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#! /usr/bin/env python
import openturns as ot
from openturns.testing import assert_almost_equal
ot.TESTPREAMBLE()
ot.PlatformInfo.SetNumericalPrecision(5)
def matrixToSample(matrix):
"""Converts a matrix into a Sample, so that we can
assert_almost_equal a Matrix"""
size = matrix.getNbRows()
dimension = matrix.getNbColumns()
sample = ot.Sample(size, dimension)
for i in range(size):
for j in range(dimension):
sample[i, j] = matrix[i, j]
return sample
def local2globalCovariance(localErrorCovariance, size, outputDimension):
"""
Converts a local covariance into a global covariance.
size = inputObservations.getSize()
outputDimension = modelObservations.getDimension()
"""
covarianceDimension = size * outputDimension
observationDimension = localErrorCovariance.getDimension()
globalErrorCovariance = ot.CovarianceMatrix(covarianceDimension)
for i in range(size):
for j in range(observationDimension):
for k in range(observationDimension):
globalErrorCovariance[
i * observationDimension + j, i * observationDimension + k
] = localErrorCovariance[j, k]
return globalErrorCovariance
"""
Thanks to - Antoine Dumas, Phimeca
References
- J. Lemaitre and J. L. Chaboche (2002) "Mechanics of solid materials" Cambridge University Press.
"""
ot.RandomGenerator.SetSeed(0)
# Create the function.
g = ot.SymbolicFunction(["strain", "R", "C", "gam"], ["R + C*(1-exp(-gam*strain))"])
# Define the random vector.
Strain = ot.Uniform(0, 0.07)
unknownR = 750
unknownC = 2750
unknownGamma = 10
R = ot.Dirac(unknownR)
C = ot.Dirac(unknownC)
Gamma = ot.Dirac(unknownGamma)
Strain.setDescription(["Strain"])
R.setDescription(["R"])
C.setDescription(["C"])
Gamma.setDescription(["Gamma"])
# Create the joint input distribution function.
inputRandomVector = ot.JointDistribution([Strain, R, C, Gamma])
# Create the Monte-Carlo sample.
sampleSize = 10
inputSample = inputRandomVector.getSample(sampleSize)
outputStress = g(inputSample)
# Generate observation noise.
stressObservationNoiseSigma = 40.0 # (Pa)
noiseSigma = ot.Normal(0.0, stressObservationNoiseSigma)
sampleNoiseH = noiseSigma.getSample(sampleSize)
observedStress = outputStress + sampleNoiseH
observedStrain = inputSample[:, 0]
# Set the calibration parameters
# Define the value of the reference values of the $\theta$ parameter.
# In the bayesian framework, this is called the mean of the *prior* gaussian distribution.
# In the data assimilation framework, this is called the *background*.
R = 700 # Exact : 750
C = 2500 # Exact : 2750
Gamma = 8.0 # Exact : 10
thetaPrior = ot.Point([R, C, Gamma])
# The following statement create the calibrated function from the model.
# The calibrated parameters Ks, Zv, Zm are at indices 1, 2, 3 in the inputs arguments of the model.
calibratedIndices = [1, 2, 3]
mycf = ot.ParametricFunction(g, calibratedIndices, thetaPrior)
# Gaussian linear calibration
# The standard deviation of the observations.
sigmaStress = 10.0 # (Pa)
# Define the covariance matrix of the output Y of the model.
localErrorCovariance = ot.CovarianceMatrix(1)
localErrorCovariance[0, 0] = sigmaStress**2
# Define the covariance matrix of the parameters $\theta$ to calibrate.
sigmaR = 0.1 * R
sigmaC = 0.1 * C
sigmaGamma = 0.1 * Gamma
priorCovariance = ot.CovarianceMatrix(3)
priorCovariance[0, 0] = sigmaR**2
priorCovariance[1, 1] = sigmaC**2
priorCovariance[2, 2] = sigmaGamma**2
methods = ["SVD", "QR", "Cholesky"]
for method in methods:
print("method=", method)
# 1. Local calibration
# The `GaussianLinearCalibration` class performs the gaussian linear
# calibration by linearizing the model in the neighbourhood of the prior.
algo = ot.GaussianLinearCalibration(
mycf,
observedStrain,
observedStress,
thetaPrior,
priorCovariance,
localErrorCovariance,
method,
)
# The `run` method computes the solution of the problem.
algo.run()
calibrationResult = algo.getResult()
# Analysis of the results
# Maximum A Posteriori estimator
thetaMAP = calibrationResult.getParameterMAP()
exactTheta = ot.Point([762.661, 3056.59, 8.52781])
assert_almost_equal(thetaMAP, exactTheta)
# Covariance matrix of theta
thetaPosterior = calibrationResult.getParameterPosterior()
covarianceThetaStar = matrixToSample(thetaPosterior.getCovariance())
exactCovarianceTheta = ot.Sample(
[
[42.4899, 288.43, -1.70502],
[288.43, 15977.1, -67.6046],
[-1.70502, -67.6046, 0.294659],
]
)
assert_almost_equal(covarianceThetaStar, exactCovarianceTheta)
print("result=", calibrationResult)
# 2. Global covariance
outputDimension = observedStress.getDimension()
globalErrorCovariance = local2globalCovariance(
localErrorCovariance, sampleSize, outputDimension
)
# The `GaussianLinearCalibration` class performs the gaussian linear
# calibration by linearizing the model in the neighbourhood of the prior.
algo = ot.GaussianLinearCalibration(
mycf,
observedStrain,
observedStress,
thetaPrior,
priorCovariance,
globalErrorCovariance,
method,
)
# The `run` method computes the solution of the problem.
algo.run()
# Check other fields
calibrationResult = algo.getResult()
# Analysis of the results
# Maximum A Posteriori estimator
thetaMAP = calibrationResult.getParameterMAP()
exactTheta = ot.Point([762.661, 3056.59, 8.52781])
assert_almost_equal(thetaMAP, exactTheta)
# Covariance matrix of theta
thetaPosterior = calibrationResult.getParameterPosterior()
covarianceThetaStar = matrixToSample(thetaPosterior.getCovariance())
exactCovarianceTheta = ot.Sample(
[
[42.4899, 288.43, -1.70502],
[288.43, 15977.1, -67.6046],
[-1.70502, -67.6046, 0.294659],
]
)
assert_almost_equal(covarianceThetaStar, exactCovarianceTheta)
# Check other fields
print("result=", calibrationResult)
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