from __future__ import print_function
import numpy as np
import scipy.sparse as sp
from discretize.utils import mkvc, sdiag
from discretize import utils
from discretize import TensorMesh, CurvilinearMesh, CylMesh
from discretize.utils.codeutils import requires
try:
from discretize.TreeMesh import TreeMesh as Tree
except ImportError as e:
Tree = None
import unittest
import inspect
# matplotlib is a soft dependencies for discretize
try:
import matplotlib
import matplotlib.pyplot as plt
except ImportError:
matplotlib = False
try:
import getpass
name = getpass.getuser()[0].upper() + getpass.getuser()[1:]
except Exception as e:
name = 'You'
happiness = [
'The test be workin!', 'You get a gold star!', 'Yay passed!',
'Happy little convergence test!', 'That was easy!',
'Testing is important.', 'You are awesome.', 'Go Test Go!',
'Once upon a time, a happy little test passed.',
'And then everyone was happy.','Not just a pretty face '+name,
'You deserve a pat on the back!', 'Well done '+name+'!',
'Awesome, '+name+', just awesome.'
]
sadness = [
'No gold star for you.', 'Try again soon.',
'Thankfully, persistence is a great substitute for talent.',
'It might be easier to call this a feature...', 'Coffee break?',
'Boooooooo :(', 'Testing is important. Do it again.',
"Did you put your clever trousers on today?",
'Just think about a dancing dinosaur and life will get better!',
'You had so much promise '+name+', oh well...', name.upper()+' ERROR!',
'Get on it '+name+'!', 'You break it, you fix it.'
]
def setupMesh(meshType, nC, nDim):
"""
For a given number of cells nc, generate a TensorMesh with uniform
cells with edge length h=1/nc.
"""
if 'TensorMesh' in meshType:
if 'uniform' in meshType:
h = [nC, nC, nC]
elif 'random' in meshType:
h1 = np.random.rand(nC)*nC*0.5 + nC*0.5
h2 = np.random.rand(nC)*nC*0.5 + nC*0.5
h3 = np.random.rand(nC)*nC*0.5 + nC*0.5
h = [hi/np.sum(hi) for hi in [h1, h2, h3]] # normalize
else:
raise Exception('Unexpected meshType')
mesh = TensorMesh(h[:nDim])
max_h = max([np.max(hi) for hi in mesh.h])
elif 'CylMesh' in meshType:
if 'uniform' in meshType:
h = [nC, nC, nC]
elif 'random' in meshType:
h1 = np.random.rand(nC)*nC*0.5 + nC*0.5
h2 = np.random.rand(nC)*nC*0.5 + nC*0.5
h3 = np.random.rand(nC)*nC*0.5 + nC*0.5
h = [hi/np.sum(hi) for hi in [h1, h2, h3]] # normalize
h[1] = h[1]*2*np.pi
else:
raise Exception('Unexpected meshType')
if nDim == 2:
mesh = CylMesh([h[0], 1, h[2]])
max_h = max([np.max(hi) for hi in [mesh.hx, mesh.hz]])
elif nDim == 3:
mesh = CylMesh(h)
max_h = max([np.max(hi) for hi in mesh.h])
elif 'Curv' in meshType:
if 'uniform' in meshType:
kwrd = 'rect'
elif 'rotate' in meshType:
kwrd = 'rotate'
else:
raise Exception('Unexpected meshType')
if nDim == 1:
raise Exception('Lom not supported for 1D')
elif nDim == 2:
X, Y = utils.exampleLrmGrid([nC, nC], kwrd)
mesh = CurvilinearMesh([X, Y])
elif nDim == 3:
X, Y, Z = utils.exampleLrmGrid([nC, nC, nC], kwrd)
mesh = CurvilinearMesh([X, Y, Z])
max_h = 1./nC
elif 'Tree' in meshType:
if Tree is None:
raise Exception(
"Tree Mesh not installed. Run 'python setup.py install'"
)
nC *= 2
if 'uniform' in meshType or 'notatree' in meshType:
h = [nC, nC, nC]
elif 'random' in meshType:
h1 = np.random.rand(nC)*nC*0.5 + nC*0.5
h2 = np.random.rand(nC)*nC*0.5 + nC*0.5
h3 = np.random.rand(nC)*nC*0.5 + nC*0.5
h = [hi/np.sum(hi) for hi in [h1, h2, h3]] # normalize
else:
raise Exception('Unexpected meshType')
levels = int(np.log(nC)/np.log(2))
mesh = Tree(h[:nDim], levels=levels)
def function(cell):
if 'notatree' in meshType:
return levels - 1
r = cell.center - 0.5
dist = np.sqrt(r.dot(r))
if dist < 0.2:
return levels
return levels - 1
mesh.refine(function)
# mesh.number()
# mesh.plotGrid(show_it=True)
max_h = max([np.max(hi) for hi in mesh.h])
return mesh, max_h
class OrderTest(unittest.TestCase):
"""
OrderTest is a base class for testing convergence orders with respect to mesh
sizes of integral/differential operators.
