def test_rotation_invariance(self):
"""test invariance to theta when epsilon=1.0"""
for type in ['FD', 'FE']:
expected = diffusion_stencil_2d(epsilon=1.0, theta=0.0, type=type)
for theta in [np.pi/8, np.pi/4, np.pi/3, np.pi/2, np.pi]:
result = diffusion_stencil_2d(epsilon=1.0,
theta=theta, type=type)
assert_almost_equal(result, expected)
def test_simple_finite_difference(self):
# isotropic
stencil = [[0.0, -1.0, 0.0], [-1.0, 4.0, -1.0], [0.0, -1.0, 0.0]]
assert_equal(diffusion_stencil_2d(epsilon=1.0, theta=0.0, type='FD'),
stencil)
# weak horizontal
stencil = [[0.0, -1.0, 0.0], [-0.5, 3.0, -0.5], [0.0, -1.0, 0.0]]
assert_almost_equal(
diffusion_stencil_2d(epsilon=0.5, theta=0.0, type='FD'), stencil)
# weak vertical
stencil = [[0.0, -0.5, 0.0], [-1.0, 3.0, -1.0], [0.0, -0.5, 0.0]]
assert_almost_equal(
diffusion_stencil_2d(epsilon=0.5, theta=pi / 2, type='FD'),
stencil)
def test_simple_finite_element(self):
# isotropic
stencil = np.array([[-1.0, -1.0, -1.0],
[-1.0, 8.0, -1.0],
[-1.0, -1.0, -1.0]]) / 3.0
assert_almost_equal(diffusion_stencil_2d(epsilon=1.0, theta=0.0,
type='FE'), stencil)
def build_poisson(n, epsilon, theta, randomize):
data = {}
h = 1. / float(n + 1)
print "Assembling diffusion using Q1 on a regular mesh with epsilon = " + \
str(epsilon) + " and theta = " + str(theta) + " ..."
stencil = diffusion_stencil_2d(type='FE', epsilon=epsilon, theta=theta)
A = stencil_grid(stencil, (n, n), format='csr')
X, Y = meshgrid(linspace(h, 1.0 - h, n), linspace(h, 1.0 - h, n))
data['X'] = X
data['Y'] = Y
if randomize:
print "Random diagonal scaling..."
D = my_rand(A.shape[0], 1)
D[D < 0.] -= 1e-3
D[D >= 0.] += 1e-3
data['D'] = D
D_inv = 1. / D
data['D_inv'] = D_inv
A = scale_rows(A, D)
A = scale_columns(A, D)
if symmetric_scale:
symmetric_rescaling(A, copy=False)
print "Ratio of largest to smallest (in magnitude) diagonal element in A: %1.3e"% \
(abs(A.diagonal()).max() / abs(A.diagonal()).min())
data['A'] = A
print 'Generate initial guess (which is random)...'
data['x0'] = my_rand(data['A'].shape[0], 1)
print 'Generate rhs (which is zero)...'
data['b'] = numpy.zeros((data['A'].shape[0], 1))
return data
def test_zero_sum(self):
"""test that stencil entries sum to zero"""
for type in ['FD', 'FE']:
for theta in [np.pi/8, np.pi/5, np.pi/4, np.pi/3, np.pi/2, np.pi]:
for epsilon in [0.001, 0.01, 1.0]:
stencil = diffusion_stencil_2d(epsilon=epsilon,
theta=theta, type=type)
assert_almost_equal(stencil.sum(), 0.0)
def test_simple_finite_difference(self):
# isotropic
stencil = [[0.0, -1.0, 0.0],
[-1.0, 4.0, -1.0],
[0.0, -1.0, 0.0]]
assert_equal(diffusion_stencil_2d(epsilon=1.0, theta=0.0,
type='FD'), stencil)
# weak horizontal
stencil = [[0.0, -1.0, 0.0],
[-0.5, 3.0, -0.5],
[0.0, -1.0, 0.0]]
assert_almost_equal(diffusion_stencil_2d(epsilon=0.5, theta=0.0,
type='FD'), stencil)
# weak vertical
stencil = [[0.0, -0.5, 0.0],
[-1.0, 3.0, -1.0],
[0.0, -0.5, 0.0]]
assert_almost_equal(diffusion_stencil_2d(epsilon=0.5, theta=np.pi/2,
type='FD'), stencil)
from copy import deepcopy
# ------------------------------------------------------------------------- #
# ------------------------------------------------------------------------- #
SOC = 'evol'
SOC_drop = 4.0
SA = 1 # Use SA type coarsening or CR
pow_G = 1
epsilon = 0.1
theta = 3.0 * math.pi / 16.0
N = 40
n = N * N
grid_dims = [N, N]
stencil = diffusion_stencil_2d(epsilon, theta)
A = stencil_grid(stencil, grid_dims, format='csr')
[d, d, A] = symmetric_rescaling(A)
A = csr_matrix(A)
B = np.kron(np.ones((A.shape[0] / 1, 1), dtype=A.dtype), np.eye(1))
