def test_conjugate_gradient(self):
np.random.seed(1)
a_np = np.random.uniform(
low=-1.0, high=1.0, size=np.prod(shape_)).reshape(shape_).astype(dtype_)
# Make a selfadjoint, positive definite.
a_np = np.dot(a_np.T, a_np)
# jacobi preconditioner
jacobi_np = np.zeros_like(a_np)
jacobi_np[range(a_np.shape[0]), range(a_np.shape[1])] = (1.0 /
a_np.diagonal())
rhs_np = np.random.uniform(
low=-1.0, high=1.0, size=shape_[0]).astype(dtype_)
x_np = np.zeros_like(rhs_np)
tol = 1e-6 if dtype_ == np.float64 else 1e-3
max_iter = 20
with self.test_session() as sess:
if use_static_shape_:
a = constant_op.constant(a_np)
rhs = constant_op.constant(rhs_np)
x = constant_op.constant(x_np)
jacobi = constant_op.constant(jacobi_np)
else:
a = array_ops.placeholder(dtype_)
rhs = array_ops.placeholder(dtype_)
x = array_ops.placeholder(dtype_)
jacobi = array_ops.placeholder(dtype_)
operator = util.create_operator(a)
preconditioners = [None, util.identity_operator(a),
util.create_operator(jacobi)]
cg_results = []
for preconditioner in preconditioners:
cg_graph = linear_equations.conjugate_gradient(
operator, rhs, preconditioner=preconditioner,
x=x, tol=tol, max_iter=max_iter)
if use_static_shape_:
cg_val = sess.run(cg_graph)
else:
cg_val = sess.run(cg_graph, feed_dict={a: a_np, rhs: rhs_np, x: x_np,
jacobi: jacobi_np})
norm_r0 = np.linalg.norm(rhs_np)
norm_r = np.linalg.norm(cg_val.r)
self.assertLessEqual(norm_r, tol * norm_r0)
# Validate that we get an equally small residual norm with numpy
# using the computed solution.
r_np = rhs_np - np.dot(a_np, cg_val.x)
norm_r_np = np.linalg.norm(r_np)
self.assertLessEqual(norm_r_np, tol * norm_r0)
cg_results.append(cg_val)
# Validate that we get same results using identity_preconditioner
# and None
self.assertEqual(cg_results[0].i, cg_results[1].i)
self.assertAlmostEqual(cg_results[0].gamma, cg_results[1].gamma)
self.assertAllClose(cg_results[0].r, cg_results[1].r, rtol=tol)
self.assertAllClose(cg_results[0].x, cg_results[1].x, rtol=tol)
self.assertAllClose(cg_results[0].p, cg_results[1].p, rtol=tol)
def _testCreateOperator(self, use_static_shape_):
for dtype in np.float32, np.float64:
a_np = np.array([[1., 2.], [3., 4.], [5., 6.]], dtype=dtype)
x_np = np.array([[2.], [-3.]], dtype=dtype)
y_np = np.array([[2], [-3.], [5.]], dtype=dtype)
with self.test_session() as sess:
if use_static_shape_:
a = constant_op.constant(a_np, dtype=dtype)
x = constant_op.constant(x_np, dtype=dtype)
y = constant_op.constant(y_np, dtype=dtype)
else:
a = array_ops.placeholder(dtype)
x = array_ops.placeholder(dtype)
y = array_ops.placeholder(dtype)
op = util.create_operator(a)
ax = op.apply(x)
aty = op.apply_adjoint(y)
op_shape = ops.convert_to_tensor(op.shape)
if use_static_shape_:
op_shape_val, ax_val, aty_val = sess.run([op_shape, ax, aty])
else:
op_shape_val, ax_val, aty_val = sess.run(
[op_shape, ax, aty], feed_dict={a: a_np,
x: x_np,
y: y_np})
self.assertAllEqual(op_shape_val, [3, 2])
self.assertAllClose(ax_val, np.dot(a_np, x_np))
self.assertAllClose(aty_val, np.dot(a_np.T, y_np))
def _testCreateOperator(self, use_static_shape_):
for dtype in np.float32, np.float64:
a_np = np.array([[1., 2.], [3., 4.], [5., 6.]], dtype=dtype)
