def test_least_interpolant( self ):
num_dims = 2
func_domain = TensorProductDomain( num_dims, [[-1,1]] )
x = -numpy.cos( numpy.linspace( 0., 1., 10 ) * numpy.pi );
[X,Y] = numpy.meshgrid( x, x )
build_points = numpy.vstack( ( X.reshape(( 1,X.shape[0]*X.shape[1])),
Y.reshape(( 1, Y.shape[0]*Y.shape[1]))))
build_values = matlab_peaks( build_points )
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
linear_solver = LeastInterpolation()
predictor = PCE( build_points.shape[0], basis = basis,
linear_solver = linear_solver )
predictor.function_domain( func_domain )
predictor.build( build_points, build_values )
result = predictor.evaluate_set( build_points )
if ( result.ndim > 1 ): result = result.reshape( result.shape[0] )
error = result - build_values
l2_error = numpy.sqrt( numpy.dot( error.T, error ) / \
build_points.shape[1] )
print 'error at nodes: %1.15e' %l2_error
assert l2_error < self.eps
poly_1d = [ JacobiPolynomial1D( 0.5, 0.5 ) ]
basis = TensorProductBasis( num_dims, poly_1d )
linear_solver = LeastInterpolation()
predictor = PCE( build_points.shape[0], basis = basis,
linear_solver = linear_solver )
predictor.function_domain( func_domain )
predictor.build( build_points, build_values )
result = predictor.evaluate_set( build_points )
if ( result.ndim > 1 ): result = result.reshape( result.shape[0] )
def interpolation_1d_test_I( f ):
seed = 2
rng = numpy.random#.RandomState( seed )
# 1D example with least interpolation
order = 3
num_dims = 1
func_domain = TensorProductDomain( num_dims, [[-1.,1.]] )
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
linear_solver = LeastInterpolation()
predictor = PCE( num_dims, basis, order = order,
linear_solver = linear_solver )
predictor.function_domain( func_domain )
x = -numpy.cos( numpy.linspace( 0., 1., order + 1 ) * numpy.pi );
build_points = x.reshape( 1, x.shape[0] )
build_values = f( build_points[0,:] )
p = predictor.build( build_points, build_values )
print 'c', predictor.coeff
def test_polynomial_chaos_expansion( self ):
# test 1D bounded domain
num_dims = 1
func_domain = TensorProductDomain( num_dims )
poly_1d = [ JacobiPolynomial1D( 0., 0. ) ]
basis = TensorProductBasis( num_dims, poly_1d )
predictor = PCE( num_dims, basis, order = 2 )
predictor.set_coefficients( numpy.ones( predictor.coeff.shape ) )
predictor.function_domain( func_domain )
num_test_points = 20
test_points = \
numpy.linspace( 0., 1., num_test_points ).reshape(1,num_test_points)
pred_vals = predictor.evaluate_set( test_points )
x = 2. * test_points - 1.
test_vals = numpy.ones( num_test_points ) + x[0,:] + \
0.5 * ( 3.*x[0,:]**2 - 1. )
assert numpy.allclose( test_vals, pred_vals )
test_mean = 1.
test_variance = 1./3. + 1./5.
assert numpy.allclose( predictor.mean(), test_mean )
assert numpy.allclose( predictor.variance(), test_variance )
# test 2D bounded domain
num_dims = 2
func_domain = TensorProductDomain( num_dims )
poly_1d = [ JacobiPolynomial1D( 0., 0. ) ]
basis = TensorProductBasis( num_dims, poly_1d )
predictor = PCE( num_dims, basis, order = 2 )
predictor.set_coefficients( numpy.ones( predictor.coeff.shape ) )
predictor.function_domain( func_domain )
num_test_points = 20
test_points = numpy.random.uniform( 0., 1., ( num_dims, num_test_points))
pred_vals = predictor.evaluate_set( test_points )
x = 2. * test_points - 1.
test_vals = numpy.ones( num_test_points ) + x[0,:] + x[1,:] + \
0.5 * ( 3.*x[0,:]**2 - 1. ) + 0.5 * ( 3.*x[1,:]**2 - 1. ) + \
x[0,:] * x[1,:]
assert numpy.allclose( test_vals, pred_vals )
test_mean = 1.
