"""Numerical integration functions incl. fixed & adaptive quadrature.
This module includes numerical integration procedures to approximate
the integral of a given function on the given mesh. The functions
can be integrated using fixed Gaussian quadrature on the elements
of the mesh, or using adaptive integration either by refining mesh
elements or increasing order of quadrature based on error criteria.
"""
import numpy as np
from numpy import array
from itertools import product
from .function import MeshFunction, MeshFunction2
from .marking import fixed_ratio_marking as marked_elements
MAX_ADAPT_ITERATIONS = 20
# TODO: Modify all integration functions, so that they work with
# vector-valued and matrix-valued functions too.
[docs]def integrate(f, mesh, quadrature_degree=None):
"""Integrate a function on a mesh using fixed Gaussian quadrature.
This procedure takes a function and a mesh, and estimates the
integral of f on mesh numerically using Gaussian quadrature.
The degree of the quadrature can be specified. If it is not given,
then the degree of the function f is used if available, otherwise
quadrature degree is 2.
Parameters
----------
f : MeshFunction or function_like
An instance of MeshFunction or a function of X where X is
a dxn array of n coordinate points in d-space).
mesh : Mesh
An instance of a mesh (possibly Curve, CurveHierarchy, Domain2d),
that gives the domain of integration.
quadrature_degree : int, optional
The degree of Gaussian quadrature to be used for integration.
Returns
-------
integral : float
The computed integral of f on given mesh.
"""
if quadrature_degree is not None:
quad_degree = quadrature_degree
else:
try:
quad_degree = f.degree()
except AttributeError:
quad_degree = 2 # assume quadratic polynomials by default
quad = mesh.ref_element.quadrature( quad_degree )
try:
integral = sum(( weight * f(mesh,pt) for pt,weight in quad.iterpoints() ))
except TypeError:
f = MeshFunction(f)
integral = sum(( weight * f(mesh,pt) for pt,weight in quad.iterpoints() ))
integral *= mesh.element_sizes()
return np.sum(integral)
[docs]def double_integrate(f, mesh1, mesh2, quadrature_degree=None):
"""Double integral of function on two meshes with Gaussian quadrature.
This procedure takes a function and a mesh, and estimates the double
integral of f on mesh1 and mesh2 numerically using Gaussian quadrature.
The degree of the quadrature can be specified. If it is not given,
then the degree of the function f is used if available, otherwise
quadrature degree is 2.
Parameters
----------
f : MeshFunction or function_like
An instance of MeshFunction or a function of X where X is
a dxn array of n coordinate points in d-space).
mesh1 : Mesh
An instance of a mesh (possibly Curve, CurveHierarchy, Domain2d),
that gives the first domain of integration.
mesh2 : Mesh
An instance of a mesh (possibly Curve, CurveHierarchy, Domain2d),
that gives the second domain of integration.
quadrature_degree : int, optional
The degree of Gaussian quadrature to be used for integration.
Returns
-------
integral : float
The computed integral of f on given mesh.
"""
if quadrature_degree is not None:
quad_degree = quadrature_degree
else:
try:
quad_degree = f.degree()
except AttributeError:
quad_degree = 2 # assume quadratic polynomials by default
quad1 = mesh1.ref_element.quadrature( quad_degree )
quad2 = mesh2.ref_element.quadrature( quad_degree )
quad_pts = product( quad1.iterpoints(), quad2.iterpoints() )
try:
integral = sum(( w1*w2 * f(mesh1,mesh2,pt1,pt2)
for (pt1,w1),(pt2,w2) in quad_pts ))
except TypeError:
f = MeshFunction2(f)
integral = sum(( w1*w2 * f(mesh1,mesh2,pt1,pt2)
for (pt1,w1),(pt2,w2) in quad_pts ))
integral *= np.outer( mesh1.element_sizes(), mesh2.element_sizes() )
return np.sum(integral)
[docs]def adapt_integrate_tolerance(f, mesh, quad0=None, quad1=None):
"""Estimates a minimum tolerance for adaptive integration for (f, mesh).
This procedure takes a function and a mesh, and estimates a minimum
tolerance for adaptive integration. The function is used to specify
a minimum length scale, which we do not want to overresolve in adaptive
integration. For example, if the function is based on an ImageFunction,
the ImageFunction provides its resolution and diameter, from which we can
compute the pixel size. The variations in function value at a scale of
the pixel size determines the minimum tolerance; we should not use many
elements with adaptive integration to resolve variations at this scale.
The function is sampled with small random elements and the average error
is scaled to the mesh size to obtain the tolerance.
Parameters
----------
f : MeshFunction or function_like
A function of X (a dxn array of n coordinate points in d-space) or
an instance of a MeshFunction. It should have member functions:
f.resolution(), f.diameter(), both of which are used to
figure out the smallest scale.
mesh : Mesh
An instance of a mesh (possibly Curve, CurveHierarchy, Domain2d),
whose ref_element is used to define the random elements.
quad0 : Quadrature
The less accurate quadrature. Default value is None, in which case
a quadrature of order=2 is used.
quad1 : Quadrature
The more accurate quadrature. Default value is None, in which case
a quadrature of order=quad0.order()+1 is used.
Returns
-------
tol : float
The estimated minimum tolerance for the given functions and mesh.
"""
if quad0 is None:
quad0 = mesh.ref_element.quadrature( order=2 )
if quad1 is None:
quad1 = mesh.ref_element.quadrature( order = quad0.order()+1 )
grid_size = f.resolution()
random_mesh = mesh.ref_element.random_elements( grid_size, f.diameter() )
try:
error = sum(( w * f(random_mesh,pt) for pt,w in quad0.iterpoints() ))
except TypeError:
f = MeshFunction(f)
error = sum(( w * f(random_mesh,pt) for pt,w in quad0.iterpoints() ))
error -= sum(( w * f(random_mesh,pt) for pt,w in quad1.iterpoints() ))
error[:] = np.abs( error[:] )
el_sizes = random_mesh.element_sizes()
error *= el_sizes
avg_error = np.mean( error )
avg_el_size = np.mean( el_sizes )
# At this point, we know the average error for a set of small random
# elements. From this, we want to infer the right tolerance for
# the given mesh and the function. The tolerance should be proportional
# to the total error if the mesh were made up of such small elements,
# so we multiply the average error by the ratio of the mesh area
# and the average size of the random elements.
if mesh.dim() == mesh.dim_of_world(): # domain in 2d or 3d
area = mesh.volume()
else: # if it is a curve or a surface
area = mesh.surface_area()
factor = 50.0 * area / avg_el_size
tol = factor * avg_error
return tol
[docs]def adapt_integrate(f, mesh, tol=None):
"""Adaptively integrate a function on a given mesh by refinement.
This procedure takes a function and a mesh, and estimates the integral
of f on mesh by numerical quadrature. This is done adaptively. Quadrature
error in each element is estimated by comparing the values of low-order
and high-order quadratures. If the total error is higher than tol,
the elements with highest error are refined. The refinements continue
until total error is below the prescribed tol.
Parameters
----------
f : MeshFunction or function_like
An instance of MeshFunction or a function of X where X is
a dxn array of n coordinate points in d-space).
mesh : Mesh
An instance of a mesh (possibly Curve, CurveHierarchy, Domain2d),
that gives the domain of integration. Some elements of
the mesh might be refined into smaller elements.
tol : float
A tolerance value for the total error in integration.
The default value for tol is None. If the user does not
specify a tolerance. This function tries to estimate a tol
based on minimum scale computed from f.diameter() and
f.resolution(). If this fails, tol is set to 100*eps,
where eps is the machine epsilon for floating point numbers.
Returns
-------
integral : float
The computed integral of f on mesh.
"""
# Error at each mesh element is given by the difference of
# the low order quadrature and the high order quadrature.
quad0 = mesh.ref_element.quadrature( order=2 )
quad1 = mesh.ref_element.quadrature( order=3 )
if tol is None:
try:
tol = adapt_integrate_tolerance( f, mesh, quad0, quad1 )
except:
tol = 100.0 * np.finfo(float).eps
try:
error = sum(( weight * f(mesh,pt) for pt,weight in quad0.iterpoints() ))
except TypeError:
f = MeshFunction(f)
error = sum(( weight * f(mesh,pt) for pt,weight in quad0.iterpoints() ))
integral = sum(( weight * f(mesh,pt) for pt,weight in quad1.iterpoints() ))
error -= integral
error[:] = np.abs( error[:] )
el_sizes = mesh.element_sizes()
integral *= el_sizes
error *= el_sizes
total_integral = np.sum( integral )
total_error = np.sum( error )
parameters = {'tolerance':tol, 'ratio':0.3}
n_iters = 0
while (total_error > tol) and (n_iters < MAX_ADAPT_ITERATIONS):
mask, other = marked_elements( error, parameters )
total_integral -= np.sum( integral[mask] )
total_error -= np.sum( error[mask] )
data_vecs = [ integral, error ] # Don't lose computed values.