x_np = np.array([[2.], [-3.]], dtype=dtype)
y_np = np.array([[2], [-3.], [5.]], dtype=dtype)
with self.test_session() as sess:
if use_static_shape_:
a = constant_op.constant(a_np, dtype=dtype)
x = constant_op.constant(x_np, dtype=dtype)
y = constant_op.constant(y_np, dtype=dtype)
else:
a = array_ops.placeholder(dtype)
x = array_ops.placeholder(dtype)
y = array_ops.placeholder(dtype)
op = util.create_operator(a)
ax = op.apply(x)
aty = op.apply_adjoint(y)
op_shape = ops.convert_to_tensor(op.shape)
if use_static_shape_:
op_shape_val, ax_val, aty_val = sess.run(
[op_shape, ax, aty])
else:
op_shape_val, ax_val, aty_val = sess.run(
[op_shape, ax, aty],
feed_dict={
a: a_np,
x: x_np,
y: y_np
})
self.assertAllEqual(op_shape_val, [3, 2])
self.assertAllClose(ax_val, np.dot(a_np, x_np))
self.assertAllClose(aty_val, np.dot(a_np.T, y_np))
def test_cgls(self):
np.random.seed(1)
a_np = np.random.uniform(
low=-1.0, high=1.0, size=np.prod(shape_)).reshape(shape_).astype(dtype_)
rhs_np = np.random.uniform(
low=-1.0, high=1.0, size=shape_[0]).astype(dtype_)
tol = 1e-12 if dtype_ == np.float64 else 1e-6
max_iter = 20
with self.test_session() as sess:
if use_static_shape_:
a = constant_op.constant(a_np)
rhs = constant_op.constant(rhs_np)
else:
a = array_ops.placeholder(dtype_)
rhs = array_ops.placeholder(dtype_)
operator = util.create_operator(a)
cgls_graph = least_squares.cgls(operator, rhs, tol=tol, max_iter=max_iter)
if use_static_shape_:
cgls_val = sess.run(cgls_graph)
else:
cgls_val = sess.run(cgls_graph, feed_dict={a: a_np, rhs: rhs_np})
# Below we use s = A^* (rhs - A x), s0 = A^* rhs
norm_s0 = np.linalg.norm(np.dot(a_np.T, rhs_np))
norm_s = np.sqrt(cgls_val.gamma)
self.assertLessEqual(norm_s, tol * norm_s0)
# Validate that we get an equally small residual norm with numpy
# using the computed solution.
r_np = rhs_np - np.dot(a_np, cgls_val.x)
norm_s_np = np.linalg.norm(np.dot(a_np.T, r_np))
self.assertLessEqual(norm_s_np, tol * norm_s0)
def test_cgls(self):
np.random.seed(1)
a_np = np.random.uniform(
low=-1.0, high=1.0, size=np.prod(shape_)).reshape(shape_).astype(dtype_)
rhs_np = np.random.uniform(
low=-1.0, high=1.0, size=shape_[0]).astype(dtype_)
tol = 1e-12 if dtype_ == np.float64 else 1e-6
max_iter = 20
with self.test_session() as sess:
if use_static_shape_:
a = tf.constant(a_np)
rhs = tf.constant(rhs_np)
else:
a = tf.placeholder(dtype_)
rhs = tf.placeholder(dtype_)
operator = util.create_operator(a)
cgls_graph = least_squares.cgls(operator, rhs, tol=tol, max_iter=max_iter)
if use_static_shape_:
cgls_val = sess.run(cgls_graph)
else:
cgls_val = sess.run(cgls_graph, feed_dict={a: a_np, rhs: rhs_np})
# Below we use s = A^* (rhs - A x), s0 = A^* rhs
norm_s0 = np.linalg.norm(np.dot(a_np.T, rhs_np))
norm_s = np.sqrt(cgls_val.gamma)
self.assertLessEqual(norm_s, tol * norm_s0)
# Validate that we get an equally small residual norm with numpy
# using the computed solution.
r_np = rhs_np - np.dot(a_np, cgls_val.x)
norm_s_np = np.linalg.norm(np.dot(a_np.T, r_np))
self.assertLessEqual(norm_s_np, tol * norm_s0)
def test_conjugate_gradient(self):
np.random.seed(1)
a_np = np.random.uniform(
low=-1.0, high=1.0, size=np.prod(shape_)).reshape(shape_).astype(dtype_)