test_variance = 2. * 1./3. + 1./9. + 2. * 1./5.
assert numpy.allclose( predictor.mean(), test_mean )
assert numpy.allclose( predictor.variance(), test_variance )
# test when domain is unbounded in one dimension
num_dims = 2
func_domain = TensorProductDomain( num_dims,
ranges = [[0.,1.],
[-numpy.inf, numpy.inf]] )
poly_1d = [ JacobiPolynomial1D( 0., 0. ), HermitePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
predictor = PCE( num_dims, basis, order = 2 )
predictor.set_coefficients( numpy.ones( predictor.coeff.shape ) )
predictor.function_domain( func_domain )
num_test_points = 20
x_1 = numpy.random.uniform( 0., 1., ( 1, 20 ) )
x_2 = numpy.random.normal( 0., 1., ( 1, 20 ) )
test_points = numpy.vstack( ( x_1, x_2 ) )
pred_vals = predictor.evaluate_set( test_points )
x = test_points
x[0,:] = 2. * x[0,:] - 1.
test_vals = numpy.ones( num_test_points ) + x[0,:] + x[1,:] + \
0.5 * ( 3.*x[0,:]**2 - 1. ) + ( x[1,:]**2 - 1. ) + \
x[0,:] * x[1,:]
assert numpy.allclose( test_vals, pred_vals )
test_mean = 1.
test_variance = 2. * 1./3. + 1./5. + 2. + 1.
assert numpy.allclose( predictor.mean(), test_mean )
assert numpy.allclose( predictor.variance(), test_variance )
class GridPointAdaptedSparseGrid(object):
def __init__( self, num_dims, max_level, quadrature_rule,
refinement_manager, sparse_grid_data, target_function ):
self.num_dims = num_dims
self.max_level = max_level
self.quadrature_rule = quadrature_rule
self.refinement_manager = refinement_manager
self.target_function = target_function
# for least interpolant
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
linear_solver = LeastInterpolation()
self.pce = PCE( self.num_dims, basis = basis,
linear_solver = linear_solver )
self.pce.function_domain( self.quadrature_rule.domain )
# grid points used in the sparse grid
self.grid_point_indices = set()
# number of grid points in the grid. This is not equal to
# len( self.grid_point_indices ). The later contains all grid points
# including candidates and those actually in the grid
self.num_grid_points = 0
# grid points that are candidates for addition to the sparse grid
# but have not been initialized and so can not be prioritized
self.uninit_grid_point_indices = set()
# grid points that are candidates for addition to the sparse grid
self.candidate_grid_point_indices_queue = []
# Python note: heap[0] has the smallest value ( inverse of priority ),
# i.e the highest prioirity
# grid points that are candidates for addition to the sparse grid
# This object will contain the same grid point indices as
# self.candidate_grid_point_indices_queue but is used to ensure a
# point is only added to the self.candidate_grid_point_indices_queue
#once
self.candidate_grid_point_indices_hash = set()
# keep track of the number of grid points generated
# ( self.num_grid_points + self.num_candidate_grid_points +
# self.num_uninit_grid_points )
self.num_grid_point_indices_generated = 0
self.data = sparse_grid_data
def build_root( self ):
"""
Initialize the grid by creating the root subspace
"""
root_grid_point_index = GridPointIndex( None )
self.insert_uninit_grid_point( root_grid_point_index )
def insert_remaining_candidate_points( self ):
if ( self.refinement_manager.candidate_point_needs_func_eval() ):
while ( len( self.candidate_grid_point_indices_queue ) > 0 ):
inv_priority, grid_point_index = \
heapq.heappop( self.candidate_grid_point_indices_queue )
self.candidate_grid_point_indices_hash.remove( grid_point_index )
self.insert_grid_point( grid_point_index )
def insert_next_grid_point( self ):
"""
Insert a subspace into the list of subspaces in the grid
"""
# if two indices have the same priority the indices themselves will be
# compared
inv_priority, grid_point_index = \
heapq.heappop( self.candidate_grid_point_indices_queue )
self.candidate_grid_point_indices_hash.remove( grid_point_index )
if ( grid_point_index not in self.grid_point_indices ):
if ( not self.refinement_manager.candidate_point_needs_func_eval() ):
self.evaluate_target_function( grid_point_index )
self.insert_grid_point( grid_point_index )
else:
# point was originally a missing parent
pass
return grid_point_index
def insert_uninit_grid_point( self, grid_point_index ):
if ( grid_point_index not in self.uninit_grid_point_indices and
grid_point_index not in self.candidate_grid_point_indices_hash and
grid_point_index not in self.grid_point_indices and
self.refinement_manager.grid_point_admissible( grid_point_index ) ):
self.uninit_grid_point_indices.add( grid_point_index )
grid_point_index.array_index = self.num_grid_point_indices_generated
self.num_grid_point_indices_generated += 1
def insert_candidate_grid_point( self, grid_point_index ):
# I think this can be removed if check is made when point is added to
# unitialized set, check for existance must be made for uninit
# candidate and sparse grid points
if ( grid_point_index not in self.candidate_grid_point_indices_hash ):
heapq.heappush( self.candidate_grid_point_indices_queue,
( 1. / grid_point_index.priority, grid_point_index ))
self.candidate_grid_point_indices_hash.add( grid_point_index )
else:
msg = 'grid point already exists'
raise Exception, msg
def insert_grid_point( self, grid_point_index ):
self.grid_point_indices.add( grid_point_index )
self.num_grid_points += 1
def refine( self ):
"""
Perform point-wise adaptive refinement.
Extract the grid points with the highest priority.
"""
grid_point_index = self.insert_next_grid_point()
if self.refinement_manager.grid_point_warrants_refinement( grid_point_index ):
for dim in xrange( self.quadrature_rule.num_dims ):
# todo would be better not to assume that a left and right child
# are both available or more are not as well
child_index = self.quadrature_rule.child_index( grid_point_index,
dim, 'left' )
if child_index is not None:
self.insert_uninit_grid_point( child_index )
child_index = self.quadrature_rule.child_index( grid_point_index,
dim, 'right' )
if child_index is not None:
self.insert_uninit_grid_point( child_index )
def compute_hierarical_surpluses( self, fn_val, coord, grid_point_index ):
self.insert_missing_parent_grid_points( grid_point_index )
subspace_index = grid_point_index.subspace_index()
return fn_val - self.evaluate_set( coord.reshape( self.num_dims,1),
subspace_index )
# evaluate all points belonging to subspaces that have indices <=
# current_subspace_index
def evaluate_set( self, pts, current_subspace_index = None ):
result = numpy.zeros( ( pts.shape[1] ), numpy.double )
for grid_point_index in self.grid_point_indices:
evaluate_grid_point = True
# this is very expensive. at least in python
#grid_point_subspace_index = grid_point_index.subspace_index()
#if ( current_subspace_index is None ):
# evaluate_grid_point = True
#elif ( grid_point_subspace_index <= current_subspace_index ):
# evaluate_grid_point = True
#else:
# evaluate_grid_point = False
if ( evaluate_grid_point ):
if ( grid_point_index.data is not None ):
#evaluate basis
result += \
self.data.hier_surplus[grid_point_index.array_index] * \
self.quadrature_rule.basis.value( pts, grid_point_index,
self.quadrature_rule.domain )
else:
result += \
self.data.hier_surplus[grid_point_index.array_index] * \
numpy.ones( ( result.shape[0] ), numpy.double )
return result
def evaluate_set1( self, pts, current_subspace_index = None ):
if ( len( self.grid_point_indices ) == 0 ):
return numpy.zeros( ( pts.shape[1] ), numpy.double )
poly_indices = \
get_polynomial_indices_of_subspace_indices(
self.refinement_manager.subspace_indices,
self.quadrature_rule )
coords = numpy.empty( ( self.num_dims, len( self.grid_point_indices ) ),
numpy.double )
values = numpy.empty( ( len( self.grid_point_indices ), 1 ),
numpy.double )
for i, grid_point_index in enumerate( self.grid_point_indices ):
coords[:,i] = \
self.quadrature_rule.coordinate( grid_point_index )
values[i,0] = self.data.fn_vals[grid_point_index.array_index]
self.pce.build( coords, values )
return self.pce.evaluate_set( pts )
def grid_point_coordinates( self ):
coords = numpy.empty( ( self.num_dims, len( self.grid_point_indices ) ),
numpy.double )
for i, grid_point_index in enumerate( self.grid_point_indices ):
coords[:,i] = \
self.quadrature_rule.coordinate( grid_point_index )
return coords
def insert_parent_grid_point( self, grid_point_index ):
