def warp_factor(n, output_nodes, scaled=True):
"""Compute warp function at order *n* and evaluate it at
the nodes *output_nodes*.
"""
from modepy.quadrature.jacobi_gauss import legendre_gauss_lobatto_nodes
warped_nodes = legendre_gauss_lobatto_nodes(n)
equi_nodes = np.linspace(-1, 1, n + 1)
from modepy.matrices import vandermonde
from modepy.modes import simplex_onb
basis = simplex_onb(1, n)
Veq = vandermonde(basis, equi_nodes) # noqa
# create interpolator from equi_nodes to output_nodes
eq_to_out = la.solve(Veq.T, vandermonde(basis, output_nodes).T).T
# compute warp factor
warp = np.dot(eq_to_out, warped_nodes - equi_nodes)
if scaled:
zerof = (abs(output_nodes) < 1.0 - 1.0e-10)
sf = 1.0 - (zerof * output_nodes)**2
warp = warp / sf + warp * (zerof - 1)
return warp
def warp_factor(n, output_nodes, scaled=True):
"""Compute warp function at order *n* and evaluate it at
the nodes *output_nodes*.
"""
from modepy.quadrature.jacobi_gauss import legendre_gauss_lobatto_nodes
warped_nodes = legendre_gauss_lobatto_nodes(n)
equi_nodes = np.linspace(-1, 1, n+1)
from modepy.matrices import vandermonde
from modepy.modes import simplex_onb
basis = simplex_onb(1, n)
Veq = vandermonde(basis, equi_nodes) # noqa
# create interpolator from equi_nodes to output_nodes
eq_to_out = la.solve(Veq.T, vandermonde(basis, output_nodes).T).T
# compute warp factor
warp = np.dot(eq_to_out, warped_nodes - equi_nodes)
if scaled:
zerof = (abs(output_nodes) < 1.0-1.0e-10)
sf = 1.0 - (zerof*output_nodes)**2
warp = warp/sf + warp*(zerof-1)
return warp
def _evaluate_lebesgue_function(n, nodes, domain):
dims = len(nodes)
huge_n = 30 * n
if domain == "simplex":
from modepy.modes import simplex_onb as domain_basis_onb
from pytools import (
generate_nonnegative_integer_tuples_summing_to_at_most as
generate_node_tuples)
elif domain == "hypercube":
from modepy.modes import (legendre_tensor_product_basis as
domain_basis_onb)
from pytools import (generate_nonnegative_integer_tuples_below as
generate_node_tuples)
else:
raise ValueError(f"unknown domain: '{domain}'")
basis = domain_basis_onb(dims, n)
equi_node_tuples = list(generate_node_tuples(huge_n, dims))
equi_nodes = (np.array(equi_node_tuples, dtype=np.float64) / huge_n * 2 -
1).T
from modepy.matrices import vandermonde
vdm = vandermonde(basis, nodes)
eq_vdm = vandermonde(basis, equi_nodes)
eq_to_out = la.solve(vdm.T, eq_vdm.T).T
lebesgue_worst = np.sum(np.abs(eq_to_out), axis=1)
return lebesgue_worst, equi_node_tuples, equi_nodes
def estimate_lebesgue_constant(n, nodes, visualize=False):
"""Estimate the
`Lebesgue constant
<https://en.wikipedia.org/wiki/Lebesgue_constant_(interpolation)>`_
of the *nodes* at polynomial order *n*.
:arg nodes: an array of shape *(dims, nnodes)* as returned by
:func:`modepy.warp_and_blend_nodes`.
:arg visualize: visualize the function that gives rise to the
returned Lebesgue constant. (2D only for now)
:return: the Lebesgue constant, a scalar
.. versionadded:: 2013.2
"""
from modepy.matrices import vandermonde
from modepy.modes import simplex_onb
dims = len(nodes)
basis = simplex_onb(dims, n)
vdm = vandermonde(basis, nodes)
from pytools import generate_nonnegative_integer_tuples_summing_to_at_most \
as gnitstam
huge_n = 30*n
equi_node_tuples = list(gnitstam(huge_n, dims))
tons_of_equi_nodes = (
np.array(equi_node_tuples, dtype=np.float64)
/ huge_n * 2 - 1).T
eq_vdm = vandermonde(basis, tons_of_equi_nodes)
eq_to_out = la.solve(vdm.T, eq_vdm.T).T
lebesgue_worst = np.sum(np.abs(eq_to_out), axis=1)
lebesgue_constant = np.max(lebesgue_worst)
if visualize:
print("Lebesgue constant: %g" % lebesgue_constant)
from modepy.tools import submesh
import mayavi.mlab as mlab
mlab.figure(bgcolor=(1, 1, 1))
mlab.triangular_mesh(
tons_of_equi_nodes[0],
tons_of_equi_nodes[1],
lebesgue_worst / lebesgue_constant,
submesh(equi_node_tuples))
x, y = np.mgrid[-1:1:20j, -1:1:20j]
mlab.mesh(x, y, 0*x, representation="wireframe", color=(0.4, 0.4, 0.4),
line_width=0.6)
mlab.show()
return lebesgue_constant
def simplex_interp_error_coefficient_estimator_matrix(unit_nodes, order,
n_tail_orders):
"""Return a matrix :math:`C` that, when multiplied by a vector of nodal values,
yields the coeffiicients belonging to the basis functions of the *n_tail_orders*
highest orders.
The 2-norm of the resulting coefficents can be used as an estimate of the
interpolation error.
.. versionadded:: 2018.1
"""
from modepy.matrices import vandermonde
from modepy.modes import simplex_onb_with_mode_ids
dim, nunit_nodes = unit_nodes.shape
mode_ids, basis = simplex_onb_with_mode_ids(dim, order)
vdm = vandermonde(basis, unit_nodes)
vdm_inv = la.inv(vdm)
order_vector = np.array([sum(mode_id) for mode_id in mode_ids])
max_order = np.max(order_vector)
assert max_order == order
return vdm_inv[order_vector > max_order - n_tail_orders]
def simplex_interp_error_coefficient_estimator_matrix(
unit_nodes, order, n_tail_orders):
"""Return a matrix :math:`C` that, when multiplied by a vector of nodal values,
yields the coeffiicients belonging to the basis functions of the *n_tail_orders*
highest orders.
The 2-norm of the resulting coefficents can be used as an estimate of the
interpolation error.
.. versionadded:: 2018.1
"""
from modepy.matrices import vandermonde
from modepy.modes import simplex_onb_with_mode_ids
dim, nunit_nodes = unit_nodes.shape
mode_ids, basis = simplex_onb_with_mode_ids(dim, order)
vdm = vandermonde(basis, unit_nodes)
vdm_inv = la.inv(vdm)
order_vector = np.array([sum(mode_id) for mode_id in mode_ids])
max_order = np.max(order_vector)
assert max_order == order
return vdm_inv[order_vector > max_order-n_tail_orders]
def warp_and_refine_until_resolved(unwarped_mesh_or_refiner, warp_callable,
est_rel_interp_tolerance):
"""Given an original ("unwarped") :class:`meshmode.mesh.Mesh` and a
warping function *warp_callable* that takes and returns a mesh and a
tolerance to which the mesh should be resolved by the mapping polynomials,
this function will iteratively refine the *unwarped_mesh* until relative
interpolation error estimates on the warped version are smaller than
*est_rel_interp_tolerance* on each element.
:returns: The refined, unwarped mesh.
.. versionadded:: 2018.1
"""
from modepy.modes import simplex_onb
from modepy.matrices import vandermonde
from modepy.modal_decay import simplex_interp_error_coefficient_estimator_matrix
from meshmode.mesh.refinement import Refiner, RefinerWithoutAdjacency
if isinstance(unwarped_mesh_or_refiner,
(Refiner, RefinerWithoutAdjacency)):
refiner = unwarped_mesh_or_refiner
unwarped_mesh = refiner.get_current_mesh()
else:
unwarped_mesh = unwarped_mesh_or_refiner
refiner = Refiner(unwarped_mesh)
iteration = 0
while True:
refine_flags = np.zeros(unwarped_mesh.nelements, dtype=bool)
warped_mesh = warp_callable(unwarped_mesh)
# test whether there are invalid values in warped mesh
if not np.isfinite(warped_mesh.vertices).all():
raise FloatingPointError(
"Warped mesh contains non-finite vertices "
"(NaN or Inf)")
for group in warped_mesh.groups:
if not np.isfinite(group.nodes).all():
raise FloatingPointError(
"Warped mesh contains non-finite nodes "
"(NaN or Inf)")
for egrp in warped_mesh.groups:
dim, nunit_nodes = egrp.unit_nodes.shape
interp_err_est_mat = simplex_interp_error_coefficient_estimator_matrix(
egrp.unit_nodes,
egrp.order,
n_tail_orders=1 if warped_mesh.dim > 1 else 2)
vdm_inv = la.inv(
vandermonde(simplex_onb(dim, egrp.order), egrp.unit_nodes))
mapping_coeffs = np.einsum("ij,dej->dei", vdm_inv, egrp.nodes)
mapping_norm_2 = np.sqrt(np.sum(mapping_coeffs**2, axis=-1))
interp_error_coeffs = np.einsum("ij,dej->dei", interp_err_est_mat,
egrp.nodes)
interp_error_norm_2 = np.sqrt(
np.sum(interp_error_coeffs**2, axis=-1))
# max over dimensions
est_rel_interp_error = np.max(interp_error_norm_2 / mapping_norm_2,
axis=0)
refine_flags[
egrp.element_nr_base:
egrp.element_nr_base+egrp.nelements] = \
est_rel_interp_error > est_rel_interp_tolerance
nrefined_elements = np.sum(refine_flags.astype(np.int32))
if nrefined_elements == 0:
break
logger.info(
"warp_and_refine_until_resolved: "
"iteration %d -> splitting %d/%d elements", iteration,
nrefined_elements, unwarped_mesh.nelements)
unwarped_mesh = refiner.refine(refine_flags)
iteration += 1
return unwarped_mesh
def warp_and_refine_until_resolved(
unwarped_mesh_or_refiner, warp_callable, est_rel_interp_tolerance):
"""Given an original ("un-warped") :class:`meshmode.mesh.Mesh` and a
warping function *warp_callable* that takes and returns a mesh and a
tolerance to which the mesh should be resolved by the mapping polynomials,
this function will iteratively refine the *unwarped_mesh* until relative
interpolation error estimates on the warped version are smaller than
*est_rel_interp_tolerance* on each element.
:returns: The refined, un-warped mesh.
.. versionadded:: 2018.1
"""
from modepy.modes import simplex_onb
from modepy.matrices import vandermonde
from modepy.modal_decay import simplex_interp_error_coefficient_estimator_matrix
from meshmode.mesh.refinement import Refiner, RefinerWithoutAdjacency
if isinstance(unwarped_mesh_or_refiner, (Refiner, RefinerWithoutAdjacency)):
refiner = unwarped_mesh_or_refiner
unwarped_mesh = refiner.get_current_mesh()
else:
unwarped_mesh = unwarped_mesh_or_refiner
refiner = Refiner(unwarped_mesh)
iteration = 0
while True:
refine_flags = np.zeros(unwarped_mesh.nelements, dtype=np.bool)
warped_mesh = warp_callable(unwarped_mesh)
# test whether there are invalid values in warped mesh
if not np.isfinite(warped_mesh.vertices).all():
raise FloatingPointError("Warped mesh contains non-finite vertices "
"(NaN or Inf)")
for group in warped_mesh.groups:
if not np.isfinite(group.nodes).all():
raise FloatingPointError("Warped mesh contains non-finite nodes "
"(NaN or Inf)")
for egrp in warped_mesh.groups:
dim, nunit_nodes = egrp.unit_nodes.shape
interp_err_est_mat = simplex_interp_error_coefficient_estimator_matrix(
egrp.unit_nodes, egrp.order,
n_tail_orders=1 if warped_mesh.dim > 1 else 2)
vdm_inv = la.inv(
vandermonde(simplex_onb(dim, egrp.order), egrp.unit_nodes))
mapping_coeffs = np.einsum("ij,dej->dei", vdm_inv, egrp.nodes)
mapping_norm_2 = np.sqrt(np.sum(mapping_coeffs**2, axis=-1))
interp_error_coeffs = np.einsum(
"ij,dej->dei", interp_err_est_mat, egrp.nodes)
interp_error_norm_2 = np.sqrt(np.sum(interp_error_coeffs**2, axis=-1))
# max over dimensions
est_rel_interp_error = np.max(interp_error_norm_2/mapping_norm_2, axis=0)
refine_flags[
egrp.element_nr_base:
egrp.element_nr_base+egrp.nelements] = \
est_rel_interp_error > est_rel_interp_tolerance
nrefined_elements = np.sum(refine_flags.astype(np.int32))
if nrefined_elements == 0:
break
logger.info("warp_and_refine_until_resolved: "
"iteration %d -> splitting %d/%d elements",
iteration, nrefined_elements, unwarped_mesh.nelements)
unwarped_mesh = refiner.refine(refine_flags)
iteration += 1
return unwarped_mesh