# Make a selfadjoint, positive definite.
a_np = np.dot(a_np.T, a_np)
rhs_np = np.random.uniform(
low=-1.0, high=1.0, size=shape_[0]).astype(dtype_)
tol = 1e-6 if dtype_ == np.float64 else 1e-3
max_iter = 20
with self.test_session() as sess:
if use_static_shape_:
a = constant_op.constant(a_np)
rhs = constant_op.constant(rhs_np)
else:
a = array_ops.placeholder(dtype_)
rhs = array_ops.placeholder(dtype_)
operator = util.create_operator(a)
cg_graph = linear_equations.conjugate_gradient(
operator, rhs, tol=tol, max_iter=max_iter)
if use_static_shape_:
cg_val = sess.run(cg_graph)
else:
cg_val = sess.run(cg_graph, feed_dict={a: a_np, rhs: rhs_np})
norm_r0 = np.linalg.norm(rhs_np)
norm_r = np.sqrt(cg_val.gamma)
self.assertLessEqual(norm_r, tol * norm_r0)
# Validate that we get an equally small residual norm with numpy
# using the computed solution.
r_np = rhs_np - np.dot(a_np, cg_val.x)
norm_r_np = np.linalg.norm(r_np)
self.assertLessEqual(norm_r_np, tol * norm_r0)
def test_conjugate_gradient(self):
np.random.seed(1)
a_np = np.random.uniform(
low=-1.0, high=1.0,
size=np.prod(shape_)).reshape(shape_).astype(dtype_)
# Make a selfadjoint, positive definite.
a_np = np.dot(a_np.T, a_np)
rhs_np = np.random.uniform(low=-1.0, high=1.0,
size=shape_[0]).astype(dtype_)
tol = 1e-6 if dtype_ == np.float64 else 1e-3
max_iter = 20
with self.test_session() as sess:
if use_static_shape_:
a = tf.constant(a_np)
rhs = tf.constant(rhs_np)
else:
a = tf.placeholder(dtype_)
rhs = tf.placeholder(dtype_)
operator = util.create_operator(a)
cg_graph = linear_equations.conjugate_gradient(operator,
rhs,
tol=tol,
max_iter=max_iter)
if use_static_shape_:
cg_val = sess.run(cg_graph)
else:
cg_val = sess.run(cg_graph, feed_dict={a: a_np, rhs: rhs_np})
norm_r0 = np.linalg.norm(rhs_np)
norm_r = np.sqrt(cg_val.gamma)
self.assertLessEqual(norm_r, tol * norm_r0)
# Validate that we get an equally small residual norm with numpy
# using the computed solution.
r_np = rhs_np - np.dot(a_np, cg_val.x)
norm_r_np = np.linalg.norm(r_np)
self.assertLessEqual(norm_r_np, tol * norm_r0)
def test_lanczos_bidiag(self):
np.random.seed(1)
a_np = np.random.uniform(
low=-1.0, high=1.0, size=np.prod(shape_)).reshape(shape_).astype(dtype_)
tol = 1e-12 if dtype_ == np.float64 else 1e-5
with self.cached_session() as sess:
if use_static_shape_:
a = constant_op.constant(a_np)
else:
a = array_ops.placeholder(dtype_)
operator = util.create_operator(a)
lbd = lanczos.lanczos_bidiag(
operator, steps_, orthogonalize=orthogonalize_)
# The computed factorization should satisfy the equations
# A * V = U * B
# A' * U[:, :-1] = V * B[:-1, :]'
av = math_ops.matmul(a, lbd.v)
ub = lanczos.bidiag_matmul(lbd.u, lbd.alpha, lbd.beta, adjoint_b=False)
atu = math_ops.matmul(a, lbd.u[:, :-1], adjoint_a=True)
vbt = lanczos.bidiag_matmul(lbd.v, lbd.alpha, lbd.beta, adjoint_b=True)
if use_static_shape_:
av_val, ub_val, atu_val, vbt_val = sess.run([av, ub, atu, vbt])
else:
av_val, ub_val, atu_val, vbt_val = sess.run([av, ub, atu, vbt],
feed_dict={a: a_np})
self.assertAllClose(av_val, ub_val, atol=tol, rtol=tol)
self.assertAllClose(atu_val, vbt_val, atol=tol, rtol=tol)
def test_lanczos_bidiag(self):
np.random.seed(1)
a_np = np.random.uniform(
low=-1.0, high=1.0, size=np.prod(shape_)).reshape(shape_).astype(dtype_)
tol = 1e-12 if dtype_ == np.float64 else 1e-5
with self.cached_session() as sess:
if use_static_shape_:
a = constant_op.constant(a_np)
else:
a = array_ops.placeholder(dtype_)
operator = util.create_operator(a)
lbd = lanczos.lanczos_bidiag(
operator, steps_, orthogonalize=orthogonalize_)
# The computed factorization should satisfy the equations
# A * V = U * B
# A' * U[:, :-1] = V * B[:-1, :]'
av = math_ops.matmul(a, lbd.v)
ub = lanczos.bidiag_matmul(lbd.u, lbd.alpha, lbd.beta, adjoint_b=False)
atu = math_ops.matmul(a, lbd.u[:, :-1], adjoint_a=True)
vbt = lanczos.bidiag_matmul(lbd.v, lbd.alpha, lbd.beta, adjoint_b=True)
if use_static_shape_:
av_val, ub_val, atu_val, vbt_val = sess.run([av, ub, atu, vbt])
else:
av_val, ub_val, atu_val, vbt_val = sess.run([av, ub, atu, vbt],
feed_dict={a: a_np})
self.assertAllClose(av_val, ub_val, atol=tol, rtol=tol)
self.assertAllClose(atu_val, vbt_val, atol=tol, rtol=tol)
def _testCreateOperator(self, use_static_shape_):
a_np = np.array([[1., 2.], [3., 4.], [5., 6.]])
