def plotKrigingPredictions(krigingMetamodel):
'''
Plot the predictions of a Kriging metamodel.
'''
# Create the mesh of the box [0., 1000.] * [0., 700.]
myInterval = ot.Interval([0., 0.], [1000., 700.])
# Define the number of intervals in each direction of the box
nx = 20
ny = 20
myIndices = [nx - 1, ny - 1]
myMesher = ot.IntervalMesher(myIndices)
myMeshBox = myMesher.build(myInterval)
# Predict
vertices = myMeshBox.getVertices()
predictions = krigingMetamodel(vertices)
# Format for plot
X = np.array(vertices[:, 0]).reshape((ny, nx))
Y = np.array(vertices[:, 1]).reshape((ny, nx))
predictions_array = np.array(predictions).reshape((ny, nx))
# Plot
plt.figure()
plt.pcolormesh(X, Y, predictions_array, shading='auto')
plt.colorbar()
plt.show()
return
def test_NonZeroMean(self):
# Create the KL result
numberOfVertices = 10
interval = ot.Interval(-1.0, 1.0)
mesh = ot.IntervalMesher([numberOfVertices - 1]).build(interval)
covariance = ot.SquaredExponential()
zeroProcess = ot.GaussianProcess(covariance, mesh)
# Define a trend function
f = ot.SymbolicFunction(["t"], ["30 * t"])
fTrend = ot.TrendTransform(f, mesh)
# Add it to the process
process = ot.CompositeProcess(fTrend, zeroProcess)
# Sample
sampleSize = 100
processSample = process.getSample(sampleSize)
threshold = 0.0
algo = ot.KarhunenLoeveSVDAlgorithm(processSample, threshold)
algo.run()
klresult = algo.getResult()
# Create the KL reduction
meanField = processSample.computeMean()
klreduce = ot.KarhunenLoeveReduction(klresult)
# Generate a trajectory and reduce it
field = process.getRealization()
values = field.getValues()
reducedValues = klreduce(values)
ott.assert_almost_equal(values, reducedValues)
def _buildMesh(self, grid_shape):
"""Builds a openturns mesh in the unit cube, based on a
comprehesive list of grid coordinates as returned by the
_getGridShape method.
Arguments
---------
grid_shape : comprehensive list
all the grid coordinates, in the unit cube.
Returns
-------
mesh : ot.Mesh
openturns Mesh object
"""
dimension = len(grid_shape)
n_intervals = [int(grid_shape[i][2]) for i in range(dimension)]
low_bounds = [grid_shape[i][0] for i in range(dimension)]
lengths = [grid_shape[i][1] for i in range(dimension)]
high_bounds = [low_bounds[i] + lengths[i] for i in range(dimension)]
mesherObj = ot.IntervalMesher(n_intervals)
grid_interval = ot.Interval(low_bounds, high_bounds)
mesh = mesherObj.build(grid_interval)
mesh.setName(str(dimension) + 'D_Grid')
return mesh
def test_KarhunenLoeveValidationMultidimensional(self):
# Create the KL result
numberOfVertices = 20
interval = ot.Interval(-1.0, 1.0)
mesh = ot.IntervalMesher([numberOfVertices - 1]).build(interval)
outputDimension = 2
univariateCovariance = ot.SquaredExponential()
covarianceCollection = [univariateCovariance] * outputDimension
multivariateCovariance = ot.TensorizedCovarianceModel(
covarianceCollection)
process = ot.GaussianProcess(multivariateCovariance, mesh)
sampleSize = 100
sampleSize = 10
processSample = process.getSample(sampleSize)
threshold = 1.0e-7
algo = ot.KarhunenLoeveSVDAlgorithm(processSample, threshold)
algo.run()
klresult = algo.getResult()
# Create the validation
validation = ot.KarhunenLoeveValidation(processSample, klresult)
# Check residuals
residualProcessSample = validation.computeResidual()
assert (type(residualProcessSample) is ot.ProcessSample)
# Check standard deviation
residualSigmaField = validation.computeResidualStandardDeviation()
zeroSample = ot.Sample(numberOfVertices, outputDimension)
ott.assert_almost_equal(residualSigmaField, zeroSample)
# Check graph
graph = validation.drawValidation()
if False:
from openturns.viewer import View
View(graph).save('validation2.png')
def __init__(self):
self.dim = 4 # number of inputs
self.outputDimension = 1 # dimension of the output
self.tmin = 0.0 # Minimum time
self.tmax = 12.0 # Maximum time
self.gridsize = 100 # Number of time steps
self.mesh = ot.IntervalMesher([self.gridsize - 1]).build(
