def test_forward_backward_with_lin_ops():
"""Test for the forward-backward solver with linear operatros.
The test is done by minimizing ||x - b||_2^2 + ||alpha * x||_2^2. The
general problem is of the form
``min_x f(x) + sum_i g_i(L_i x) + h(x)``
and here we take f = 0, g = ||.||_2^2, L = alpha * IndentityOperator,
and h = ||. - b||_2^2.
"""
space = odl.rn(10)
alpha = 0.1
b = noise_element(space)
lin_ops = [alpha * odl.IdentityOperator(space)]
g = [odl.solvers.L2NormSquared(space)]
f = odl.solvers.ZeroFunctional(space)
# Gradient of two-norm square
h = odl.solvers.L2NormSquared(space).translated(b)
x = noise_element(space)
# Explicit solution: x_hat = (I^T * I + (alpha*I)^T * (alpha*I))^-1 * (I*b)
x_global_min = b / (1 + alpha**2)
forward_backward_pd(x, f, g, lin_ops, h, tau=0.5, sigma=[1.0], niter=20)
assert all_almost_equal(x, x_global_min, places=LOW_ACCURACY)
def _reconstruct(self, observation, out):
observation = self.observation_space.element(observation)
out[:] = self.x0
gradient = Gradient(self.op.domain)
L = [self.op, gradient]
f = ZeroFunctional(self.op.domain)
l2_norm = 0.5 * L2NormSquared(self.op.range).translated(observation)
l12_norm = self.lam * GroupL1Norm(gradient.range)
g = [l2_norm, l12_norm]
op_norm = power_method_opnorm(self.op, maxiter=20)
gradient_norm = power_method_opnorm(gradient, maxiter=20)
sigma_ray_trafo = 45.0 / op_norm**2
sigma_gradient = 45.0 / gradient_norm**2
sigma = [sigma_ray_trafo, sigma_gradient]
h = ZeroFunctional(self.op.domain)
forward_backward_pd(out,
f,
g,
L,
h,
self.tau,
sigma,
self.niter,
callback=self.callback)
return out
def test_forward_backward_basic():
"""Test for the forward-backward solver by minimizing ||x||_2^2.
The general problem is of the form
``min_x f(x) + sum_i g_i(L_i x) + h(x)``
and here we take f(x) = g(x) = 0, h(x) = ||x||_2^2 and L is the
zero-operator.
"""
space = odl.rn(10)
lin_ops = [odl.ZeroOperator(space)]
g = [odl.solvers.ZeroFunctional(space)]
f = odl.solvers.ZeroFunctional(space)
h = odl.solvers.L2NormSquared(space)
x = noise_element(space)
x_global_min = space.zero()
forward_backward_pd(x, f, g, lin_ops, h, tau=0.5,
sigma=[1.0], niter=10)
assert all_almost_equal(x, x_global_min, places=HIGH_ACCURACY)
def test_forward_backward_with_li_and_h():
"""Test for the forward-backward solver with infimal convolution.
The test is done by minimizing the functional ``(g @ l)(x) + h(x)``, where
``(g @ l)(x) = inf_y { g(y) + l(x-y) }``,
g is the indicator function on [-3, -1], and l(x) = h(x) = 1/2||x||_2^2.
The optimal solution to this problem is given by x = -0.5.
"""
# Parameter values for the box constraint
upper_lim = -1
lower_lim = -3
space = odl.rn(1)
lin_ops = [odl.IdentityOperator(space)]
g = [odl.solvers.IndicatorBox(space, lower=lower_lim, upper=upper_lim)]
f = odl.solvers.ZeroFunctional(space)
h = 0.5 * odl.solvers.L2NormSquared(space)
l = [0.5 * odl.solvers.L2NormSquared(space)]
# Creating an element not to far away from -0.5, in order to converge in
# a few number of iterations.
x = space.element(10)
forward_backward_pd(x, f, g, lin_ops, h, tau=0.5,
sigma=[1.0], niter=20, l=l)
expected_result = -0.5
assert almost_equal(x[0], expected_result, places=LOW_ACCURACY)
def test_forward_backward_with_lin_ops():
"""Test for the forward-backward solver with linear operatros.
The test is done by minimizing ||x - b||_2^2 + ||alpha * x||_2^2. The
general problem is of the form
``min_x f(x) + sum_i g_i(L_i x) + h(x)``
and here we take f = 0, g = ||.||_2^2, L = alpha * IndentityOperator,
and h = ||. - b||_2^2.
"""
space = odl.rn(10)
alpha = 0.1
b = noise_element(space)
lin_ops = [alpha * odl.IdentityOperator(space)]
g = [odl.solvers.L2NormSquared(space)]
f = odl.solvers.ZeroFunctional(space)
# Gradient of two-norm square
h = odl.solvers.L2NormSquared(space).translated(b)
x = noise_element(space)
# Explicit solution: x_hat = (I^T * I + (alpha*I)^T * (alpha*I))^-1 * (I*b)
x_global_min = b / (1 + alpha ** 2)
forward_backward_pd(x, f, g, lin_ops, h, tau=0.5,
sigma=[1.0], niter=20)
assert all_almost_equal(x, x_global_min, places=LOW_ACCURACY)
def test_forward_backward_with_li_and_h():
"""Test for the forward-backward solver with infimal convolution.
