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#! /usr/bin/env python
import openturns as ot
import math as m
ot.TESTPREAMBLE()
# Problem parameters
dimension = 3
a = 7.0
b = 0.1
# Reference analytical values
meanTh = a / 2
covTh = (b**2 * m.pi**8) / 18.0 + (b * m.pi**4) / 5.0 + (a**2) / 8.0 + 1.0 / 2.0
sob_1 = [
(b * m.pi**4 / 5.0 + b**2 * m.pi**8 / 50.0 + 1.0 / 2.0) / covTh,
(a**2 / 8.0) / covTh,
0.0,
]
sob_2 = [0.0, (b**2 * m.pi**8 / 18.0 - b**2 * m.pi**8 / 50.0) / covTh, 0.0]
sob_3 = [0.0]
sob_T1 = [
sob_1[0] + sob_2[0] + sob_2[1] + sob_3[0],
sob_1[1] + sob_2[0] + sob_2[2] + sob_3[0],
sob_1[2] + sob_2[1] + sob_2[2] + sob_3[0],
]
sob_T2 = [
sob_2[0] + sob_2[1] + sob_3[0],
sob_2[0] + sob_2[2] + sob_3[0],
sob_2[1] + sob_2[2] + sob_3[0],
]
sob_T3 = [sob_3[0]]
# Create the Ishigami function
inputVariables = ["xi1", "xi2", "xi3"]
formula = ot.Description(1)
formula[0] = (
"sin(xi1) + (" + str(a) + ") * (sin(xi2)) ^ 2 + (" + str(b) + ") * xi3^4 * sin(xi1)"
)
model = ot.SymbolicFunction(inputVariables, formula)
# Create the input distribution
distribution = ot.JointDistribution([ot.Uniform(-m.pi, m.pi)] * dimension)
# Create the orthogonal basis
enumerateFunction = ot.LinearEnumerateFunction(dimension)
productBasis = ot.OrthogonalProductPolynomialFactory(
[ot.LegendreFactory()] * dimension, enumerateFunction
)
# Create the projection strategy
samplingSize = 250
listProjectionStrategy = list()
# Monte Carlo sampling
inputSample = ot.LowDiscrepancyExperiment(
ot.SobolSequence(), distribution, samplingSize
).generate()
outputSample = model(inputSample)
# From here, the model is no more needed
listProjectionStrategy.append(ot.LeastSquaresStrategy())
listProjectionStrategy.append(
ot.LeastSquaresStrategy(
ot.LeastSquaresMetaModelSelectionFactory(ot.LARS(), ot.CorrectedLeaveOneOut())
)
)
listProjectionStrategy.append(ot.IntegrationStrategy())
# Create the adaptive strategy
# We can choose amongst several strategies
# First, the most efficient (but more complex!) strategy
degree = 6
listAdaptiveStrategy = list()
indexMax = enumerateFunction.getBasisSizeFromTotalDegree(degree)
basisDimension = enumerateFunction.getBasisSizeFromTotalDegree(degree // 2)
threshold = 1.0e-6
listAdaptiveStrategy.append(
ot.CleaningStrategy(productBasis, indexMax, basisDimension, threshold)
)
# Second, the most used (and most basic!) strategy
listAdaptiveStrategy.append(
ot.FixedStrategy(
productBasis, enumerateFunction.getBasisSizeFromTotalDegree(degree)
)
)
# Check LeastSquaresStrategy
projectionStrategy = ot.LeastSquaresStrategy()
assert projectionStrategy.isLeastSquares()
assert not projectionStrategy.involvesModelSelection()
# Check LeastSquaresMetaModelSelectionFactory
projectionStrategy = ot.LeastSquaresStrategy(
ot.LeastSquaresMetaModelSelectionFactory(ot.LARS(), ot.CorrectedLeaveOneOut())
)
assert projectionStrategy.isLeastSquares()
assert projectionStrategy.involvesModelSelection()
# Check IntegrationStrategy
projectionStrategy = ot.IntegrationStrategy()
assert not projectionStrategy.isLeastSquares()
assert not projectionStrategy.involvesModelSelection()
# Check CleaningStrategy
adaptiveStrategy = ot.CleaningStrategy(
