KarhunenLoeveP1Algorithm¶

class KarhunenLoeveP1Algorithm(*args)¶

Computation of Karhunen-Loeve decomposition using P1 approximation.

Parameters

meshMesh: The mesh $\cD_N$ that discretizes the domain $\cD$ .
covarianceCovarianceModel: The covariance function to decompose.
sfloat, $\geq0$: The threshold used to select the most significant eigenmodes, defined in KarhunenLoeveAlgorithm.

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 $G_{i\ell}= \int_{\cD} \theta_i(\vect{s})\theta_\ell(\vect{s})\, d\vect{s}$ , $\mat{\Lambda}=diag(\vect{\lambda})$ .

Though the eigenvalues computation is performed by LAPACK by default, several other solvers can be used. The solver to use must be specified by setting KarhunenLoeveP1Algorithm-EigenvaluesSolver key in ResourceMap. Some solvers set limits to the number of eigenvalues they can compute with regards to the size of the problem. The following table summarizes the supported solvers and their maximum number of eigenvalues for a problem of size $n$ :

Solver	ResourceMap key	Maximum EV number
Lapack	LAPACK	$n$
Spectra	SPECTRA	$n-2$

For large dimension problems, it can be useful to generate the covariance matrix of the problem as a HMatrix instead of a standard dense matrix. In this case, one can set KarhunenLoeveP1Algorithm-CovarianceMatrixStorage key in ResourceMap to HMAT (instead of DENSE).

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))
>>> s = 0.01
>>> model = ot.AbsoluteExponential([1.0]*2)
>>> algorithm = ot.KarhunenLoeveP1Algorithm(mesh, model, s)

Run it!

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

Methods

`getClassName`(self)	Accessor to the object’s name.
`getCovarianceModel`(self)	Accessor to the covariance model.
`getId`(self)	Accessor to the object’s id.
`getMesh`(self)	Accessor to the mesh.
`getName`(self)	Accessor to the object’s name.
`getNbModes`(self)	Accessor to number of modes to compute.
`getResult`(self)	Get the result structure.
`getShadowedId`(self)	Accessor to the object’s shadowed id.
`getThreshold`(self)	Accessor to the threshold used to select the most significant eigenmodes.
`getVisibility`(self)	Accessor to the object’s visibility state.
`hasName`(self)	Test if the object is named.
`hasVisibleName`(self)	Test if the object has a distinguishable name.
`run`(self)	Computation of the eigenvalues and eigenfunctions values at nodes.
`setCovarianceModel`(self, covariance)	Accessor to the covariance model.
`setName`(self, name)	Accessor to the object’s name.
`setNbModes`(self, nbModes)	Accessor to number of modes to compute.
`setShadowedId`(self, id)	Accessor to the object’s shadowed id.
`setThreshold`(self, threshold)	Accessor to the limit ratio on eigenvalues.
`setVisibility`(self, visible)	Accessor to the object’s visibility state.

__init__(self, \*args)¶: Initialize self. See help(type(self)) for accurate signature.

getClassName(self)¶

Accessor to the object’s name.

Returns

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

getCovarianceModel(self)¶

Accessor to the covariance model.

Returns

covModelCovarianceModel: The covariance model.

getId(self)¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getMesh(self)¶

Accessor to the mesh.

Returns

meshMesh: The mesh $\cD_N$ that discretizes the domain $\cD$ .

getName(self)¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getNbModes(self)¶

Accessor to number of modes to compute.

Returns

nint: The number of modes to compute.

getResult(self)¶

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.

getShadowedId(self)¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getThreshold(self)¶

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

Returns

sfloat, positive: The threshold $s$ .

Notes

OpenTURNS truncates the sequence $(\lambda_k, \vect{\varphi}_k)_{k \geq 1}$ at the index $K$ defined in (3).

getVisibility(self)¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

hasName(self)¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName(self)¶

Test if the object has a distinguishable name.

Returns

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

run(self)¶

Computation of the eigenvalues and eigenfunctions values at nodes.

Notes

Runs the algorithm and creates the result structure KarhunenLoeveResult.

setCovarianceModel(self, covariance)¶

Accessor to the covariance model.

Parameters

covModelCovarianceModel: The covariance model.

setName(self, name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setNbModes(self, nbModes)¶

Accessor to number of modes to compute.

Parameters

nint: The number of modes to compute.

setShadowedId(self, id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setThreshold(self, threshold)¶

Accessor to the limit ratio on eigenvalues.

Parameters

sfloat, $\geq 0$: The threshold $s$ defined in (3).

setVisibility(self, visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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