def basis_evaluation(self, order, ps, entity=None, coordinate_mapping=None):
"""Return code for evaluating the element at known points on the
reference element.
:param order: return derivatives up to this order.
:param ps: the point set object.
:param entity: the cell entity on which to tabulate.
"""
# Spatial dimension
dimension = self.cell.get_spatial_dimension()
# Shape of the tabulation matrix
shape = tuple(index.extent for index in ps.indices) + self.index_shape + self.value_shape
result = {}
for derivative in range(order + 1):
for alpha in mis(dimension, derivative):
name = str.format("rt_{}_{}_{}_{}_{}_{}",
self.variant,
self.degree,
''.join(map(str, alpha)),
self.shift_axes,
'c' if self.continuous else 'd',
{None: "",
'+': "p",
'-': "m"}[self.restriction])
result[alpha] = gem.partial_indexed(gem.Variable(name, shape), ps.indices)
return result
def basis_evaluation(self, order, ps, entity=None):
"""Return code for evaluating the element at known points on the
reference element.
:param order: return derivatives up to this order.
:param ps: the point set object.
:param entity: the cell entity on which to tabulate.
"""
# Spatial dimension
dimension = self.cell.get_spatial_dimension()
# Shape of the tabulation matrix
shape = tuple(
index.extent
for index in ps.indices) + self.index_shape + self.value_shape
result = {}
for derivative in range(order + 1):
for alpha in mis(dimension, derivative):
name = str.format("rt_{}_{}_{}_{}_{}_{}", self.variant,
self.degree, ''.join(map(str, alpha)),
self.shift_axes,
'c' if self.continuous else 'd', {
None: "",
'+': "p",
'-': "m"
}[self.restriction])
result[alpha] = gem.partial_indexed(gem.Variable(name, shape),
ps.indices)
return result
def coefficient(self, ufl_coefficient, restriction):
"""A function that maps :class:`ufl.Coefficient`s to GEM
expressions."""
kernel_arg = self.coefficient_map[ufl_coefficient]
if ufl_coefficient.ufl_element().family() == 'Real':
return kernel_arg
else:
return gem.partial_indexed(kernel_arg, {None: (), '+': (0,), '-': (1,)}[restriction])
def _selector(self, callback, opts, restriction):
"""Helper function for selecting code for the correct entity
at run-time."""
if len(opts) == 1:
return callback(opts[0])
else:
results = gem.ListTensor(list(map(callback, opts)))
f = self.facet_number(restriction)
return gem.partial_indexed(results, (f,))
def _coefficient(self, coefficient, name):
element = create_element(coefficient.ufl_element())
shape = self.shape + element.index_shape
size = numpy.prod(shape, dtype=int)
funarg = ast.Decl(SCALAR_TYPE, ast.Symbol(name), pointers=[("restrict", )],
qualifiers=["const"])
expression = gem.reshape(gem.Variable(name, (size, )), shape)
expression = gem.partial_indexed(expression, self.indices)
self.coefficient_map[coefficient] = expression
return funarg
def callback(key):
table = ctx.tabulation_manager[key]
if len(table.shape) == 1:
# Cellwise constant
row = table
else:
table = ctx.index_selector(lambda i: as_gem(table.array[i]),
mt.restriction)
row = gem.partial_indexed(table, (ctx.point_index,))
return gem.Indexed(row, (argument_index,))
def _coefficient(self, coefficient, name):
element = create_element(coefficient.ufl_element())
shape = self.shape + element.index_shape
size = numpy.prod(shape, dtype=int)
funarg = ast.Decl(SCALAR_TYPE, ast.Symbol(name), pointers=[("restrict", )],
qualifiers=["const"])
expression = gem.reshape(gem.Variable(name, (size, )), shape)
expression = gem.partial_indexed(expression, self.indices)
self.coefficient_map[coefficient] = expression
return funarg
def basis_evaluation(self, order, ps, entity=None):
'''Return code for evaluating the element at known points on the
reference element.
:param order: return derivatives up to this order.
:param ps: the point set.
:param entity: the cell entity on which to tabulate.
'''
space_dimension = self._element.space_dimension()
value_size = np.prod(self._element.value_shape(), dtype=int)
fiat_result = self._element.tabulate(order, ps.points, entity)
result = {}
for alpha, fiat_table in iteritems(fiat_result):
if isinstance(fiat_table, Exception):
result[alpha] = gem.Failure(
self.index_shape + self.value_shape, fiat_table)
continue
derivative = sum(alpha)
table_roll = fiat_table.reshape(space_dimension, value_size,
len(ps.points)).transpose(1, 2, 0)
exprs = []
for table in table_roll:
if derivative < self.degree:
point_indices = ps.indices
point_shape = tuple(index.extent
for index in point_indices)
exprs.append(
gem.partial_indexed(
gem.Literal(
table.reshape(point_shape + self.index_shape)),
point_indices))
elif derivative == self.degree:
# Make sure numerics satisfies theory
assert np.allclose(table, table.mean(axis=0,
keepdims=True))
exprs.append(gem.Literal(table[0]))
else:
# Make sure numerics satisfies theory
assert np.allclose(table, 0.0)
exprs.append(gem.Zero(self.index_shape))
if self.value_shape:
beta = self.get_indices()
zeta = self.get_value_indices()
result[alpha] = gem.ComponentTensor(
gem.Indexed(
gem.ListTensor(
np.array([
gem.Indexed(expr, beta) for expr in exprs
]).reshape(self.value_shape)), zeta), beta + zeta)
else:
expr, = exprs
result[alpha] = expr
return result
def dual_basis(self):
Q, x = self.wrappee.dual_basis
beta = self.get_indices()
zeta = self.get_value_indices()
# Index out the basis indices from wrapee's Q, to get
# something of wrappee.value_shape, then promote to new shape
# with the same transform as done for basis evaluation
Q = gem.ListTensor(self.transform(gem.partial_indexed(Q, beta)))
# Finally wrap up Q in shape again (now with some extra
# value_shape indices)
return gem.ComponentTensor(Q[zeta], beta + zeta), x
def test_cellwise_constant(cell, degree):
dim = cell.get_spatial_dimension()
element = finat.Lagrange(cell, degree)
index = gem.Index()
point = gem.partial_indexed(gem.Variable('X', (17, dim)), (index,))
order = 2
for alpha, table in element.point_evaluation(order, point).items():
if sum(alpha) < degree:
assert table.free_indices == (index,)
else:
assert table.free_indices == ()
def test_cellwise_constant(cell, degree):
dim = cell.get_spatial_dimension()
element = finat.Lagrange(cell, degree)
index = gem.Index()
point = gem.partial_indexed(gem.Variable('X', (17, dim)), (index, ))
order = 2
for alpha, table in element.point_evaluation(order, point).items():
if sum(alpha) < degree:
assert table.free_indices == (index, )
else:
assert table.free_indices == ()
def translate_cell_coordinate(terminal, mt, ctx):
if ctx.integration_dim == ctx.fiat_cell.get_dimension():
return ctx.point_expr
# This destroys the structure of the quadrature points, but since
# this code path is only used to implement CellCoordinate in facet
# integrals, hopefully it does not matter much.
ps = ctx.point_set
point_shape = tuple(index.extent for index in ps.indices)
def callback(entity_id):
t = ctx.fiat_cell.get_entity_transform(ctx.integration_dim, entity_id)
data = numpy.asarray(list(map(t, ps.points)))
return gem.Literal(data.reshape(point_shape + data.shape[1:]))
return gem.partial_indexed(ctx.entity_selector(callback, mt.restriction),
ps.indices)
def translate_cell_coordinate(terminal, mt, ctx):
if ctx.integration_dim == ctx.fiat_cell.get_dimension():
return ctx.point_expr
# This destroys the structure of the quadrature points, but since
# this code path is only used to implement CellCoordinate in facet
# integrals, hopefully it does not matter much.
ps = ctx.point_set
point_shape = tuple(index.extent for index in ps.indices)
def callback(entity_id):
t = ctx.fiat_cell.get_entity_transform(ctx.integration_dim, entity_id)
data = numpy.asarray(list(map(t, ps.points)))
return gem.Literal(data.reshape(point_shape + data.shape[1:]))
return gem.partial_indexed(ctx.entity_selector(callback, mt.restriction),
ps.indices)
def callback(key):
table = ctx.tabulation_manager[key]
if len(table.shape) == 1:
# Cellwise constant
row = table
if numpy.count_nonzero(table.array) <= 2:
assert row.shape == vec.shape
return reduce(gem.Sum,
[gem.Product(gem.Indexed(row, (i,)), gem.Indexed(vec, (i,)))
for i in range(row.shape[0])],
gem.Zero())
else:
table = ctx.index_selector(lambda i: as_gem(table.array[i]),
mt.restriction)
row = gem.partial_indexed(table, (ctx.point_index,))
r = ctx.index_cache[terminal.ufl_element()]
return gem.IndexSum(gem.Product(gem.Indexed(row, (r,)),
gem.Indexed(vec, (r,))), r)
def basis_evaluation(self,
order,
ps,
entity=None,
coordinate_mapping=None):
'''Return code for evaluating the element at known points on the
reference element.
:param order: return derivatives up to this order.
:param ps: the point set.
:param entity: the cell entity on which to tabulate.
'''
# Build everything in sympy
vs, xx, _ = self._basis
# and convert -- all this can be used for each derivative!
phys_verts = coordinate_mapping.physical_vertices()
phys_points = gem.partial_indexed(
coordinate_mapping.physical_points(ps, entity=entity), ps.indices)
repl = dict(
(vs[idx], phys_verts[idx]) for idx in numpy.ndindex(vs.shape))
repl.update(zip(xx, phys_points))
mapper = gem.node.Memoizer(sympy2gem)
mapper.bindings = repl
result = {}
for i in range(order + 1):
alphas = mis(2, i)
for alpha in alphas:
dphis = self._basis_deriv(xx, alpha)
result[alpha] = gem.ListTensor(list(map(mapper, dphis)))
return result
def basis_evaluation(self,
order,
ps,
entity=None,
coordinate_mapping=None):
'''Return code for evaluating the element at known points on the
reference element.
:param order: return derivatives up to this order.
:param ps: the point set.
:param entity: the cell entity on which to tabulate.
'''
space_dimension = self._element.space_dimension()
value_size = np.prod(self._element.value_shape(), dtype=int)
fiat_result = self._element.tabulate(order, ps.points, entity)
result = {}
# In almost all cases, we have
# self.space_dimension() == self._element.space_dimension()
# But for Bell, FIAT reports 21 basis functions,
# but FInAT only 18 (because there are actually 18
# basis functions, and the additional 3 are for
# dealing with transformations between physical
# and reference space).
index_shape = (self._element.space_dimension(), )
for alpha, fiat_table in fiat_result.items():
if isinstance(fiat_table, Exception):
result[alpha] = gem.Failure(
self.index_shape + self.value_shape, fiat_table)
continue
derivative = sum(alpha)
table_roll = fiat_table.reshape(space_dimension, value_size,
len(ps.points)).transpose(1, 2, 0)
exprs = []
for table in table_roll:
if derivative < self.degree:
point_indices = ps.indices
point_shape = tuple(index.extent
for index in point_indices)
exprs.append(
gem.partial_indexed(
gem.Literal(
table.reshape(point_shape + index_shape)),
point_indices))
elif derivative == self.degree:
# Make sure numerics satisfies theory
exprs.append(gem.Literal(table[0]))
else:
# Make sure numerics satisfies theory
assert np.allclose(table, 0.0)
exprs.append(gem.Zero(self.index_shape))
if self.value_shape:
# As above, this extent may be different from that
# advertised by the finat element.
beta = tuple(gem.Index(extent=i) for i in index_shape)
assert len(beta) == len(self.get_indices())
zeta = self.get_value_indices()
result[alpha] = gem.ComponentTensor(
gem.Indexed(
gem.ListTensor(
np.array([
gem.Indexed(expr, beta) for expr in exprs
]).reshape(self.value_shape)), zeta), beta + zeta)
else:
expr, = exprs
result[alpha] = expr
return result
def promote(table):
v = gem.partial_indexed(table, beta)
u = gem.ListTensor(self.transform(v))
return gem.ComponentTensor(gem.Indexed(u, zeta), beta + zeta)
def expression(self):
return gem.partial_indexed(gem.Literal(self.points), self.indices)
def expression(self):
return gem.partial_indexed(self._points_expr, self.indices)
def compile_expression_dual_evaluation(expression,
to_element,
*,
domain=None,
interface=None,
parameters=None,
coffee=False):
"""Compile a UFL expression to be evaluated against a compile-time known reference element's dual basis.
Useful for interpolating UFL expressions into e.g. N1curl spaces.
:arg expression: UFL expression
:arg to_element: A FInAT element for the target space
:arg domain: optional UFL domain the expression is defined on (required when expression contains no domain).
:arg interface: backend module for the kernel interface
:arg parameters: parameters object
:arg coffee: compile coffee kernel instead of loopy kernel
"""
import coffee.base as ast
import loopy as lp
# Just convert FInAT element to FIAT for now.
# Dual evaluation in FInAT will bring a thorough revision.
to_element = to_element.fiat_equivalent
if any(len(dual.deriv_dict) != 0 for dual in to_element.dual_basis()):
raise NotImplementedError(
"Can only interpolate onto dual basis functionals without derivative evaluation, sorry!"
)
if parameters is None:
parameters = default_parameters()
else:
_ = default_parameters()
_.update(parameters)
parameters = _
# Determine whether in complex mode
complex_mode = is_complex(parameters["scalar_type"])
# Find out which mapping to apply
try:
mapping, = set(to_element.mapping())
except ValueError:
raise NotImplementedError(
"Don't know how to interpolate onto zany spaces, sorry")
expression = apply_mapping(expression, mapping, domain)
# Apply UFL preprocessing
expression = ufl_utils.preprocess_expression(expression,
complex_mode=complex_mode)
# Initialise kernel builder
if interface is None:
if coffee:
import tsfc.kernel_interface.firedrake as firedrake_interface_coffee
interface = firedrake_interface_coffee.ExpressionKernelBuilder
else:
# Delayed import, loopy is a runtime dependency
import tsfc.kernel_interface.firedrake_loopy as firedrake_interface_loopy
interface = firedrake_interface_loopy.ExpressionKernelBuilder
builder = interface(parameters["scalar_type"])
arguments = extract_arguments(expression)
argument_multiindices = tuple(
builder.create_element(arg.ufl_element()).get_indices()
for arg in arguments)
# Replace coordinates (if any) unless otherwise specified by kwarg
if domain is None:
domain = expression.ufl_domain()
assert domain is not None
# Collect required coefficients
first_coefficient_fake_coords = False
coefficients = extract_coefficients(expression)
if has_type(expression, GeometricQuantity) or any(
fem.needs_coordinate_mapping(c.ufl_element())
for c in coefficients):