Mathematical Problem:
Given are an operator A and its discretization A[h]. For a given test function f
and h --> 0 we compare:
.. math::
error(h) = \| A[h](f) - A(f) \|_{\infty}
Note that you can provide any norm.
Test is passed when estimated rate order of convergence is at least within the specified tolerance of the
estimated rate supplied by the user.
Minimal example for a curl operator::
class TestCURL(OrderTest):
name = "Curl"
def getError(self):
# For given Mesh, generate A[h], f and A(f) and return norm of error.
fun = lambda x: np.cos(x) # i (cos(y)) + j (cos(z)) + k (cos(x))
sol = lambda x: np.sin(x) # i (sin(z)) + j (sin(x)) + k (sin(y))
Ex = fun(self.M.gridEx[:, 1])
Ey = fun(self.M.gridEy[:, 2])
Ez = fun(self.M.gridEz[:, 0])
f = np.concatenate((Ex, Ey, Ez))
Fx = sol(self.M.gridFx[:, 2])
Fy = sol(self.M.gridFy[:, 0])
Fz = sol(self.M.gridFz[:, 1])
Af = np.concatenate((Fx, Fy, Fz))
# Generate DIV matrix
Ah = self.M.edgeCurl
curlE = Ah*E
err = np.linalg.norm((Ah*f -Af), np.inf)
return err
def test_order(self):
# runs the test
self.orderTest()
See also: test_operatorOrder.py
"""
name = "Order Test"
expectedOrders = 2. # This can be a list of orders, must be the same length as meshTypes
tolerance = 0.85 # This can also be a list, must be the same length as meshTypes
meshSizes = [4, 8, 16, 32]
meshTypes = ['uniformTensorMesh']
_meshType = meshTypes[0]
meshDimension = 3
def setupMesh(self, nC):
mesh, max_h = setupMesh(self._meshType, nC, self.meshDimension)
self.M = mesh
return max_h
def getError(self):
"""For given h, generate A[h], f and A(f) and return norm of error."""
return 1.
def orderTest(self):
"""
For number of cells specified in meshSizes setup mesh, call getError
and prints mesh size, error, ratio between current and previous error,
and estimated order of convergence.
"""
if not isinstance(self.meshTypes, list):
raise TypeError('meshTypes must be a list')
if type(self.tolerance) is not list:
self.tolerance = np.ones(len(self.meshTypes))*self.tolerance
# if we just provide one expected order, repeat it for each mesh type
if type(self.expectedOrders) == float or type(self.expectedOrders) == int:
self.expectedOrders = [self.expectedOrders for i in self.meshTypes]
if isinstance(self.expectedOrders, np.ndarray):
self.expectedOrders = list(self.expectedOrders)
assert type(self.expectedOrders) == list, 'expectedOrders must be a list'
assert len(self.expectedOrders) == len(self.meshTypes), 'expectedOrders must have the same length as the meshTypes'
for ii_meshType, meshType in enumerate(self.meshTypes):
self._meshType = meshType
self._tolerance = self.tolerance[ii_meshType]
self._expectedOrder = self.expectedOrders[ii_meshType]
order = []
err_old = 0.
max_h_old = 0.
for ii, nc in enumerate(self.meshSizes):
max_h = self.setupMesh(nc)
err = self.getError()
if ii == 0:
print('')
print(self._meshType + ': ' + self.name)
print('_____________________________________________')
print(' h | error | e(i-1)/e(i) | order')
print('~~~~~~|~~~~~~~~~~~~~|~~~~~~~~~~~~~|~~~~~~~~~~')
print('{0:4d} | {1:8.2e} |'.format(nc, err))
else:
order.append(np.log(err/err_old)/np.log(max_h/max_h_old))
print('{0:4d} | {1:8.2e} | {2:6.4f} | {3:6.4f}'.format(nc, err, err_old/err, order[-1]))
err_old = err
max_h_old = max_h
print('---------------------------------------------')
passTest = np.mean(np.array(order)) > self._tolerance*self._expectedOrder
if passTest:
print(happiness[np.random.randint(len(happiness))])
else:
print('Failed to pass test on ' + self._meshType + '.')
