Computation of Karhunen-Loeve decomposition using Quadrature approximation.

Available constructors:

KarhunenLoeveQuadratureAlgorithm(domain, bounds, covariance, experiment, basis, basisSize, mustScale, s)

Parameters:
domainDomain

The domain on which the covariance model and the Karhunen-Loeve eigenfunctions (modes) are discretized.

boundsInterval

Numerical bounds of the domain.

covarianceCovarianceModel

The covariance function to decompose.

experimentWeightedExperiment

The points and weights used in the quadrature approximation.

basissequence of Function

The basis in which the eigenfunctions are projected.

marginalDegreeint

The maximum degree to take into account in the tensorized Legendre basis.

mustScaleboolean

Flag to tell if the bounding box of the weighted experiment and the domain have to be maped or not.

sfloat,

The threshold used to select the most significant eigenmodes, defined in KarhunenLoeveAlgorithm.

Notes

The Karhunen-Loeve quadrature algorithm solves the Fredholm problem associated to the covariance function : see KarhunenLoeveAlgorithm to get the notations.

The Karhunen-Loeve quadrature approximation consists in replacing the integral by a quadrature approximation: if is the weighted experiment (see WeightedExperiment) associated to the measure , then for all functions measurable wrt , we have:

If we note , we build a more general quadrature approximation such that:

where only the points are considered.

We introduce the matrices such that , and such that .

The normalisation constraint and the orthogonality of the in leads to:

(1)

The Galerkin approach leads to the following generalized eigenvalue problem:

(2)

where and .

The collocation approach leads to the following generalized eigenvalue problem:

(3)

Equations (2) and (3) are equivalent when is invertible.

OpenTURNS solves the equation (2).

The second constructor is a short-hand to the first one, where basis is the tensorized Legendre basis (see OrthogonalProductPolynomialFactory and LegendreFactory), experiment is a tensorized Gauss-Legendre quadrature (see GaussProductExperiment), basisSize is equal to marginalDegree to the power the dimension of domain and mustScale is set to True.

Examples

Discretize the domain and create a covariance model:

>>> import openturns as ot
>>> bounds = ot.Interval([-1.0]*2, [1.0]*2)
>>> domain = ot.MeshDomain(ot.IntervalMesher([10]*2).build(bounds))
>>> s = 0.01
>>> model = ot.AbsoluteExponential([1.0]*2)


Give the basis used to decompose the eigenfunctions:

here, the 10 first Legendre polynomials family:

>>> basis = ot.OrthogonalProductPolynomialFactory([ot.LegendreFactory()]*2)
>>> functions = [basis.build(i) for i in range(10)]


Create the weighted experiment of the quadrature approximation: here, a Monte Carlo experiment from the measure orthogonal wrt the Legendre polynomials family:

>>> experiment = ot.MonteCarloExperiment(basis.getMeasure(), 1000)


>>> algorithm = ot.KarhunenLoeveQuadratureAlgorithm(domain, bounds, model, experiment, functions, True, s)


Run it!

>>> algorithm.run()
>>> result = algorithm.getResult()


Methods

 Accessor to the functional basis. Accessor to the object's name. Accessor to the covariance model. Accessor to the domain. Accessor to the points and weights of the quadrature approximation. Accessor to the object's id. Accessor to scale option. Accessor to the object's name. Accessor to number of modes to compute. Get the result structure. Accessor to the object's shadowed id. Accessor to the threshold used to select the most significant eigenmodes. Accessor to the object's visibility state. Test if the object is named. Test if the object has a distinguishable name. Computation of the eigenvalues and eigenfunctions values at the quadrature points. setCovarianceModel(covariance) Accessor to the covariance model. setName(name) Accessor to the object's name. setNbModes(nbModes) Accessor to the maximum number of modes to compute. Accessor to the object's shadowed id. setThreshold(threshold) Accessor to the limit ratio on eigenvalues. setVisibility(visible) Accessor to the object's visibility state.
__init__(*args)
getBasis()

Accessor to the functional basis.

Returns:
basisBasis

The basis in which the eigenfunctions are projected.

getClassName()

Accessor to the object’s name.

Returns:
class_namestr

The object class name (object.__class__.__name__).

getCovarianceModel()

Accessor to the covariance model.

Returns:
covModelCovarianceModel

The covariance model.

getDomain()

Accessor to the domain.

Returns:
domainDomain
The domain that discretizes the domin .
getExperiment()

Accessor to the points and weights of the quadrature approximation.

Returns:
experimentWeightedExperiment

The points and weights used in the quadrature approximation.

getId()

Accessor to the object’s id.

Returns:
idint

Internal unique identifier.

getMustScale()

Accessor to scale option.

Returns:
mustScaleboolean

Flag to tell if the bounding box of the weighted experiment and the domain have to be maped or not.

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getNbModes()

Accessor to number of modes to compute.

Returns:
nint

The maximum number of modes to compute. The actual number of modes also depends on the threshold criterion.

getResult()

Get the result structure.

Returns:
resKLKarhunenLoeveResult

The structure containing all the results of the Fredholm problem.

Notes

The structure contains all the results of the Fredholm problem.

Accessor to the object’s shadowed id.

Returns:
idint

Internal unique identifier.

getThreshold()

Accessor to the threshold used to select the most significant eigenmodes.

Returns:
sfloat, positive

The threshold .

Notes

OpenTURNS truncates the sequence at the index defined in (3).

getVisibility()

Accessor to the object’s visibility state.

Returns:
visiblebool

Visibility flag.

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:
hasVisibleNamebool

True if the name is not empty and not the default one.

run()

Computation of the eigenvalues and eigenfunctions values at the quadrature points.

Notes

Runs the algorithm and creates the result structure KarhunenLoeveResult.

setCovarianceModel(covariance)

Accessor to the covariance model.

Parameters:
covModelCovarianceModel

The covariance model.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

setNbModes(nbModes)

Accessor to the maximum number of modes to compute.

Parameters:
nint

The maximum number of modes to compute. The actual number of modes also depends on the threshold criterion.

Accessor to the object’s shadowed id.

Parameters:
idint

Internal unique identifier.

setThreshold(threshold)

Accessor to the limit ratio on eigenvalues.

Parameters:
sfloat,

The threshold defined in (3).

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:
visiblebool

Visibility flag.