data_vecs, mask = mesh.refine_coarsen( mask, data_vecs )
integral, error = data_vecs
error[mask] = sum(( weight * f(mesh, pt, mask)
for pt,weight in quad0.iterpoints() ))
integral[mask] = sum(( weight * f(mesh, pt, mask)
for pt,weight in quad1.iterpoints() ))
error[mask] -= integral[mask]
error[mask] = np.abs( error[mask] )
el_sizes = mesh.element_sizes()
integral[mask] *= el_sizes[mask]
error[mask] *= el_sizes[mask]
total_integral += np.sum( integral[mask] )
total_error += np.sum( error[mask] )
n_iters += 1
return (total_integral, total_error)
[docs]def adapt_order_integrate(f, mesh, tol=None):
"""Adaptively integrate a function by increasing quadrature order.
This procedure takes a function and a mesh, and estimates the integral
of f on mesh using numerical quadrature. This is done adaptively.
Quadrature error in each element is estimated by comparing the values
of low-order and high-order quadratures. If the total error is higher
than tol, a higher order quadrature is applied to the elements with
highest error in order to integrate those elements with more accuracy.
This continues until total error is below the prescribed tol.
Parameters
----------
f : MeshFunction or function_like
An instance of MeshFunction or a function of X where X is
a dxn array of n coordinate points in d-space).
mesh : Mesh
An instance of a mesh (possibly Curve, CurveHierarchy, Domain2d),
that gives the domain of integration. Some elements of
the mesh might be refined into smaller elements.
tol : float
A tolerance value for the total error in integration.
Returns
-------
integral : float
The computed integral of f on mesh.
"""
# Error at each mesh element is given by the difference of
# the low order quadrature and the high order quadrature.
get_quadrature = mesh.ref_element.quadrature # shortcut function
quad_order = 3
quad = get_quadrature( order=quad_order-1 )
try:
error = sum(( weight*f(mesh,pt) for pt,weight in quad.iterpoints() ))
except TypeError:
f = MeshFunction(f)
error = sum(( weight*f(mesh,pt) for pt,weight in quad.iterpoints() ))
quad = get_quadrature( order=quad_order )
integral = sum(( weight*f(mesh,pt) for pt,weight in quad.iterpoints() ))
error -= integral
error[:] = np.abs(error[:])
el_sizes = mesh.element_sizes()
integral *= el_sizes
error *= el_sizes
total_integral = np.sum( integral )
total_error = np.sum( error )
# Recompute the integrals of those elements using higher order quadrature.
parameters = {'tolerance':tol, 'ratio':0.3}
n_iters = 0
while (total_error > tol) and (quad_order < quad.max_order()) \
and (n_iters < MAX_ADAPT_ITERATIONS):
quad_order += 1
quad = get_quadrature( order=quad_order )
mask, other = marked_elements( error, parameters )
total_integral -= np.sum( integral[mask] )
total_error -= np.sum( error[mask] )
error[mask] = integral[mask]
integral[mask] = sum(( weight*f(mesh, pt, mask)
for pt,weight in quad.iterpoints() ))
integral[mask] *= el_sizes[mask]
error[mask] -= integral[mask]
error[mask] = np.abs( error[mask] )
total_integral += np.sum( integral[mask] )
total_error += np.sum( error[mask] )
n_iters += 1
return (total_integral, total_error)
[docs]def L2_norm(f, mesh, quadrature_degree=None):
"""L2 norm of f on mesh using fixed Gaussian quadrature.
This procedure takes a function and a mesh, and estimates the
L2 norm of f on mesh numerically using Gaussian quadrature.
The degree of the quadrature can be specified. If it is not given,
then the degree of the function f is used if available, otherwise
quadrature degree is 2.
