SG++
PCE with Combigrids (Python)

This simple example shows how to create a Polynomial Chaos Expansion from an adaptively refined combigrid.

import numpy as np
import pysgpp
import time

First we have to define a model to approximate.

def expModel(x):
return np.exp(-x[0] * x[1])

Then we can create a refined combigrid

def ct_to_pce():
start_time = time.time()
# initialize model function
func = pysgpp.multiFunc(expModel)
numDims = 2
# regular sparse grid level q
q = 6
# create polynomial basis
config = pysgpp.OrthogonalPolynomialBasis1DConfiguration()
config.polyParameters.type_ = pysgpp.OrthogonalPolynomialBasisType_LEGENDRE
basisFunction = pysgpp.OrthogonalPolynomialBasis1D(config)
# create sprarse grid interpolation operation
op = pysgpp.CombigridOperation.createExpClenshawCurtisPolynomialInterpolation(numDims, func)
# start with regular level q and add some levels adaptively
op.getLevelManager().addRegularLevels(q)
op.getLevelManager().addLevelsAdaptiveByNumLevels(5)

and construct a PCE representation to easily calculate statistical features of our model.

# create polynomial chaos surrogate from sparse grid
surrogateConfig = pysgpp.CombigridSurrogateModelConfiguration()
surrogateConfig.type = pysgpp.CombigridSurrogateModelsType_POLYNOMIAL_CHAOS_EXPANSION
surrogateConfig.loadFromCombigridOperation(op)
surrogateConfig.basisFunction = basisFunction
pce = pysgpp.createCombigridSurrogateModel(surrogateConfig)
# compute sobol indices
sobol_indices = pysgpp.DataVector(1)
total_indices = pysgpp.DataVector(1)
pce.getComponentSobolIndices(sobol_indices)
pce.getTotalSobolIndices(total_indices)
# print results
print("Mean: {} Variance: {}".format(pce.mean(), pce.variance()))
print("Sobol indices {}".format(sobol_indices.toString()))
print("Total Sobol indices {}".format(total_indices.toString()))
print("Sum {}\n".format(sobol_indices.sum()))
print("Elapsed time: {} s".format(time.time() - start_time))

Output:

 Mean: 0.796599599298 Variance: 0.0250607565267
 Sobol indices [4.46123454530588048339e-01, 4.46123454530588825495e-01, 1.07753090938823264944e-01]
 Total Sobol indices [5.53876545469411341038e-01, 5.53876545469412118194e-01]
 Sum 1.0