The test is done by minimizing the functional ``(g @ l)(x) + h(x)``, where
``(g @ l)(x) = inf_y { g(y) + l(x-y) }``,
g is the indicator function on [-3, -1], and l(x) = h(x) = 1/2||x||_2^2.
The optimal solution to this problem is given by x = -0.5.
"""
# Parameter values for the box constraint
upper_lim = -1
lower_lim = -3
space = odl.rn(1)
lin_ops = [odl.IdentityOperator(space)]
g = [odl.solvers.IndicatorBox(space, lower=lower_lim, upper=upper_lim)]
f = odl.solvers.ZeroFunctional(space)
h = 0.5 * odl.solvers.L2NormSquared(space)
l = [0.5 * odl.solvers.L2NormSquared(space)]
# Creating an element not to far away from -0.5, in order to converge in
# a few number of iterations.
x = space.element(10)
forward_backward_pd(x,
f,
g,
lin_ops,
h,
tau=0.5,
sigma=[1.0],
niter=20,
l=l)
expected_result = -0.5
assert almost_equal(x[0], expected_result, places=LOW_ACCURACY)
def test_forward_backward_basic():
"""Test for the forward-backward solver by minimizing ||x||_2^2.
The general problem is of the form
``min_x f(x) + sum_i g_i(L_i x) + h(x)``
and here we take f(x) = g(x) = 0, h(x) = ||x||_2^2 and L is the
zero-operator.
"""
space = odl.rn(10)
lin_ops = [odl.ZeroOperator(space)]
g = [odl.solvers.ZeroFunctional(space)]
f = odl.solvers.ZeroFunctional(space)
h = odl.solvers.L2NormSquared(space)
x = noise_element(space)
x_global_min = space.zero()
forward_backward_pd(x, f, g, lin_ops, h, tau=0.5, sigma=[1.0], niter=10)
assert all_almost_equal(x, x_global_min, places=HIGH_ACCURACY)
def test_forward_backward_input_handling():
"""Test to see that input is handled correctly."""
space1 = odl.uniform_discr(0, 1, 10)
lin_ops = [odl.ZeroOperator(space1), odl.ZeroOperator(space1)]
g = [odl.solvers.ZeroFunctional(space1),
odl.solvers.ZeroFunctional(space1)]
f = odl.solvers.ZeroFunctional(space1)
h = odl.solvers.ZeroFunctional(space1)
# Check that the algorithm runs. With the above operators, the algorithm
# returns the input.
x0 = noise_element(space1)
x = x0.copy()
niter = 3
forward_backward_pd(x, f, g, lin_ops, h, tau=1.0,
sigma=[1.0, 1.0], niter=niter)
assert x == x0
# Testing that sizes needs to agree:
# Too few sigma_i:s
with pytest.raises(ValueError):
forward_backward_pd(x, f, g, lin_ops, h, tau=1.0,
sigma=[1.0], niter=niter)
# Too many operators
g_too_many = [odl.solvers.ZeroFunctional(space1),
odl.solvers.ZeroFunctional(space1),
odl.solvers.ZeroFunctional(space1)]
with pytest.raises(ValueError):
forward_backward_pd(x, f, g_too_many, lin_ops, h,
tau=1.0, sigma=[1.0, 1.0], niter=niter)
# Test for correct space
space2 = odl.uniform_discr(1, 2, 10)
x = noise_element(space2)
with pytest.raises(ValueError):
forward_backward_pd(x, f, g, lin_ops, h, tau=1.0,
sigma=[1.0, 1.0], niter=niter)
def test_forward_backward_input_handling():
"""Test to see that input is handled correctly."""
space1 = odl.uniform_discr(0, 1, 10)
lin_ops = [odl.ZeroOperator(space1), odl.ZeroOperator(space1)]
g = [
odl.solvers.ZeroFunctional(space1),
odl.solvers.ZeroFunctional(space1)
]
f = odl.solvers.ZeroFunctional(space1)
h = odl.solvers.ZeroFunctional(space1)
# Check that the algorithm runs. With the above operators, the algorithm
# returns the input.
x0 = noise_element(space1)
x = x0.copy()
niter = 3
forward_backward_pd(x,
f,
g,
lin_ops,
h,
tau=1.0,
sigma=[1.0, 1.0],
niter=niter)
assert x == x0
# Testing that sizes needs to agree:
# Too few sigma_i:s
with pytest.raises(ValueError):
forward_backward_pd(x,
f,
g,
lin_ops,
h,
tau=1.0,
sigma=[1.0],
niter=niter)
# Too many operators
g_too_many = [
odl.solvers.ZeroFunctional(space1),
odl.solvers.ZeroFunctional(space1),
odl.solvers.ZeroFunctional(space1)
]
with pytest.raises(ValueError):
forward_backward_pd(x,
f,
g_too_many,
lin_ops,
h,
tau=1.0,
sigma=[1.0, 1.0],
niter=niter)
# Test for correct space
space2 = odl.uniform_discr(1, 2, 10)
x = noise_element(space2)
with pytest.raises(ValueError):
forward_backward_pd(x,
f,
g,
lin_ops,
h,
tau=1.0,
sigma=[1.0, 1.0],
niter=niter)