Parameters
----------
f : MeshFunction or function_like
An instance of MeshFunction or a function of X where X is
a dxn array of n coordinate points in d-space).
mesh : Mesh
An instance of a mesh (possibly Curve, CurveHierarchy, Domain2d),
that gives the domain of integration.
quadrature_degree : int, optional
The degree of Gaussian quadrature to be used for integration.
Returns
-------
L2_norm_est : float
The estimated L2 norm of f on given mesh.
"""
if quadrature_degree is not None:
quad_degree = quadrature_degree
else:
try:
quad_degree = 2*f.degree()
except AttributeError:
quad_degree = 4 # default: square of quadratic polynomials
quad = mesh.ref_element.quadrature( quad_degree )
try:
integral = sum(( weight*f(mesh,pt)**2 for pt,weight in quad.iterpoints() ))
except TypeError:
f = MeshFunction(f)
integral = sum(( weight*f(mesh,pt)**2 for pt,weight in quad.iterpoints() ))
integral *= mesh.element_sizes()
return np.sum(integral)**0.5
[docs]def max_norm(f, mesh, quadrature_degree=None):
"""Max norm of f on mesh using fixed Gaussian quadrature.
This procedure takes a function and a mesh, and estimates the
maximum norm of f on mesh numerically using Gaussian quadrature.
The degree of the quadrature can be specified. If it is not given,
then the degree of the function f is used if available, otherwise
quadrature degree is 2.
Parameters
----------
f : MeshFunction or function_like
An instance of MeshFunction or a function of X where X is
a dxn array of n coordinate points in d-space).
mesh : Mesh
An instance of a mesh (possibly Curve, CurveHierarchy, Domain2d),
that gives the domain of integration.
quadrature_degree : int, optional
The degree of Gaussian quadrature to be used for integration.
Returns
-------
max_norm_est : float
The estimated maximum norm of f on given mesh.
"""
if quadrature_degree is not None:
quad_degree = quadrature_degree
else:
try:
quad_degree = 2*f.degree()
except AttributeError:
quad_degree = 4 # default: square of quadratic polynomials
quad = mesh.ref_element.quadrature( quad_degree )
try:
max_val = max(( np.max(np.abs(f(mesh,pt))) for pt in quad.points() ))
except TypeError:
f = MeshFunction(f)
max_val = max(( np.max(np.abs(f(mesh,pt))) for pt in quad.points() ))
return max_val
[docs]def L2_error(f1, f2, mesh, quadrature_degree=None):
"""L2 error between f1, f2 on mesh using fixed Gaussian quadrature.
This procedure takes a function and a mesh, and estimates the L2 error
between f1 and f2 on mesh numerically using Gaussian quadrature.
The degree of the quadrature can be specified. If it is not given,
then the maximum of the degree of the functions f1 and f2 is used if
their degrees are available, otherwise quadrature degree is 2.
Parameters
----------
f1 : MeshFunction or function_like
An instance of MeshFunction or a function of X where X is
a dxn array of n coordinate points in d-space).
f2 : MeshFunction or function_like
An instance of MeshFunction or a function of X where X is
a dxn array of n coordinate points in d-space).
mesh : Mesh
An instance of a mesh (possibly Curve, CurveHierarchy, Domain2d),
that gives the domain of integration.
quadrature_degree : int, optional
The degree of Gaussian quadrature to be used for integration.
Returns
-------
L2_error_est : float
The estimated L2 error between f1 and f2 on given mesh.
"""
if quadrature_degree is not None:
quad_degree = quadrature_degree
else:
try:
quad_degree = 2*max( f1.degree(), f2.degree() )
except AttributeError:
quad_degree = 4 # assume quadratic polynomials by default
quad = mesh.ref_element.quadrature( quad_degree )
integral = np.zeros(mesh.size())
for pt, weight in quad.iterpoints():
try:
F1 = f1( mesh, pt )
except TypeError as e:
f1 = MeshFunction(f1)
F1 = f1( mesh, pt )
try:
F2 = f2( mesh, pt )
except TypeError:
f2 = MeshFunction(f2)
F2 = f2( mesh, pt )
diff = (F1 - F2)**2
if diff.ndim == 1: # the case of scalar-valued mesh function
integral += weight * diff
elif diff.ndim == 2: # the case of vector-valued mesh function
integral += weight * np.sum( diff, 0 )
elif diff.ndim == 3: # the case of matrix-valued mesh function
integral += weight * np.sum(np.sum( diff, 0 ), 0)
integral *= mesh.element_sizes()
return np.sum(integral)**0.5
[docs]def max_error(f1, f2, mesh, quadrature_degree=None):
"""Max error between f1, f2 on mesh using fixed Gaussian quadrature.