ot.Interval(self.tmin, self.tmax))
self.vertices = self.mesh.getVertices()
# Marginals
self.distZ0 = ot.Uniform(100.0, 150.0)
self.distV0 = ot.Normal(55.0, 10.0)
self.distM = ot.Normal(80.0, 8.0)
self.distC = ot.Uniform(0.0, 30.0)
# Joint distribution
self.distribution = ot.ComposedDistribution(
[self.distZ0, self.distV0, self.distM, self.distC])
# Exact solution
self.alti = ot.PythonPointToFieldFunction(self.dim, self.mesh,
self.outputDimension,
AltiFunc)
def test_KarhunenLoeveValidation(self):
# Create the KL result
numberOfVertices = 20
interval = ot.Interval(-1.0, 1.0)
mesh = ot.IntervalMesher([numberOfVertices - 1]).build(interval)
covariance = ot.SquaredExponential()
process = ot.GaussianProcess(covariance, mesh)
sampleSize = 100
processSample = process.getSample(sampleSize)
threshold = 1.0e-7
algo = ot.KarhunenLoeveSVDAlgorithm(processSample, threshold)
algo.run()
klresult = algo.getResult()
# Create validation
validation = ot.KarhunenLoeveValidation(processSample, klresult)
# Check residuals
residualProcessSample = validation.computeResidual()
assert (type(residualProcessSample) is ot.ProcessSample)
# Check standard deviation
residualSigmaField = validation.computeResidualStandardDeviation()
exact = ot.Sample(numberOfVertices, 1)
#ott.assert_almost_equal(residualSigmaField, exact)
# Check mean
residualMean = validation.computeResidualMean()
exact = ot.Sample(numberOfVertices, 1)
#ott.assert_almost_equal(residualMean, exact)
# Check graph
graph0 = validation.drawValidation()
graph1 = residualProcessSample.drawMarginal(0)
graph2 = residualMean.drawMarginal(0)
graph3 = residualSigmaField.drawMarginal(0)
graph4 = validation.drawObservationWeight(0)
graph5 = validation.drawObservationQuality()
if 0:
from openturns.viewer import View
View(graph0).save('validation1.png')
View(graph1).save('validation1-residual.png')
View(graph2).save('validation1-residual-mean.png')
View(graph3).save('validation1-residual-stddev.png')
View(graph4).save('validation1-indiv-weight.png')
View(graph5).save('validation1-indiv-quality.png')
def test_ZeroMean(self):
# Create the KL result
numberOfVertices = 10
interval = ot.Interval(-1.0, 1.0)
mesh = ot.IntervalMesher([numberOfVertices - 1]).build(interval)
covariance = ot.SquaredExponential()
process = ot.GaussianProcess(covariance, mesh)
sampleSize = 10
processSample = process.getSample(sampleSize)
threshold = 0.0
algo = ot.KarhunenLoeveSVDAlgorithm(processSample, threshold)
algo.run()
klresult = algo.getResult()
# Create the KL reduction
meanField = processSample.computeMean()
klreduce = ot.KarhunenLoeveReduction(klresult)
# Generate a trajectory and reduce it
field = process.getRealization()
values = field.getValues()
reducedValues = klreduce(values)
ott.assert_almost_equal(values, reducedValues)
def test_trend(self):
N = 100
M = 1000
P = 10
mean = ot.SymbolicFunction("x", "sign(x)")
cov = ot.SquaredExponential([1.0], [0.1])
mesh = ot.IntervalMesher([N]).build(ot.Interval(-2.0, 2.0))
process = ot.GaussianProcess(ot.TrendTransform(mean, mesh), cov, mesh)
sample = process.getSample(M)
algo = ot.KarhunenLoeveSVDAlgorithm(sample, 1e-6)
algo.run()
result = algo.getResult()
trend = ot.TrendTransform(
ot.P1LagrangeEvaluation(sample.computeMean()), mesh)
sample2 = process.getSample(P)
sample2.setName('reduction of sign(x) w/o trend')
reduced1 = ot.KarhunenLoeveReduction(result)(sample2)
reduced2 = ot.KarhunenLoeveReduction(result, trend)(sample2)
g = sample2.drawMarginal(0)
g.setColors(["red"])
g1 = reduced1.drawMarginal(0)
g1.setColors(["blue"])
drs = g1.getDrawables()
for i, d in enumerate(drs):
d.setLineStyle("dashed")
drs[i] = d
g1.setDrawables(drs)
g.add(g1)
g2 = reduced2.drawMarginal(0)
g2.setColors(["green"])
drs = g2.getDrawables()
for i, d in enumerate(drs):
d.setLineStyle("dotted")
drs[i] = d
g2.setDrawables(drs)
g.add(g2)
if 0:
from openturns.viewer import View
View(g).save('reduction.png')
def readProcessSample(fname):
"""
Return a ProcessSample from a text file.
Assume the mesh is regular [0,1].
"""
# Dataset
data = np.loadtxt(fname)
# Create the mesh
n_nodes = data.shape[1]
mesher = ot.IntervalMesher([n_nodes - 1])
Interval = ot.Interval([0.0], [1.0])
mesh = mesher.build(Interval)
# Create the ProcessSample from the data
n_fields = data.shape[0]
dim_fields = 1
processSample = ot.ProcessSample(mesh, n_fields, dim_fields)
for i in range(n_fields):
trajectory = ot.Sample([[x] for x in data[i, :]])
processSample[i] = ot.Field(mesh, trajectory)
return processSample
def dummyFunction2Wrap(field_10x10, field_100x1, scalar_0):
## Function doing some operation on the 2 field and a scalar and returning a field
outDim = 1
NElem = [10]
mesher = ot.IntervalMesher(NElem)
lowerBound = [0]
upperBound = [10]
interval = ot.Interval(lowerBound, upperBound)
mesh = mesher.build(interval)
outField = ot.Field(mesh, [[0]] * mesh.getVerticesNumber())
for i in range(10):
for j in range(10):
if field_10x10[i][0] * field_100x1[i + j][0] > scalar_0[0][0]:
outField.setValueAtIndex(i, [
field_10x10[i][0] * field_100x1[(i + 1) * (j + 1) - 1][0] -
scalar_0[0][0]
])
else:
outField.setValueAtIndex(
i, [(field_10x10[j][0] - scalar_0[0][0]) /
field_100x1[(i + 1) * (j + 1) - 1][0]])
return outField
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
# Create a bivariate normal process
myMesh = ot.IntervalMesher([39, 39]).build(ot.Interval([0.0] * 2, [1.0] * 2))
myCov = ot.GeneralizedExponential(2, 0.1, 1.3)
myProcess = ot.TemporalNormalProcess(myCov, myMesh)
myField = myProcess.getRealization()
graph = myField.drawMarginal(0, False)
fig = plt.figure(figsize=(8, 4))
plt.suptitle("A field")
axis = fig.add_subplot(111)
axis.set_xlim(auto=True)
View(graph, figure=fig, axes=[axis], add_legend=True)
else:
y = 2.0*m.sin(7.0*xx)
return y
XX_input = ot.Sample([[0.1, 0], [0.32, 0], [0.6, 0], [0.9, 0], [
0.07, 1], [0.1, 1], [0.4, 1], [0.5, 1], [0.85, 1]])
y_output = ot.Sample(len(XX_input), 1)
for i in range(len(XX_input)):
y_output[i, 0] = fun_mixte(XX_input[i])
def C(s, t):
return m.exp(-4.0 * abs(s - t) / (1 + (s * s + t * t)))
N = 32
a = 4.0
myMesh = ot.IntervalMesher([N]).build(ot.Interval(-a, a))
myCovariance = ot.CovarianceMatrix(myMesh.getVerticesNumber())
for k in range(myMesh.getVerticesNumber()):
t = myMesh.getVertices()[k]
for l in range(k + 1):
s = myMesh.getVertices()[l]
myCovariance[k, l] = C(s[0], t[0])
covModel_discrete = ot.UserDefinedCovarianceModel(myMesh, myCovariance)
f_ = ot.SymbolicFunction(["tau", "theta", "sigma"], [
"(tau!=0) * exp(-1/theta) * sigma * sigma + (tau==0) * exp(0) * sigma * sigma"])
rho = ot.ParametricFunction(f_, [1, 2], [0.2, 0.3])
covModel_discrete = ot.StationaryFunctionalCovarianceModel([1.0], [
1.0], rho)
covModel_continuous = ot.SquaredExponential([1.0], [1.0])
""" Aggregate processes =================== """ # %% # In this example we are going to concatenate several processes that share the same mesh. # %% import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) # %% # Create processes to aggregate myMesher = ot.IntervalMesher([100, 10]) lowerbound = [0.0, 0.0] upperBound = [2.0, 4.0] myInterval = ot.Interval(lowerbound, upperBound) myMesh = myMesher.build(myInterval) myProcess1 = ot.WhiteNoise(ot.Normal(), myMesh) myProcess2 = ot.WhiteNoise(ot.Triangular(), myMesh) # %% # Draw values of a realization of the 2nd process marginal = ot.HistogramFactory().build(myProcess1.getRealization().getValues()) graph = marginal.drawPDF() view = viewer.View(graph) # %% # Create an aggregated process
