KarhunenLoeveQuadratureAlgorithm¶
(Source code
, png
)
- class KarhunenLoeveQuadratureAlgorithm(*args)¶
Computation of Karhunen-Loeve decomposition using Quadrature approximation.
- Available constructors:
KarhunenLoeveQuadratureAlgorithm(domain, bounds, covariance, experiment, basis, basisSize, mustScale, s)
KarhunenLoeveQuadratureAlgorithm(domain, bounds, covariance, marginalDegree, s)
- Parameters:
- domain
Domain
The domain on which the covariance model and the Karhunen-Loeve eigenfunctions (modes) are discretized.
- bounds
Interval
Numerical bounds of the domain.
- covariance
CovarianceModel
The covariance function to decompose.
- experiment
WeightedExperiment
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
.
- domain
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
andLegendreFactory
), experiment is a tensorized Gauss-Legendre quadrature (seeGaussProductExperiment
), 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)
Create the Karhunen-Loeve Quadrature algorithm:
>>> algorithm = ot.KarhunenLoeveQuadratureAlgorithm(domain, bounds, model, experiment, functions, True, s)
Run it!
>>> algorithm.run() >>> result = algorithm.getResult()
Methods
getBasis
()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.
getId
()Accessor to the object's id.
Accessor to scale option.
getName
()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.
hasName
()Test if the object is named.
Test if the object has a distinguishable name.
run
()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.
setShadowedId
(id)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:
- basis
Basis
The basis in which the eigenfunctions are projected.
- basis
- getClassName()¶
Accessor to the object’s name.
- Returns:
- class_namestr
The object class name (object.__class__.__name__).
- getCovarianceModel()¶
Accessor to the covariance model.
- Returns:
- covModel
CovarianceModel
The covariance model.
- covModel
- getExperiment()¶
Accessor to the points and weights of the quadrature approximation.
- Returns:
- experiment
WeightedExperiment
The points and weights used in the quadrature approximation.
- experiment
- 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:
- resKL
KarhunenLoeveResult
The structure containing all the results of the Fredholm problem.
- resKL
Notes
The structure contains all the results of the Fredholm problem.
- getShadowedId()¶
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:
- covModel
CovarianceModel
The covariance model.
- covModel
- 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.
- setShadowedId(id)¶
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.