This procedure takes a function and a mesh, and estimates the maximum
error between f1 and f2 on mesh numerically using Gaussian quadrature.
The degree of the quadrature can be specified. If it is not given,
then the maximum of the degree of the functions f1 and f2 is used if
their degrees are available, otherwise quadrature degree is 2.
Parameters
----------
f1 : MeshFunction or function_like
An instance of MeshFunction or a function of X where X is
a dxn array of n coordinate points in d-space).
f2 : MeshFunction or function_like
An instance of MeshFunction or a function of X where X is
a dxn array of n coordinate points in d-space).
mesh : Mesh
An instance of a mesh (possibly Curve, CurveHierarchy, Domain2d),
that gives the domain of integration.
quadrature_degree : int, optional
The degree of Gaussian quadrature to be used for integration.
Returns
-------
max_error_est : float
The estimated L2 error between f1 and f2 on given mesh.
"""
if quadrature_degree is not None:
quad_degree = quadrature_degree
else:
try:
quad_degree = max( f1.degree(), f2.degree() )
except AttributeError:
quad_degree = 2 # assume quadratic polynomials by default
quad = mesh.ref_element.quadrature( quad_degree )
max_error = 0.0
for pt in quad.points():
try:
F1 = f1( mesh, pt )
except TypeError:
f1 = MeshFunction(f1)
F1 = f1( mesh, pt )
try:
F2 = f2( mesh, pt )
except TypeError:
f2 = MeshFunction(f2)
F2 = f2( mesh, pt )
diff = np.abs(F1 - F2)
max_error = max( max_error, np.max( diff ) )
return max_error
[docs]class Quadrature(object):
def __init__(self):
self._degree = 0
self._order = 0
self._points = np.array([])
self._weights = np.array([])
def __len__(self):
return len(self._points)
[docs] def order(self):
return self._order
[docs] def max_order(self):
raise NotImplementedError
[docs] def degree(self):
return self._degree
[docs] def max_degree(self):
raise NotImplementedError
[docs] def points(self):
return self._points
[docs] def weights(self):
return self._weights
QUADRATURE1D_MAX_ORDER = 19
QUADRATURE1D_MAX_DEGREE = 2*QUADRATURE1D_MAX_ORDER - 1
[docs]class Quadrature1d(Quadrature):
def __init__(self, degree=None, order=None):
if degree is not None:
if degree < 0:
raise ValueError("Minimum degree is 0.")
elif degree > QUADRATURE1D_MAX_DEGREE:
raise ValueError("Degree cannot be larger than %d." % QUADRATURE1D_MAX_DEGREE)
self._degree = degree
self._order = degree/2
elif order is not None:
if order < 0:
raise ValueError("Minimum order is 0.")
elif order > QUADRATURE1D_MAX_ORDER:
raise ValueError("Order cannot be larger than %d." % QUADRATURE1D_MAX_ORDER)
self._order = order
self._degree = 2*order + 1
else: # both order and degree are None
raise ValueError("Either degree or order should be specified.")
self._points = _quadrature1d_points[self._order]
self._weights = _quadrature1d_weights[self._order]
[docs] def integrate(self,fct):
return np.dot( self._weights, fct(self._points) )
[docs] def max_order(self):
return QUADRATURE1D_MAX_ORDER
[docs] def max_degree(self):
return QUADRATURE1D_MAX_DEGREE
[docs] def iterpoints(self):
return zip( self._points, self._weights )
QUADRATURE2D_MAX_ORDER = 19
QUADRATURE2D_MAX_DEGREE = QUADRATURE2D_MAX_ORDER
[docs]class Quadrature2d(Quadrature):
def __init__(self, degree=None, order=None):
if degree is not None:
if degree < 0:
raise ValueError("Minimum degree is 0.")
elif degree > QUADRATURE2D_MAX_DEGREE:
raise ValueError("Degree cannot be larger than %d." % QUADRATURE2D_MAX_DEGREE)
self._degree = degree
self._order = max(0,degree-1)
elif order is not None:
if order < 0:
raise ValueError("Minimum order is 0.")
elif order > QUADRATURE2D_MAX_ORDER:
raise ValueError("Order cannot be larger than %d." % QUADRATURE2D_MAX_ORDER)
self._order = order
self._degree = order+1
else: # both order and degree are None
raise ValueError("Either degree or order should be specified.")
self._points = _quadrature2d_points[self._order]
self._weights = _quadrature2d_weights[self._order]
[docs] def integrate(self,fct):
return 0.5 * np.dot( self._weights, fct(self._points) )
[docs] def max_order(self):
return QUADRATURE2D_MAX_ORDER
[docs] def max_degree(self):
return QUADRATURE2D_MAX_DEGREE
[docs] def iterpoints(self):
weights = self._weights
pts = self._points
n = len(weights)
return ( ( (pts[0,k],pts[1,k]), weights[k] ) for k in range(n) )
_quadrature1d_points = [
array([ 0.5 ]),
array([ 0.2113248654051871, 0.7886751345948129 ]),
array([ 0.1127016653792583, 0.5, 0.8872983346207417 ]),
array([ 0.06943184420297371, 0.3300094782075719, 0.6699905217924281,
0.9305681557970262 ]),
array([ 0.04691007703066802, 0.2307653449471584, 0.5,
0.7692346550528415, 0.9530899229693319 ]),
array([ 0.03376524289842397, 0.1693953067668678, 0.3806904069584016,
0.6193095930415985, 0.8306046932331322, 0.966234757101576 ]),
array([ 0.02544604382862076, 0.1292344072003028, 0.2970774243113014,
0.5, 0.7029225756886985, 0.8707655927996972,
0.9745539561713792 ]),
array([ 0.01985507175123191, 0.1016667612931866, 0.2372337950418355,
0.4082826787521751, 0.5917173212478249, 0.7627662049581645,
0.8983332387068134, 0.9801449282487681 ]),
array([ 0.01591988024618696, 0.08198444633668212, 0.1933142836497048,
0.3378732882980955, 0.5, 0.6621267117019045,
0.8066857163502952, 0.9180155536633179, 0.984080119753813 ]),
array([ 0.01304673574141413, 0.06746831665550773, 0.1602952158504878,
0.2833023029353764, 0.4255628305091844, 0.5744371694908156,
0.7166976970646236, 0.8397047841495122, 0.9325316833444923,
0.9869532642585859 ]),
array([ 0.01088567092697151, 0.05646870011595234, 0.1349239972129753,
0.2404519353965941, 0.3652284220238275, 0.5,
0.6347715779761725, 0.7595480646034058, 0.8650760027870247,
0.9435312998840477, 0.9891143290730284 ]),
array([ 0.009219682876640378, 0.04794137181476255, 0.1150486629028477,
0.2063410228566913, 0.3160842505009099, 0.4373832957442655,
0.5626167042557344, 0.6839157494990901, 0.7936589771433087,
0.8849513370971523, 0.9520586281852375, 0.9907803171233596 ]),
array([ 0.007908472640705932, 0.04120080038851104, 0.09921095463334506,
0.1788253302798299, 0.2757536244817765, 0.3847708420224326,
0.5, 0.6152291579775674, 0.7242463755182235,
0.8211746697201701, 0.9007890453666549, 0.958799199611489,
0.9920915273592941 ]),
array([ 0.006858095651593843, 0.03578255816821324, 0.08639934246511749,
0.1563535475941573, 0.242375681820923, 0.3404438155360551,
0.4459725256463282, 0.5540274743536718, 0.6595561844639448,
0.757624318179077, 0.8436464524058427, 0.9136006575348825,
0.9642174418317868, 0.9931419043484062 ]),
array([ 0.006003740989757311, 0.03136330379964702, 0.0758967082947864,
0.137791134319915, 0.2145139136957306, 0.3029243264612183,
0.3994029530012827, 0.5, 0.6005970469987173,
0.6970756735387817, 0.7854860863042694, 0.862208865680085,
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