#! /usr/bin/env python
import openturns as ot
ot.TESTPREAMBLE()
# A 1D->1D field
mesh = ot.IntervalMesher([10]).build(ot.Interval(-2.0, 2.0))
function = ot.SymbolicFunction("x", "x")
field = ot.Field(mesh, function(mesh.getVertices()))
graph = field.draw()
graph = field.drawMarginal(0, False)
graph = field.drawMarginal(0, True)
# A 2D->1D field
mesh = ot.IntervalMesher([10] * 2).build(ot.Interval([-2.0] * 2, [2.0] * 2))
function = ot.SymbolicFunction(["x0", "x1"], ["x0 - x1"])
field = ot.Field(mesh, function(mesh.getVertices()))
graph = field.draw()
graph = field.drawMarginal(0, False)
graph = field.drawMarginal(0, True)
# A 2D->2D field
function = ot.SymbolicFunction(["x0", "x1"], ["x0", "x1"])
field = ot.Field(mesh, function(mesh.getVertices()))
graph = field.draw()
def test_model(myModel, test_partial_grad=True, x1=None, x2=None):
inputDimension = myModel.getInputDimension()
dimension = myModel.getOutputDimension()
if x1 is None and x2 is None:
x1 = ot.Point(inputDimension)
x2 = ot.Point(inputDimension)
for j in range(inputDimension):
x1[j] = -1.0 - j
x2[j] = 3.0 + 2.0 * j
else:
x1 = ot.Point(x1)
x2 = ot.Point(x2)
if myModel.isStationary():
ott.assert_almost_equal(myModel(x1 - x2), myModel(x1, x2), 1e-14,
1e-14)
ott.assert_almost_equal(myModel(x2 - x1), myModel(x1, x2), 1e-14,
1e-14)
eps = 1e-3
mesh = ot.IntervalMesher([7] * inputDimension).build(
ot.Interval([-10] * inputDimension, [10] * inputDimension))
C = myModel.discretize(mesh)
if dimension == 1:
# Check that discretize & computeAsScalar provide the
# same values
vertices = mesh.getVertices()
for j in range(len(vertices)):
for i in range(j, len(vertices)):
ott.assert_almost_equal(
C[i, j], myModel.computeAsScalar(vertices[i], vertices[j]),
1e-14, 1e-14)
else:
# Check that discretize & operator() provide the same values
vertices = mesh.getVertices()
localMatrix = ot.SquareMatrix(dimension)
for j in range(len(vertices)):
for i in range(j, len(vertices)):
for localJ in range(dimension):
for localI in range(dimension):
localMatrix[localI, localJ] = C[i * dimension + localI,
j * dimension + localJ]
ott.assert_almost_equal(localMatrix,
myModel(vertices[i], vertices[j]),
1e-14, 1e-14)
# Now we suppose that discretize is ok
# we look at crossCovariance of (vertices, vertices) which should return the same values
C.getImplementation().symmetrize()
crossCov = myModel.computeCrossCovariance(vertices, vertices)
ott.assert_almost_equal(
crossCov, C, 1e-14, 1e-14,
"in " + myModel.getClassName() + "::computeCrossCovariance")
# Now crossCovariance(sample, sample) is ok
# Let us validate crossCovariance(Sample, point) with 1st column(s) of previous calculations
crossCovSamplePoint = myModel.computeCrossCovariance(vertices, vertices[0])
crossCovCol = crossCov.reshape(crossCov.getNbRows(), dimension)
ott.assert_almost_equal(
crossCovSamplePoint, crossCovCol, 1e-14, 1e-14,
"in " + myModel.getClassName() + "::computeCrossCovarianceSamplePoint")
if test_partial_grad:
grad = myModel.partialGradient(x1, x2)
if (dimension == 1):
gradfd = ot.Matrix(inputDimension, 1)
for j in range(inputDimension):
x1_g = ot.Point(x1)
x1_d = ot.Point(x1)
x1_g[j] = x1_d[j] + eps
x1_d[j] = x1_d[j] - eps
gradfd[j, 0] = (myModel.computeAsScalar(x1_g, x2) -
myModel.computeAsScalar(x1_d, x2)) / (2 * eps)
else:
gradfd = ot.Matrix(inputDimension, dimension * dimension)
covarianceX1X2 = myModel(x1, x2)
centralValue = ot.Point(covarianceX1X2.getImplementation())
# Loop over the shifted points
for i in range(inputDimension):
currentPoint = ot.Point(x1)
currentPoint[i] += eps
localCovariance = myModel(currentPoint, x2)
currentValue = ot.Point(localCovariance.getImplementation())
for j in range(currentValue.getSize()):
gradfd[i, j] = (currentValue[j] - centralValue[j]) / eps
ott.assert_almost_equal(grad, gradfd, 1e-5, 1e-5,
"in " + myModel.getClassName() + " grad")
#! /usr/bin/env python
from __future__ import print_function
import openturns as ot
ot.TESTPREAMBLE()
for diamond in [False, True]:
mesher1D = ot.IntervalMesher([5])
print("mesher1D=", mesher1D)
mesh1D = mesher1D.build(ot.Interval(-1.0, 2.0), diamond)
print("mesh1D=", mesh1D)
mesher2D = ot.IntervalMesher([5, 5])
print("mesher2D=", mesher2D)
mesh2D = mesher2D.build(ot.Interval([-1.0, -1.0], [2.0, 2.0]), diamond)
print("mesh2D=", mesh2D)
mesher3D = ot.IntervalMesher([5]*3)
print("mesher3D=", mesher3D)
try:
mesh3D = mesher3D.build(ot.Interval(3), diamond)
print("mesh3D=", mesh3D)
except RuntimeError:
print('notyetimpl')
# # In this example we are going to assess a Karhunen-Loeve decomposition # # %% from __future__ import print_function import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) # %% # Create a Gaussian process numberOfVertices = 20 interval = ot.Interval(-1.0, 1.0) mesh = ot.IntervalMesher([numberOfVertices - 1]).build(interval) covariance = ot.SquaredExponential() process = ot.GaussianProcess(covariance, mesh) # %% # decompose it using KL-SVD sampleSize = 100 processSample = process.getSample(sampleSize) threshold = 1.0e-7 algo = ot.KarhunenLoeveSVDAlgorithm(processSample, threshold) algo.run() klresult = algo.getResult() # %% # Instanciate the validation service validation = ot.KarhunenLoeveValidation(processSample, klresult)
#! /usr/bin/env python
from __future__ import print_function
import openturns as ot
ot.TESTPREAMBLE()
# Create a KarhunenLoeveResult
mesh = ot.IntervalMesher([9]).build(ot.Interval(-1.0, 1.0))
cov1D = ot.AbsoluteExponential([1.0])
algo = ot.KarhunenLoeveP1Algorithm(mesh, cov1D, 0.0)
algo.run()
result = algo.getResult()
projection = ot.KarhunenLoeveProjection(result)
# Construction based on a FieldFunction followed by a FieldToPointFunction
fieldFunction = ot.ValueFunction(ot.SymbolicFunction("x", "x"), mesh)
# Create an instance
myFunc = ot.FieldToPointConnection(projection, fieldFunction)
print("myFunc=", myFunc)
# Get the input and output description
print("myFunc input description=", myFunc.getInputDescription())
print("myFunc output description=", myFunc.getOutputDescription())
# Get the input and output dimension
print("myFunc input dimension=", myFunc.getInputDimension())
print("myFunc output dimension=", myFunc.getOutputDimension())
# Connection on a field
field = result.getModesAsProcessSample().computeMean()
print("field=", field)
print("myFunc(field)=", myFunc(field.getValues()))
print("called ", myFunc.getCallsNumber(), " times")
def get_fem_vertices(min_vertices, max_vertices, n_elements):
interval = ot.Interval([min_vertices],[max_vertices])
mesher = ot.IntervalMesher([n_elements])
fem_vertices = mesher.build(interval)
return fem_vertices
