinear solver for computing frequencies
"Maximum Nonlinear Solver Iterations" – [int] (100) Maximum number of nonlinear solver iterations for computing the frequencies.

Definition at line 119 of file Stokhos_KL_OneDExponentialCovarianceFunction.hpp.

Member Typedef Documentation

template<typename value_type>

typedef ExponentialOneDEigenFunction<value_type> Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::eigen_function_type

Definition at line 122 of file Stokhos_KL_OneDExponentialCovarianceFunction.hpp.

template<typename value_type>

typedef OneDEigenPair<eigen_function_type> Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::eigen_pair_type

Definition at line 123 of file Stokhos_KL_OneDExponentialCovarianceFunction.hpp.

template<typename value_type>

typedef Teuchos::ScalarTraits<value_type>::magnitudeType Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::magnitude_type

protected

Definition at line 157 of file Stokhos_KL_OneDExponentialCovarianceFunction.hpp.

Constructor & Destructor Documentation

template<typename value_type >

Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::OneDExponentialCovarianceFunction	(	int	M,
		const value_type &	a,
		const value_type &	b,
		const value_type &	L,
		const int	dim_name,
		Teuchos::ParameterList &	solverParams
	)

Constructor.

Definition at line 46 of file Stokhos_KL_OneDExponentialCovarianceFunctionImp.hpp.

template<typename value_type>

Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::~OneDExponentialCovarianceFunction ( )

inline

Destructor.

Definition at line 134 of file Stokhos_KL_OneDExponentialCovarianceFunction.hpp.

template<typename value_type>

Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::OneDExponentialCovarianceFunction ( const OneDExponentialCovarianceFunction< value_type > & )

private

Prohibit copying.

Member Function Documentation

template<typename value_type>

value_type Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::evaluateCovariance	(	const value_type &	x,
		const value_type &	xp
	)		const

inline

Evaluate covariance.

Definition at line 137 of file Stokhos_KL_OneDExponentialCovarianceFunction.hpp.

template<typename value_type>

const Teuchos::Array<eigen_pair_type>& Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::getEigenPairs ( ) const

inline

Get eigenpairs.

Definition at line 143 of file Stokhos_KL_OneDExponentialCovarianceFunction.hpp.

template<typename value_type>

OneDExponentialCovarianceFunction& Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::operator= ( const OneDExponentialCovarianceFunction< value_type > & )

private

Prohibit copying.

template<typename value_type >

template<class Func >

value_type Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::newton	(	const Func &	func,
		const value_type &	a,
		const value_type &	b,
		magnitude_type	tol,
		int	max_num_its
	)

protected

A basic root finder based on Newton's method.

Definition at line 100 of file Stokhos_KL_OneDExponentialCovarianceFunctionImp.hpp.

template<typename value_type >

template<class Func >

value_type Stokhos::KL::OneDExponentialCovarianceFunction< value_type >::bisection	(	const Func &	func,
		const value_type &	a,
		const value_type &	b,
		magnitude_type	tol,
		int	max_num_its
	)