# If the parent is missing from self.sparse_grid_indices
# always promote it to self.sparse grid_indices.
# If the parent is in self.candidate_grid_point_indices
# keep it there so it will be refined. Doing this means that
# in insert_next_grid_point we must include a check that does not
# allow points that are already in self.sparse_grid_indices to
# be added again. We must also evaluate
# the target function if it has not been evaluated already.
# If the parent is in self.uninit_grid_point_indices
# then promote it to the candidate set. We must always calculate
# the priority and evaluate the function if not done so already.
self.insert_missing_parent_grid_points( grid_point_index )
# Check if grid_point_index is in a candidate point
array_index = None
grid_point_index_ref=find( self.candidate_grid_point_indices_hash,
grid_point_index )
if ( grid_point_index_ref is not None ):
# grid_point_index is not a candidate point.
# grid_point_index_ref contains is the original index
# contained in self.candidate_grid_point_indices_hash.
# grid_point_index will not have priority or aray_index set.
# only evaluate function if not done so already
if ( not self.refinement_manager.candidate_point_needs_func_eval() ):
self.evaluate_target_function( grid_point_index_ref )
# promote point to the sparse grid
self.insert_grid_point( grid_point_index_ref )
else:
# grid_point_index is not a candidate point so check if it is
# an uninit point
grid_point_index_ref = \
find( self.uninit_grid_point_indices, grid_point_index )
if ( grid_point_index_ref is not None ):
# grid_point_index is an uninit point
# grid_point_index_ref contains is the original index
# contained in self.uninit_grid_point_indices.
self.uninit_grid_point_indices.remove(grid_point_index_ref)
self.insert_candidate_grid_point( grid_point_index_ref )
else:
# grid_point_index is not an uninit point or candidate
# point
grid_point_index_ref = grid_point_index
grid_point_index_ref.array_index = \
self.num_grid_point_indices_generated
self.num_grid_point_indices_generated += 1
self.evaluate_target_function( grid_point_index_ref )
self.refinement_manager.prioritize_grid_point( grid_point_index_ref )
self.insert_candidate_grid_point( grid_point_index_ref )
self.insert_grid_point( grid_point_index )
def insert_missing_parent_grid_points( self, grid_point_index ):
for d in xrange( grid_point_index.num_effect_dims() ):
dim = grid_point_index.dimension_from_array_index ( d )
parent_index = self.quadrature_rule.parent_index( grid_point_index,
dim )
if ( parent_index not in self.grid_point_indices ):
self.insert_parent_grid_point( parent_index )
def evaluate_target_function( self, grid_point_index ):
coord = self.quadrature_rule.coordinate( grid_point_index )
coord = coord.reshape( coord.shape[0], 1 )
array_index = grid_point_index.array_index
self.data.set_fn_vals( self.target_function( coord ), array_index )
hier_surplus = \
self.compute_hierarical_surpluses( self.data.fn_vals[array_index],
coord, grid_point_index )
self.data.set_hier_surplus( hier_surplus, array_index )
self.refinement_manager.num_model_runs +=1
def prioritize( self ):
for grid_point_index in self.uninit_grid_point_indices:
if ( self.refinement_manager.candidate_point_needs_func_eval() ):
self.evaluate_target_function( grid_point_index )
self.refinement_manager.prioritize_grid_point( grid_point_index )
self.insert_candidate_grid_point( grid_point_index )
self.uninit_grid_point_indices = set()