=================== """ # %% # In this example we are going to concatenate several processes that share the same mesh. # %% from __future__ import print_function import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) # %% # Create processes to aggregate myMesher = ot.IntervalMesher(ot.Indices([100, 10])) lowerbound = [0.0, 0.0] upperBound = [2.0, 4.0] myInterval = ot.Interval(lowerbound, upperBound) myMesh = myMesher.build(myInterval) myProcess1 = ot.WhiteNoise(ot.Normal(), myMesh) myProcess2 = ot.WhiteNoise(ot.Triangular(), myMesh) # %% # Draw values of a realization of the 2nd process marginal = ot.HistogramFactory().build(myProcess1.getRealization().getValues()) graph = marginal.drawPDF() view = viewer.View(graph) # %% # Create an aggregated process
print("Nearest index(", points, ")=", tree.query(points))
print("P1 gram=\n", mesh3D.computeP1Gram())
rotation = ot.SquareMatrix(3)
rotation[0, 0] = m.cos(m.pi / 3.0)
rotation[0, 1] = m.sin(m.pi / 3.0)
rotation[1, 0] = -m.sin(m.pi / 3.0)
rotation[1, 1] = m.cos(m.pi / 3.0)
rotation[2, 2] = 1.0
# isregular bug
time_grid = ot.RegularGrid(0.0, 0.2, 40963)
mesh = ot.Mesh(time_grid)
print(mesh.isRegular())
# numerical limit testcase
m1 = ot.IntervalMesher([1] * 2).build(ot.Interval([0.0] * 2, [1.0] * 2))
simplex = 0
point = [0.8, 0.2]
found, coordinates = m1.checkPointInSimplexWithCoordinates(point, simplex)
assert found, "not inside"
# Fix https://github.com/openturns/openturns/issues/1547
# We force the checking
try:
vertices = [[2.1], [2.8], [3.5], [4.2], [4.9], [5.6], [6.3], [7.0]]
simplices = [[3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10]]
mesh = ot.Mesh(vertices, simplices, True)
weights = mesh.computeWeights()
except Exception:
print('ok')
# %% from __future__ import print_function import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt import numpy as np ot.Log.Show(ot.Log.NONE) # %% # We first define the time grid associated with the model. # %% tmin = 0.0 # Minimum time tmax = 12. # Maximum time gridsize = 100 # Number of time steps mesh = ot.IntervalMesher([gridsize-1]).build(ot.Interval(tmin, tmax)) # %% vertices = mesh.getVertices() # %% # Creation of the input distribution. # %% distZ0 = ot.Uniform(100.0, 150.0) distV0 = ot.Normal(55.0, 10.0) distM = ot.Normal(80.0, 8.0) distC = ot.Uniform(0.0, 30.0) distribution = ot.ComposedDistribution([distZ0, distV0, distM, distC]) # %%
def __call__(self, i, j):
pt1 = self.vertices[i]
pt2 = self.vertices[j]
difference = pt1 - pt2
val = m.exp(-difference.norm() / self.scaling)
return val
ot.ResourceMap.SetAsBool('HMatrix-ForceSequential', True)
ot.ResourceMap.SetAsUnsignedInteger('HMatrix-MaxLeafSize', 10)
ot.PlatformInfo.SetNumericalPrecision(3)
n = 2
indices = [n, n]
intervalMesher = ot.IntervalMesher(indices)
interval = ot.Interval([0.0] * 2, [1.0] * 2)
mesh2D = intervalMesher.build(interval)
vertices = mesh2D.getVertices()
factory = ot.HMatrixFactory()
parameters = ot.HMatrixParameters()
parameters.setAssemblyEpsilon(1.e-6)
parameters.setRecompressionEpsilon(1.e-6)
# HMatrix must be symmetric in order to perform Cholesky decomposition
hmat = factory.build(vertices, 1, True, parameters)
simpleAssembly = TestHMatrixRealAssemblyFunction(vertices, 0.1)
hmat.assembleReal(simpleAssembly, 'L')
hmatRef = ot.HMatrix(hmat)
# We note :math:`(\underline{t}_0, \dots, \underline{t}_{N-1})` the vertices of :math:`\mathcal{M}` and :math:`(\underline{x}_0, \dots, \underline{x}_{N-1})` the associated values in :math:`\mathbb{R}^d`.