print(sadness[np.random.randint(len(sadness))])
print('')
self.assertTrue(passTest)
def Rosenbrock(x, return_g=True, return_H=True):
"""Rosenbrock function for testing GaussNewton scheme"""
f = 100*(x[1]-x[0]**2)**2+(1-x[0])**2
g = np.array([2*(200*x[0]**3-200*x[0]*x[1]+x[0]-1), 200*(x[1]-x[0]**2)])
H = sp.csr_matrix(np.array([[-400*x[1]+1200*x[0]**2+2, -400*x[0]], [-400*x[0], 200]]))
out = (f,)
if return_g:
out += (g,)
if return_H:
out += (H,)
return out if len(out) > 1 else out[0]
def checkDerivative(fctn, x0, num=7, plotIt=True, dx=None, expectedOrder=2, tolerance=0.85, eps=1e-10, ax=None):
"""
Basic derivative check
Compares error decay of 0th and 1st order Taylor approximation at point
x0 for a randomized search direction.
:param callable fctn: function handle
:param numpy.ndarray x0: point at which to check derivative
:param int num: number of times to reduce step length, h
:param bool plotIt: if you would like to plot
:param numpy.ndarray dx: step direction
:param int expectedOrder: The order that you expect the derivative to yield.
:param float tolerance: The tolerance on the expected order.
:param float eps: What is zero?
:rtype: bool
:return: did you pass the test?!
.. plot::
:include-source:
from discretize import Tests, utils
import numpy as np
def simplePass(x):
return np.sin(x), utils.sdiag(np.cos(x))
Tests.checkDerivative(simplePass, np.random.randn(5))
"""
print("{0!s} checkDerivative {1!s}".format('='*20, '='*20))
print("iter h |ft-f0| |ft-f0-h*J0*dx| Order\n{0!s}".format(('-'*57)))
f0, J0 = fctn(x0)
x0 = mkvc(x0)
if dx is None:
dx = np.random.randn(len(x0))
h = np.logspace(-1, -num, num)
E0 = np.ones(h.shape)
E1 = np.ones(h.shape)
def l2norm(x):
# because np.norm breaks if they are scalars?
return np.sqrt(np.real(np.vdot(x, x)))
for i in range(num):
# Evaluate at test point
ft, Jt = fctn( x0 + h[i]*dx )
# 0th order Taylor
E0[i] = l2norm( ft - f0 )
# 1st order Taylor
if inspect.isfunction(J0):
E1[i] = l2norm( ft - f0 - h[i]*J0(dx) )
else:
# We assume it is a numpy.ndarray
E1[i] = l2norm( ft - f0 - h[i]*J0.dot(dx) )
order0 = np.log10(E0[:-1]/E0[1:])
order1 = np.log10(E1[:-1]/E1[1:])
print(" {0:d} {1:1.2e} {2:1.3e} {3:1.3e} {4:1.3f}".format(i, h[i], E0[i], E1[i], np.nan if i == 0 else order1[i-1]))
# Ensure we are about precision
order0 = order0[E0[1:] > eps]
order1 = order1[E1[1:] > eps]
belowTol = (order1.size == 0 and order0.size >= 0)
# Make sure we get the correct order
correctOrder = order1.size > 0 and np.mean(order1) > tolerance * expectedOrder
passTest = belowTol or correctOrder
if passTest:
print("{0!s} PASS! {1!s}".format('='*25, '='*25))
print(happiness[np.random.randint(len(happiness))]+'\n')
else:
print("{0!s}\n{1!s} FAIL! {2!s}\n{3!s}".format('*'*57, '<'*25, '>'*25, '*'*57))
print(sadness[np.random.randint(len(sadness))]+'\n')
@requires({'matplotlib': matplotlib})
def plot_it(ax):
if plotIt:
if ax is None:
ax = plt.subplot(111)
ax.loglog(h, E0, 'b')
ax.loglog(h, E1, 'g--')
ax.set_title(
'Check Derivative - {0!s}'.format(('PASSED :)'
if passTest else 'FAILED :('))
)
ax.set_xlabel('h')
ax.set_ylabel('Error')
leg = ax.legend(
['$\mathcal{O}(h)$', '$\mathcal{O}(h^2)$'], loc='best',
title="$f(x + h\Delta x) - f(x) - h g(x) \Delta x - \mathcal{O}(h^2) = 0$",
frameon=False)
plt.setp(leg.get_title(), fontsize=15)
plt.show()
plot_it(ax)
return passTest
def getQuadratic(A, b, c=0):
"""
Given A, b and c, this returns a quadratic, Q
.. math::
\mathbf{Q( x ) = 0.5 x A x + b x} + c
"""
def Quadratic(x, return_g=True, return_H=True):
f = 0.5 * x.dot( A.dot(x)) + b.dot( x ) + c
out = (f,)
if return_g:
g = A.dot(x) + b
out += (g,)
if return_H:
H = A
out += (H,)
return out if len(out) > 1 else out[0]
return Quadratic
