KarhunenLoeveP1Algorithm

(Source code, png)

../../_images/KarhunenLoeveP1Algorithm.png
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.

Methods

getClassName()

Accessor to the object's name.

getCovarianceModel()

Accessor to the covariance model.

getMesh()

Accessor to the mesh.

getName()

Accessor to the object's name.

getNbModes()

Accessor to number of modes to compute.

getResult()

Get the result structure.

getThreshold()

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

hasName()

Test if the object is named.

run()

Computation of the eigenvalues and eigenfunctions values at nodes.

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.

setThreshold(threshold)

Accessor to the limit ratio on eigenvalues.

Notes

The Karhunen-Loeve P_1 algorithm solves the Fredholm problem associated to the covariance function C: see KarhunenLoeveAlgorithm for 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, whose 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 the 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 the 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()
__init__(*args)
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.

getMesh()

Accessor to the mesh.

Returns:
meshMesh

The mesh \cD_N that discretizes the domain \cD.

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.

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).

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

run()

Computation of the eigenvalues and eigenfunctions values at nodes.

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.

setThreshold(threshold)

Accessor to the limit ratio on eigenvalues.

Parameters:
sfloat, \geq 0

The threshold s defined in (3).

Examples using the class

Metamodel of a field function

Metamodel of a field function