#
# A field is stored in the *Field* object that stores the mesh and the values at each vertex of the mesh.
# It can be built from a mesh and values or as a realization of a stochastic process.
# %%
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
import math as m
ot.Log.Show(ot.Log.NONE)
# %%
# First, we define a regular 2-d mesh
discretization = [10, 5]
mesher = ot.IntervalMesher(discretization)
lowerBound = [0.0, 0.0]
upperBound = [2.0, 1.0]
interval = ot.Interval(lowerBound, upperBound)
mesh = mesher.build(interval)
graph = mesh.draw()
graph.setTitle('Regular 2-d mesh')
view = viewer.View(graph)
# %%
# We now create a field from a mesh and some values
values = ot.Normal([0.0] * 2, [1.0] * 2,
ot.CorrelationMatrix(2)).getSample(len(mesh.getVertices()))
for i in range(len(values)):
x = values[i]
values[i] = 0.05 * x / x.norm()
return [dY0, dY1]
f = ot.PythonFunction(3, 2, flow)
phi = ot.ParametricFunction(f, [2], [0.0])
solver = ot.RungeKutta(phi)
initialState = [2.0, 2.0]
nt = 47
dt = 0.1
timeGrid = ot.RegularGrid(0.0, dt, nt)
result = solver.solve(initialState, timeGrid)
xMin = result.getMin()
xMax = result.getMax()
delta = 0.2 * (xMax - xMin)
mesh = ot.IntervalMesher([12] * 2).build(
ot.Interval(xMin - delta, xMax + delta))
field = ot.Field(mesh, phi(mesh.getVertices()))
ot.ResourceMap.SetAsScalar("Field-ArrowScaling", 0.1)
graph = field.draw()
cloud = ot.Cloud(mesh.getVertices())
cloud.setColor("black")
graph.add(cloud)
curve = ot.Curve(result)
curve.setColor("red")
curve.setLineWidth(2)
graph.add(curve)
fig = plt.figure()
ax = fig.add_subplot(111)
View(graph, figure=fig)
plt.suptitle("Lotka-Volterra ODE system")
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
from math import sqrt
mesh = ot.IntervalMesher([128]).build(ot.Interval(-1.0, 1.0))
threshold = 0.001
model = ot.AbsoluteExponential([1.0])
algo = ot.KarhunenLoeveP1Algorithm(mesh, model, threshold)
algo.run()
ev = algo.getResult().getEigenvalues()
modes = algo.getResult().getScaledModesAsProcessSample()
g = modes.drawMarginal(0)
g.setXTitle("$t$")
g.setYTitle("$\sqrt{\lambda_n}\phi_n$")
g.setTitle("P1 approx. of KL expansion for $C(s,t)=e^{-|s-t|}$")
fig = plt.figure(figsize=(6, 4))
axis = fig.add_subplot(111)
axis.set_xlim(auto=True)
View(g, figure=fig, axes=[axis], add_legend=False)
#! /usr/bin/env python
import openturns as ot
ot.TESTPREAMBLE()
dim = 2
interval = ot.Interval([0.0] * dim, [10.0] * dim)
mesh = ot.IntervalMesher([30] * dim).build(interval)
f = ot.SymbolicFunction(["x", "y"], ["x + 0.5*sin(y)", "y-0.1*x*sin(x)"])
mesh.setVertices(f(mesh.getVertices()))
simplices = mesh.getSimplices()
nrSimplices = len(simplices)
naive = ot.NaiveEnclosingSimplex(mesh.getVertices(), simplices)
print("naive=", naive)
ot.RandomGenerator.SetSeed(0)
test = ot.ComposedDistribution([ot.Uniform(-1.0, 11.0)] * dim).getSample(100)