productBasis, indexMax, basisDimension, threshold
)
assert adaptiveStrategy.involvesModelSelection()
# Check FixedStrategy
adaptiveStrategy = ot.FixedStrategy(
productBasis, enumerateFunction.getBasisSizeFromTotalDegree(degree)
)
assert not adaptiveStrategy.involvesModelSelection()
for adaptiveStrategyIndex in range(len(listAdaptiveStrategy)):
adaptiveStrategy = listAdaptiveStrategy[adaptiveStrategyIndex]
for projectionStrategyIndex in range(len(listProjectionStrategy)):
projectionStrategy = listProjectionStrategy[projectionStrategyIndex]
# Create the polynomial chaos algorithm
maximumResidual = 1.0e-10
algo = ot.FunctionalChaosAlgorithm(
inputSample,
outputSample,
distribution,
adaptiveStrategy,
projectionStrategy,
)
algo.setMaximumResidual(maximumResidual)
ot.RandomGenerator.SetSeed(0)
algo.run()
# Examine the results
result = algo.getResult()
print("###################################")
print(adaptiveStrategy)
print(algo.getProjectionStrategy())
residuals = result.getResiduals()
print("residuals=", residuals)
relativeErrors = result.getRelativeErrors()
print("relativeErrors=", relativeErrors)
print("isLeastSquares= ", result.isLeastSquares())
isLeastSquaresReference = (
projectionStrategy.getClassName() == "LeastSquaresStrategy"
)
assert result.isLeastSquares() == isLeastSquaresReference
print("involvesModelSelection= ", result.involvesModelSelection())
modelSelectionReference = (
projectionStrategy.involvesModelSelection()
or adaptiveStrategy.involvesModelSelection()
)
assert result.involvesModelSelection() == modelSelectionReference
# Post-process the results
vector = ot.FunctionalChaosRandomVector(result)
mean = vector.getMean()[0]
print("mean=%.8f" % mean, "absolute error=%.10f" % abs(mean - meanTh))
variance = vector.getCovariance()[0, 0]
print(
"variance=%.8f" % variance, "absolute error=%.10f" % abs(variance - covTh)
)
sensitivity = ot.FunctionalChaosSobolIndices(result)
for i in range(dimension):
value = sensitivity.getSobolIndex(i)
print(
"Sobol index",
i,
"= %.8f" % value,
"absolute error=%.10f" % abs(value - sob_1[i]),
)
indices = ot.Indices(2)
k = 0
for i in range(dimension):
indices[0] = i
for j in range(i + 1, dimension):
indices[1] = j
value = sensitivity.getSobolIndex(indices)
print(
"Sobol index",
indices,
"=%.8f" % value,
"absolute error=%.10f" % abs(value - sob_2[k]),
)
k = k + 1
indices = ot.Indices([0, 1, 2])
value = sensitivity.getSobolIndex(indices)
print(
"Sobol index",
indices,
"=%.8f" % value,
"absolute error=%.10f" % abs(value - sob_3[0]),
)
for i in range(dimension):
value = sensitivity.getSobolTotalIndex(i)
print(
"Sobol total index",
i,
"=%.8f" % value,
"absolute error=%.10f" % abs(value - sob_T1[i]),
)
indices = ot.Indices(2)
k = 0
for i in range(dimension):
indices[0] = i
for j in range(i + 1, dimension):
indices[1] = j
value = sensitivity.getSobolIndex(indices)
print(
"Sobol total index",
indices,
"=%.8f" % value,
"absolute error=%.10f" % abs(value - sob_2[k]),
)
k = k + 1
indices = ot.Indices([0, 1, 2])
value = sensitivity.getSobolTotalIndex(indices)
print(
"Sobol total index ",
indices,
"=%.8f" % value,
"absolute error=%.10f" % abs(value - sob_3[0]),
)
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