x = np.array([[2.], [-3.]])
y = np.array([[2], [-3.], [5.]])
with self.test_session() as sess:
if use_static_shape_:
a = tf.constant(a_np, dtype=tf.float32)
else:
a = tf.placeholder(tf.float32)
op = util.create_operator(a)
ax = op.apply(x)
aty = op.apply_adjoint(y)
op_shape = tf.convert_to_tensor(op.shape)
if use_static_shape_:
op_shape_val, ax_val, aty_val = sess.run([op_shape, ax, aty])
else:
op_shape_val, ax_val, aty_val = sess.run([op_shape, ax, aty],
feed_dict={a: a_np})
self.assertAllEqual(op_shape_val, [3, 2])
self.assertAllClose(ax_val, [[-4], [-6], [-8]])
self.assertAllClose(aty_val, [[18], [22]])
def test_conjugate_gradient(self):
np.random.seed(1)
a_np = np.random.uniform(
low=-1.0, high=1.0, size=np.prod(shape_)).reshape(shape_).astype(dtype_)
# Make a selfadjoint, positive definite.
a_np = np.dot(a_np.T, a_np)
# jacobi preconditioner
jacobi_np = np.zeros_like(a_np)
jacobi_np[range(a_np.shape[0]), range(a_np.shape[1])] = (
1.0 / a_np.diagonal())
rhs_np = np.random.uniform(
low=-1.0, high=1.0, size=shape_[0]).astype(dtype_)
x_np = np.zeros_like(rhs_np)
tol = 1e-6 if dtype_ == np.float64 else 1e-3
max_iter = 20
with self.cached_session() as sess:
if use_static_shape_:
a = constant_op.constant(a_np)
rhs = constant_op.constant(rhs_np)
x = constant_op.constant(x_np)
jacobi = constant_op.constant(jacobi_np)
else:
a = array_ops.placeholder(dtype_)
rhs = array_ops.placeholder(dtype_)
x = array_ops.placeholder(dtype_)
jacobi = array_ops.placeholder(dtype_)
operator = util.create_operator(a)
preconditioners = [
None, util.identity_operator(a),
util.create_operator(jacobi)
]
cg_results = []
for preconditioner in preconditioners:
cg_graph = linear_equations.conjugate_gradient(
operator,
rhs,
preconditioner=preconditioner,
x=x,
tol=tol,
max_iter=max_iter)
if use_static_shape_:
cg_val = sess.run(cg_graph)
else:
cg_val = sess.run(
cg_graph,
feed_dict={
a: a_np,
rhs: rhs_np,
x: x_np,
jacobi: jacobi_np
})
norm_r0 = np.linalg.norm(rhs_np)
norm_r = np.linalg.norm(cg_val.r)
self.assertLessEqual(norm_r, tol * norm_r0)
# Validate that we get an equally small residual norm with numpy
# using the computed solution.
r_np = rhs_np - np.dot(a_np, cg_val.x)
norm_r_np = np.linalg.norm(r_np)
self.assertLessEqual(norm_r_np, tol * norm_r0)
cg_results.append(cg_val)
# Validate that we get same results using identity_preconditioner
# and None
self.assertEqual(cg_results[0].i, cg_results[1].i)
self.assertAlmostEqual(cg_results[0].gamma, cg_results[1].gamma)
self.assertAllClose(cg_results[0].r, cg_results[1].r, rtol=tol)
self.assertAllClose(cg_results[0].x, cg_results[1].x, rtol=tol)
self.assertAllClose(cg_results[0].p, cg_results[1].p, rtol=tol)