for i, vertex in enumerate(test):
index = naive.query(vertex)
if index >= nrSimplices:
print(i, "is outside")
else:
found, coordinates = mesh.checkPointInSimplexWithCoordinates(
vertex, index)
if not found:
print("Wrong simplex found for", vertex, "(index=", index,
simplices[index], "barycentric coordinates=", coordinates)
point = ot.Point([123.456, 125.43, 3975.4567])
point2 = ot.Point(3, 789.123)
point3 = ot.Point(3, 1673.456)
point4 = ot.Point(3, 789.654123)
sample = ot.Sample(1, point)
sample.add(point2)
sample.add(point3)
sample.add(point4)
sample.add(point2)
sample.add(point4)
sample.add(point3)
print(sample)
study.add('sample', sample)
mesh = ot.IntervalMesher([50] * 3).build(ot.Interval(3))
study.add('mesh', mesh)
study.save()
study2 = ot.Study()
study2.setStorageManager(ot.XMLH5StorageManager(fileName))
study2.load()
sample2 = ot.Sample()
study2.fillObject('sample', sample2)
print(sample2)
assert sample == sample2, "wrong sample"
mesh2 = ot.Mesh()
study2.fillObject('mesh', mesh2)
import openturns as ot from matplotlib import pyplot as plt from openturns.viewer import View # Create a process X: R^2 --> R^2 # Define a bi dimensional mesh as a box myIndices = ot.Indices([40, 20]) myMesher = ot.IntervalMesher(myIndices) lowerBound = [0.0, 0.0] upperBound = [2.0, 1.0] myInterval = ot.Interval(lowerBound, upperBound) myMesh = myMesher.build(myInterval) # Define a scalar temporal Gaussian process on the mesh # this process is stationary # myXproc R^2 --> R amplitude = [1.0] scale = [0.2, 0.2] myCovModel = ot.ExponentialModel(scale, amplitude) myXproc = ot.GaussianProcess(myCovModel, myMesh) # Transform myXproc to make its variance depend on the vertex (s,t) # and to get a positive process # thanks to the spatial function g # myXtProcess R --> R g = ot.SymbolicFunction(['x1'], ['exp(x1)']) myDynTransform = ot.ValueFunction(g, 2) myXtProcess = ot.CompositeProcess(myDynTransform, myXproc) myField = myXtProcess.getRealization()
#! /usr/bin/env python
from __future__ import print_function
import openturns as ot
try:
mesh = ot.IntervalMesher(ot.Indices(1, 9)).build(ot.Interval(-1.0, 1.0))
factory = ot.KarhunenLoeveP1Factory(mesh, 0.0)
eigenValues = ot.NumericalPoint()
KLModes = factory.buildAsProcessSample(ot.AbsoluteExponential([1.0]),
eigenValues)
print("KL modes=", KLModes)
print("KL eigenvalues=", eigenValues)
cov1D = ot.AbsoluteExponential([1.0])
KLFunctions = factory.build(cov1D, eigenValues)
print("KL functions=", KLFunctions)
print("KL eigenvalues=", eigenValues)
R = ot.CorrelationMatrix(2, [1.0, 0.5, 0.5, 1.0])
scale = [1.0]
amplitude = [1.0, 2.0]
cov2D = ot.ExponentialModel(scale, amplitude, R)
KLFunctions = factory.build(cov2D, eigenValues)
print("KL functions=", KLFunctions)
print("KL eigenvalues=", eigenValues)
except:
import sys
print("t_KarhunenLoeveP1Factory_std.py",
sys.exc_info()[0],
sys.exc_info()[1])