# Create a fake coordinate coefficient for a domain.
coords_coefficient = ufl.Coefficient(
ufl.FunctionSpace(domain, domain.ufl_coordinate_element()))
builder.domain_coordinate[domain] = coords_coefficient
builder.set_cell_sizes(domain)
coefficients = [coords_coefficient] + coefficients
first_coefficient_fake_coords = True
builder.set_coefficients(coefficients)
# Split mixed coefficients
expression = ufl_utils.split_coefficients(expression,
builder.coefficient_split)
# Translate to GEM
kernel_cfg = dict(
interface=builder,
ufl_cell=domain.ufl_cell(),
# FIXME: change if we ever implement
# interpolation on facets.
integral_type="cell",
argument_multiindices=argument_multiindices,
index_cache={},
scalar_type=parameters["scalar_type"])
if all(
isinstance(dual, PointEvaluation)
for dual in to_element.dual_basis()):
# This is an optimisation for point-evaluation nodes which
# should go away once FInAT offers the interface properly
qpoints = []
# Everything is just a point evaluation.
for dual in to_element.dual_basis():
ptdict = dual.get_point_dict()
qpoint, = ptdict.keys()
(qweight, component), = ptdict[qpoint]
assert allclose(qweight, 1.0)
assert component == ()
qpoints.append(qpoint)
point_set = PointSet(qpoints)
config = kernel_cfg.copy()
config.update(point_set=point_set)
# Allow interpolation onto QuadratureElements to refer to the quadrature
# rule they represent
if isinstance(to_element, FIAT.QuadratureElement):
assert allclose(asarray(qpoints), asarray(to_element._points))
quad_rule = QuadratureRule(point_set, to_element._weights)
config["quadrature_rule"] = quad_rule
expr, = fem.compile_ufl(expression, **config, point_sum=False)
# In some cases point_set.indices may be dropped from expr, but nothing
# new should now appear
assert set(expr.free_indices) <= set(
chain(point_set.indices, *argument_multiindices))
shape_indices = tuple(gem.Index() for _ in expr.shape)
basis_indices = point_set.indices
ir = gem.Indexed(expr, shape_indices)
else:
# This is general code but is more unrolled than necssary.
dual_expressions = [] # one for each functional
broadcast_shape = len(expression.ufl_shape) - len(
to_element.value_shape())
shape_indices = tuple(gem.Index()
for _ in expression.ufl_shape[:broadcast_shape])
expr_cache = {} # Sharing of evaluation of the expression at points
for dual in to_element.dual_basis():
pts = tuple(sorted(dual.get_point_dict().keys()))
try:
expr, point_set = expr_cache[pts]
except KeyError:
point_set = PointSet(pts)
config = kernel_cfg.copy()
config.update(point_set=point_set)
expr, = fem.compile_ufl(expression, **config, point_sum=False)
# In some cases point_set.indices may be dropped from expr, but
# nothing new should now appear
assert set(expr.free_indices) <= set(
chain(point_set.indices, *argument_multiindices))
expr = gem.partial_indexed(expr, shape_indices)
expr_cache[pts] = expr, point_set
weights = collections.defaultdict(list)
for p in pts:
for (w, cmp) in dual.get_point_dict()[p]:
weights[cmp].append(w)
qexprs = gem.Zero()
for cmp in sorted(weights):
qweights = gem.Literal(weights[cmp])
qexpr = gem.Indexed(expr, cmp)
qexpr = gem.index_sum(
gem.Indexed(qweights, point_set.indices) * qexpr,
point_set.indices)
qexprs = gem.Sum(qexprs, qexpr)
assert qexprs.shape == ()
assert set(qexprs.free_indices) == set(
chain(shape_indices, *argument_multiindices))
dual_expressions.append(qexprs)
basis_indices = (gem.Index(), )
ir = gem.Indexed(gem.ListTensor(dual_expressions), basis_indices)
# Build kernel body
return_indices = basis_indices + shape_indices + tuple(
chain(*argument_multiindices))
return_shape = tuple(i.extent for i in return_indices)
return_var = gem.Variable('A', return_shape)
if coffee:
return_arg = ast.Decl(parameters["scalar_type"],
ast.Symbol('A', rank=return_shape))
else:
return_arg = lp.GlobalArg("A",
dtype=parameters["scalar_type"],
shape=return_shape)
return_expr = gem.Indexed(return_var, return_indices)