tol = 1e-12 # Drop tolerance for singular values
if SA:
if SOC == 'evol':
C = evolution_strength_of_connection(A, B, epsilon=SOC_drop, k=2)
else:
SOC = 'symm'
C = symmetric_strength_of_connection(A, theta=SOC_drop)
AggOp, Cpts = standard_aggregation(C)
else:
#------------------------------------------------------------------ # Step 1: import scipy and pyamg packages #------------------------------------------------------------------ from numpy import meshgrid, linspace from scipy import rand, pi from scipy.linalg import norm from pyamg import * from pyamg.gallery import stencil_grid from pyamg.gallery.diffusion import diffusion_stencil_2d #------------------------------------------------------------------ # Step 2: setup up the system using pyamg.gallery #------------------------------------------------------------------ n=200 X,Y = meshgrid(linspace(0,1,n),linspace(0,1,n)) stencil = diffusion_stencil_2d(type='FE',epsilon=0.001,theta=pi/3) A = stencil_grid(stencil, (n,n), format='csr') b = rand(A.shape[0]) # pick a random right hand side #------------------------------------------------------------------ # Step 3: setup of the multigrid hierarchy #------------------------------------------------------------------ ml = smoothed_aggregation_solver(A) # construct the multigrid hierarchy #------------------------------------------------------------------ # Step 4: solve the system #------------------------------------------------------------------ res1 = [] x = ml.solve(b, tol=1e-12, residuals=res1)# solve Ax=b to a tolerance of 1e-12 #------------------------------------------------------------------
# task1.3
from pyamg.gallery.diffusion import diffusion_stencil_2d
from pyamg.gallery import stencil_grid
sten = diffusion_stencil_2d(type='FD', \
epsilon=0.001, theta=3.1416/3.0)
A = stencil_grid(sten, (100, 100), format='csr')
# task1.4
from pyamg import *
ml = smoothed_aggregation_solver(A)
# task1.5
from numpy import ones
b = ones((A.shape[0], 1))
res = []
x = ml.solve(b, tol=1e-8, \
residuals=res)
from pylab import *
semilogy(res[1:])
xlabel('iteration')
ylabel('residual norm')
title('Residual History')
show()
'sweep': 'symmetric',
'iterations': 1
})
smooth = ('energy', {'maxiter': 9, 'degree': 3}) # Prolongation Smoother
classic_theta = 0.0 # Classic Strength Measure
# Drop Tolerance
evolution_theta = 4.0 # evolution Strength Measure
# Drop Tolerance
for n in nlist:
nx = n
ny = n
print "Running Grid = (%d x %d)" % (nx, ny)
# Rotated Anisotropic Diffusion Operator
stencil = diffusion_stencil_2d(type='FE', epsilon=epsilon, theta=theta)
A = stencil_grid(stencil, (nx, ny), format='csr')
# Random initial guess, zero RHS
x0 = scipy.rand(A.shape[0])
b = numpy.zeros((A.shape[0], ))
# Classic SA strength measure
ml = smoothed_aggregation_solver(A,
max_coarse=mcoarse,
coarse_solver='pinv2',
presmoother=prepost,
postsmoother=prepost,
smooth=smooth,
strength=('symmetric', {
'theta': classic_theta
#------------------------------------------------------------------ # Step 1: import scipy and pyamg packages #------------------------------------------------------------------ from numpy import meshgrid, linspace from scipy import rand, pi from scipy.linalg import norm from pyamg import * from pyamg.gallery import stencil_grid from pyamg.gallery.diffusion import diffusion_stencil_2d #------------------------------------------------------------------ # Step 2: setup up the system using pyamg.gallery #------------------------------------------------------------------ n=200 X,Y = meshgrid(linspace(0,1,n),linspace(0,1,n)) stencil = diffusion_stencil_2d(type='FE',epsilon=0.001,theta=pi/3) A = stencil_grid(stencil, (n,n), format='csr') b = rand(A.shape[0]) # pick a random right hand side #------------------------------------------------------------------ # Step 3: setup of the multigrid hierarchy #------------------------------------------------------------------ ml = smoothed_aggregation_solver(A) # construct the multigrid hierarchy #------------------------------------------------------------------ # Step 4: solve the system #------------------------------------------------------------------ res1 = [] x = ml.solve(b, tol=1e-12, residuals=res1)# solve Ax=b to a tolerance of 1e-12 #------------------------------------------------------------------
from pyamg.gallery.diffusion import diffusion_stencil_2d from pyamg.gallery import stencil_grid from numpy import set_printoptions set_printoptions(precision=2) sten = diffusion_stencil_2d(type="FD", epsilon=0.001, theta=3.1416 / 3.0) A = stencil_grid(sten, (100, 100), format="csr") # print the matrix stencil print(A[5050, :].data) print(sten)
# task1.3
from pyamg.gallery.diffusion import diffusion_stencil_2d
from pyamg.gallery import stencil_grid
sten = diffusion_stencil_2d(type='FD', \
epsilon=0.001, theta=3.1416/3.0)
A = stencil_grid(sten, (100,100), format='csr')
# task1.4
from pyamg import *
ml = smoothed_aggregation_solver(A)
# task1.5
from numpy import ones
b = ones((A.shape[0],1))
res = []
x = ml.solve(b, tol=1e-8, \
residuals=res)
from pylab import *
semilogy(res[1:])
xlabel('iteration')
ylabel('residual norm')
title('Residual History')
show()
epsilon = 0.001 # Anisotropic coefficient
mcoarse = 10 # Max coarse grid size
prepost = ("gauss_seidel", {"sweep": "symmetric", "iterations": 1}) # pre/post smoother
smooth = ("energy", {"maxiter": 9, "degree": 3}) # Prolongation Smoother
classic_theta = 0.0 # Classic Strength Measure
# Drop Tolerance
evolution_theta = 4.0 # evolution Strength Measure
# Drop Tolerance
for n in nlist:
nx = n
ny = n
print "Running Grid = (%d x %d)" % (nx, ny)
# Rotated Anisotropic Diffusion Operator
stencil = diffusion_stencil_2d(type="FE", epsilon=epsilon, theta=theta)
A = stencil_grid(stencil, (nx, ny), format="csr")
# Random initial guess, zero RHS
x0 = scipy.rand(A.shape[0])
b = numpy.zeros((A.shape[0],))
# Classic SA strength measure
ml = smoothed_aggregation_solver(
A,
max_coarse=mcoarse,
coarse_solver="pinv2",
presmoother=prepost,
postsmoother=prepost,
smooth=smooth,
strength=("symmetric", {"theta": classic_theta}),
CSR matrix for basic rotated anisotropic diffusion
BSR matrix for basic 2D linearized elasticity
CSR matrix for a nonsymmetric recirculating flow problem
Many more solver parameters may be specified than outlined in the below
examples. Only the most basic are shown.
Run with
>>> python demo.py
and examine the on-screen output and file output.
'''
from scipy import pi
from pyamg import gallery
from pyamg.gallery import diffusion
stencil = diffusion.diffusion_stencil_2d(type='FE',
epsilon=0.001,
theta=2 * pi / 16.0)
A = gallery.stencil_grid(stencil, (50, 50), format='csr')
from pyamg import gallery
from solver_diagnostics import solver_diagnostics
from scipy import pi
from pyamg.gallery import diffusion
choice = input('\nThere are four different test problems. Enter \n' + \
'1: Isotropic diffusion example\n' + \
'2: Anisotropic diffusion example\n' + \
'3: Elasticity example\n' + \
'4: Nonsymmetric flow example\n\n ')
if choice == 1: