KarhunenLoeveQuadratureAlgorithm¶

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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

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, $\geq0$: 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 $C$ : see KarhunenLoeveAlgorithm to get the notations.

The Karhunen-Loeve quadrature approximation consists in replacing the integral by a quadrature approximation: if $(\omega_\ell, \vect{\xi}_\ell)_{1 \leq \ell \leq L}$ is the weighted experiment (see WeightedExperiment) associated to the measure $\mu$ , then for all functions measurable wrt $\mu$ , we have:

$\int_{\Rset^n} f(\vect{x}) \mu(\vect{x})\, d\vect{x} \simeq \sum_{\ell=1}^{\ell=L} \omega_\ell f(\vect{\xi}_\ell)$

If we note $\eta_{\ell}=\omega_{\ell}\dfrac{1_{\cD}(\vect{\xi}_{\ell})}{\mu(\vect{\xi}_{\ell})}$ , we build a more general quadrature approximation $(\eta_\ell, \xi_\ell)_{1 \leq l \leq L}$ such that:

$\int_{\cD} f(\vect{t}) \, d\vect{t} \simeq \sum_{\ell=1}^L \eta_\ell f(\vect{\xi}_\ell)$

where only the points $\vect{\xi}_\ell \in \cD$ are considered.

We introduce the matrices $\mat{\Theta}=(\mat{\Theta}_{ij})_{1 \leq i \leq L, 1 \leq j \leq P} \in \cM_{Ld, Pd}(\Rset)$ such that $\mat{\Theta}_{ij} = \theta_{j}(\vect{\xi}_i)\mat{I}_d$ , $\mat{C}_{\ell, \ell'} = \mat{C}(\vect{\xi}_{\ell},\vect{\xi}_{\ell'})$ and $\mat{W}= \mathrm{diag} (\mat{W}_{ii})_{1 \leq i \leq L} \in \cM_{Ld, Ld}(\Rset)$ such that $\mat{W}_{ii} = \eta_i \mat{I}_d$ .

The normalisation constraint $\|\vect{\varphi}_k\|^2_{L^2(\cD, \Rset^d)}=1$ ang the orthogonality of the $\vect{\varphi}_k$ in $L^2(\cD, \Rset^d)$ leads to:

(1)¶ $\Tr{\vect{\Phi}} \,\Tr{\mat{\Theta}} \,\mat{W}\, \mat{\Theta} \,\vect{\Phi}=\mat{I}$

The Galerkin approach leads to the following generalized eigenvalue problem:

(2)¶ $\Tr{\mat{\Theta}} \,\mat{W} \,\mat{C} \,\mat{W} \,\mat{\Theta}\,\vect{\Phi}_k = \lambda_k \Tr{\mat{\Theta}}\, \mat{W}\,\mat{\Theta}\,\vect{\Phi}_k$

where $\Tr{\mat{\Theta}}\, \mat{W} \,\mat{C} \, \mat{W}\, \mat{\Theta}$ and $\Tr{\mat{\Theta}}\, \mat{W}\, \mat{\Theta}\in \cM_{Pd,Pd}(\Rset)$ .

The collocation approach leads to the following generalized eigenvalue problem:

(3)¶ $\mat{C}\, \mat{W} \,\mat{\Theta}\, \vect{\Phi}_k = \lambda_k \mat{\Theta}\,\vect{\Phi}_k$

Equations (2) and (3) are equivalent when $\mat{\Theta}$ 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 $\cD$ and create a covariance model:

>>> import openturns as ot
>>> bounds = ot.Interval([-1.0]*2, [1.0]*2)
>>> domain = 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.
`getClassName`()	Accessor to the object's name.
`getCovarianceModel`()	Accessor to the covariance model.
`getDomain`()	Accessor to the domain.
`getExperiment`()	Accessor to the points and weights of the quadrature approximation.
`getId`()	Accessor to the object's id.
`getMustScale`()	Accessor to scale option.
`getName`()	Accessor to the object's name.
`getNbModes`()	Accessor to number of modes to compute.
`getResult`()	Get the result structure.
`getShadowedId`()	Accessor to the object's shadowed id.
`getThreshold`()	Accessor to the threshold used to select the most significant eigenmodes.
`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 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

basisBasis: The basis in wich 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 $\cD_N$ that discretizes the domin $\cD$ .

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.

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 $s$ .

Notes

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

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

setVisibility(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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