# TODO: one should apply some GEM optimisations as in assembly,
# but we don't for now.
ir, = impero_utils.preprocess_gem([ir])
impero_c = impero_utils.compile_gem([(return_expr, ir)], return_indices)
index_names = dict(
(idx, "p%d" % i) for (i, idx) in enumerate(basis_indices))
# Handle kernel interface requirements
builder.register_requirements([ir])
# Build kernel tuple
return builder.construct_kernel(return_arg, impero_c, index_names,
first_coefficient_fake_coords)
def basis_evaluation(self, order, ps, entity=None, coordinate_mapping=None):
'''Return code for evaluating the element at known points on the
reference element.
:param order: return derivatives up to this order.
:param ps: the point set.
:param entity: the cell entity on which to tabulate.
'''
space_dimension = self._element.space_dimension()
value_size = np.prod(self._element.value_shape(), dtype=int)
fiat_result = self._element.tabulate(order, ps.points, entity)
result = {}
# In almost all cases, we have
# self.space_dimension() == self._element.space_dimension()
# But for Bell, FIAT reports 21 basis functions,
# but FInAT only 18 (because there are actually 18
# basis functions, and the additional 3 are for
# dealing with transformations between physical
# and reference space).
index_shape = (self._element.space_dimension(),)
for alpha, fiat_table in fiat_result.items():
if isinstance(fiat_table, Exception):
result[alpha] = gem.Failure(self.index_shape + self.value_shape, fiat_table)
continue
derivative = sum(alpha)
table_roll = fiat_table.reshape(
space_dimension, value_size, len(ps.points)
).transpose(1, 2, 0)
exprs = []
for table in table_roll:
if derivative < self.degree:
point_indices = ps.indices
point_shape = tuple(index.extent for index in point_indices)
exprs.append(gem.partial_indexed(
gem.Literal(table.reshape(point_shape + index_shape)),
point_indices
))
elif derivative == self.degree:
# Make sure numerics satisfies theory
exprs.append(gem.Literal(table[0]))
else:
# Make sure numerics satisfies theory
assert np.allclose(table, 0.0)
exprs.append(gem.Zero(self.index_shape))
if self.value_shape:
beta = self.get_indices()
zeta = self.get_value_indices()
result[alpha] = gem.ComponentTensor(
gem.Indexed(
gem.ListTensor(np.array(
[gem.Indexed(expr, beta) for expr in exprs]
).reshape(self.value_shape)),
zeta),
beta + zeta
)
else:
expr, = exprs
result[alpha] = expr
return result
def promote(table):
v = gem.partial_indexed(table, beta)
u = gem.ListTensor(self._transform(v))
return gem.ComponentTensor(gem.Indexed(u, zeta), beta + zeta)
def cell_size(self, restriction):
f = {None: (), '+': (0, ), '-': (1, )}[restriction]
# cell_sizes expression must have been set up by now.
return gem.partial_indexed(self._cell_sizes, f)
def translate_cell_coordinate(terminal, mt, params):
return gem.partial_indexed(
params.index_selector(lambda i: gem.Literal(params.entity_points[i]), mt.restriction), (params.point_index,)
)
def translate_facet_coordinate(terminal, mt, params):
assert params.integration_dim != params.fiat_cell.get_dimension()
points = params.points
return gem.partial_indexed(gem.Literal(points), (params.point_index,))
def cell_size(self, restriction):
if not hasattr(self, "_cell_sizes"):
raise RuntimeError("Haven't called set_cell_sizes")
f = {None: (), '+': (0, ), '-': (1, )}[restriction]
# cell_sizes expression must have been set up by now.
return gem.partial_indexed(self._cell_sizes, f)
def expression(self):
return gem.partial_indexed(gem.Literal(self.points), self.indices)