KarhunenLoeveP1Algorithm

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../../_images/KarhunenLoeveP1Algorithm.png
class KarhunenLoeveP1Algorithm(*args)

Computation of Karhunen-Loeve decomposition using P1 approximation.

Parameters:

mesh : Mesh

The mesh \cD_N that discretizes the domain \cD.

covariance : CovarianceModel

The covariance function to decompose.

threshold : float

The minimal relative amplitude of the eigenvalues to consider in the decomposition wrt the sum of the preceeding eigenvalues. The default value is 0.

Notes

The Karhunen-Loeve P_1 algorithm solves the Fredholm problem associated to the covariance function C: see KarhunenLoeveAlgorithm to get the notations.

The Karhunen-Loeve P_1 approximation uses the P_1 functional basis (\theta_p)_{1 \leq p \leq N} where \theta_p: \cD_N \mapsto \Rset are the basis functions of the P_1 finite element space associated to \cD_N, which vertices are (\vect{s}_i)_{1 \leq i \leq N}.

The covariance function \mat{C} is approximated by its P_1 approximation \hat{\mat{C}} on \cD_N:

\hat{\mat{C}}(\vect{s},\vect{t})=\sum_{\vect{s}_i,\vect{s}_j\in\cV_N}\mat{C}(\vect{s}_i,\vect{s}_j)\theta_i(\vect{s})\theta_j(\vect{t}), \quad  \forall \vect{s},\vect{t}\in\cD_N

The Galerkin approach and the collocation one are equivalent in the P_1 approach and both lead to the following formulation:

\mat{C}\, \mat{G}\, \mat{\Phi}  =   \mat{\Phi}\, \mat{\Lambda}

where \mat{G} = (G_{ij})_{1\leq i,j \leq N} with G_{i\ell}= \int_{\cD} \theta_i(\vect{s})\theta_\ell(\vect{s})\,  d\vect{s}, \mat{\Lambda}=diag(\vect{\lambda}).

Examples

Create a Karhunen-Loeve P1 algorithm:

>>> import openturns as ot
>>> mesh = ot.IntervalMesher([10]*2).build(ot.Interval([-1.0]*2, [1.0]*2))
>>> threshold = 0.01
>>> model = ot.AbsoluteExponential([1.0]*2)
>>> algorithm = ot.KarhunenLoeveP1Algorithm(mesh, model, threshold)

Methods

getClassName() Accessor to the object’s name.
getCovarianceModel() Accessor to the covariance model.
getId() Accessor to the object’s id.
getMesh() Accessor to the mesh.
getName() Accessor to the object’s name.
getResult() Get the result structure.
getShadowedId() Accessor to the object’s shadowed id.
getThreshold() Accessor to the limit ratio on eigenvalues.
getVisibility() Accessor to the object’s visibility state.
hasName() Test if the object is named.
hasVisibleName() Test if the object has a distinguishable name.
run() Computation of the eigenvalues and eigen functions values at nodes.
setCovarianceModel(covariance) Accessor to the covariance model.
setName(name) Accessor to the object’s name.
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)
getClassName()

Accessor to the object’s name.

Returns:

class_name : str

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

getCovarianceModel()

Accessor to the covariance model.

Returns:

covModel : CovarianceModel

The covariance model.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getMesh()

Accessor to the mesh.

Returns:

mesh : Mesh

The mesh \cD_N that discretizes the domain \cD.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getResult()

Get the result structure.

Returns:

resKL : KarhunenLoeveResult

The structure containing all the results of the Fredholm problem.

Notes

The structure contains all the results of the Fredholm problem.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getThreshold()

Accessor to the limit ratio on eigenvalues.

Returns:

s : float, positive

The minimal relative amplitude of the eigenvalues to consider in the decomposition wrt the maximum eigenvalue.

Notes

OpenTURNS truncates the sequence (\lambda_k,  \vect{\varphi}_k)_{k \geq 1} at the highest index K such that \lambda_K \geq s \lambda_1.

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

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

run()

Computation of the eigenvalues and eigen functions values at nodes.

Examples

>>> import openturns as ot
>>> mesh = ot.IntervalMesher([10]*2).build(ot.Interval([-1.0]*2, [1.0]*2))
>>> threshold = 0.01
>>> model = ot.AbsoluteExponential([1.0]*2)
>>> algorithm = ot.KarhunenLoeveP1Algorithm(mesh, model, threshold)
>>> algorithm.run()
>>> result = algorithm.getResult()
setCovarianceModel(covariance)

Accessor to the covariance model.

Parameters:

covModel : CovarianceModel

The covariance model.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setThreshold(threshold)

Accessor to the limit ratio on eigenvalues.

Parameters:

s : float, positive

The minimal relative amplitude of the eigenvalues to consider in the decomposition wrt the maximum eigenvalue.

Notes

OpenTURNS truncates the sequence (\lambda_k,  \vect{\varphi}_k)_{k \geq 1} at the highest index K such that \lambda_K \geq s \lambda_1.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.