def cg_step(i, state): # pylint: disable=missing-docstring
z = operator.apply(state.p)
alpha = state.gamma / util.dot(state.p, z)
x = state.x + alpha * state.p
r = state.r - alpha * z
if preconditioner is None:
gamma = util.dot(r, r)
beta = gamma / state.gamma
p = r + beta * state.p
else:
q = preconditioner.apply(r)
gamma = util.dot(r, q)
beta = gamma / state.gamma
p = q + beta * state.p
return i + 1, cg_state(i + 1, x, r, p, gamma)
def cg_step(i, state): # pylint: disable=missing-docstring
z = operator.apply(state.p)
alpha = state.gamma / util.dot(state.p, z)
x = state.x + alpha * state.p
r = state.r - alpha * z
if preconditioner is None:
gamma = util.dot(r, r)
beta = gamma / state.gamma
p = r + beta * state.p
else:
q = preconditioner.apply(r)
gamma = util.dot(r, q)
beta = gamma / state.gamma
p = q + beta * state.p
return i + 1, cg_state(i + 1, x, r, p, gamma)
def cg_step(i, state): z = operator.apply(state.p) alpha = state.gamma / util.dot(state.p, z) x = state.x + alpha * state.p r = state.r - alpha * z gamma = util.l2norm_squared(r) beta = gamma / state.gamma p = r + beta * state.p return i + 1, cg_state(i + 1, x, r, p, gamma)
def cg_step(i, state):
z = operator.apply(state.p)
alpha = state.gamma / util.dot(state.p, z)
x = state.x + alpha * state.p
r = state.r - alpha * z
gamma = util.l2norm_squared(r)
beta = gamma / state.gamma
p = r + beta * state.p
return i + 1, cg_state(i + 1, x, r, p, gamma)
def conjugate_gradient(operator,
rhs,
preconditioner=None,
x=None,
tol=1e-4,
max_iter=20,
name="conjugate_gradient"):
r"""Conjugate gradient solver.
Solves a linear system of equations `A*x = rhs` for selfadjoint, positive
definite matrix `A` and right-hand side vector `rhs`, using an iterative,
matrix-free algorithm where the action of the matrix A is represented by
`operator`. The iteration terminates when either the number of iterations
exceeds `max_iter` or when the residual norm has been reduced to `tol`
times its initial value, i.e. \\(||rhs - A x_k|| <= tol ||rhs||\\).
Args:
operator: An object representing a linear operator with attributes:
- shape: Either a list of integers or a 1-D `Tensor` of type `int32` of
length 2. `shape[0]` is the dimension on the domain of the operator,
`shape[1]` is the dimension of the co-domain of the operator. On other
words, if operator represents an N x N matrix A, `shape` must contain
`[N, N]`.
- dtype: The datatype of input to and output from `apply`.
- apply: Callable object taking a vector `x` as input and returning a
vector with the result of applying the operator to `x`, i.e. if
`operator` represents matrix `A`, `apply` should return `A * x`.
rhs: A rank-1 `Tensor` of shape `[N]` containing the right-hand size vector.
preconditioner: An object representing a linear operator, see `operator`
for detail. The preconditioner should approximate the inverse of `A`.
An efficient preconditioner could dramatically improve the rate of
convergence. If `preconditioner` represents matrix `M`(`M` approximates
`A^{-1}`), the algorithm uses `preconditioner.apply(x)` to estimate
`A^{-1}x`. For this to be useful, the cost of applying `M` should be
much lower than computing `A^{-1}` directly.
x: A rank-1 `Tensor` of shape `[N]` containing the initial guess for the
solution.
tol: A float scalar convergence tolerance.
max_iter: An integer giving the maximum number of iterations.
name: A name scope for the operation.
Returns:
output: A namedtuple representing the final state with fields:
- i: A scalar `int32` `Tensor`. Number of iterations executed.
- x: A rank-1 `Tensor` of shape `[N]` containing the computed solution.
- r: A rank-1 `Tensor` of shape `[M]` containing the residual vector.
- p: A rank-1 `Tensor` of shape `[N]`. `A`-conjugate basis vector.
- gamma: \\(r \dot M \dot r\\), equivalent to \\(||r||_2^2\\) when
`preconditioner=None`.
"""
# ephemeral class holding CG state.
cg_state = collections.namedtuple("CGState", ["i", "x", "r", "p", "gamma"])
def stopping_criterion(i, state):
return math_ops.logical_and(i < max_iter,
linalg_ops.norm(state.r) > tol)
def cg_step(i, state): # pylint: disable=missing-docstring
z = operator.apply(state.p)
alpha = state.gamma / util.dot(state.p, z)
x = state.x + alpha * state.p
r = state.r - alpha * z
if preconditioner is None:
gamma = util.dot(r, r)
beta = gamma / state.gamma
p = r + beta * state.p
else:
q = preconditioner.apply(r)
gamma = util.dot(r, q)
beta = gamma / state.gamma
p = q + beta * state.p
return i + 1, cg_state(i + 1, x, r, p, gamma)
with ops.name_scope(name):
n = operator.shape[1:]
rhs = array_ops.expand_dims(rhs, -1)
if x is None:
x = array_ops.expand_dims(
array_ops.zeros(n, dtype=rhs.dtype.base_dtype), -1)
r0 = rhs
else:
x = array_ops.expand_dims(x, -1)
r0 = rhs - operator.apply(x)
if preconditioner is None:
p0 = r0
else:
p0 = preconditioner.apply(r0)
gamma0 = util.dot(r0, p0)
tol *= linalg_ops.norm(r0)
i = constant_op.constant(0, dtype=dtypes.int32)
state = cg_state(i=i, x=x, r=r0, p=p0, gamma=gamma0)
_, state = control_flow_ops.while_loop(stopping_criterion, cg_step,
[i, state])
return cg_state(state.i,
x=array_ops.squeeze(state.x),
r=array_ops.squeeze(state.r),
p=array_ops.squeeze(state.p),
gamma=state.gamma)
def conjugate_gradient(operator,
rhs,
preconditioner=None,
x=None,
tol=1e-4,
max_iter=20,
name="conjugate_gradient"):
r"""Conjugate gradient solver.
Solves a linear system of equations `A*x = rhs` for selfadjoint, positive
definite matrix `A` and right-hand side vector `rhs`, using an iterative,
matrix-free algorithm where the action of the matrix A is represented by
`operator`. The iteration terminates when either the number of iterations
exceeds `max_iter` or when the residual norm has been reduced to `tol`
times its initial value, i.e. \\(||rhs - A x_k|| <= tol ||rhs||\\).
Args:
operator: An object representing a linear operator with attributes:
- shape: Either a list of integers or a 1-D `Tensor` of type `int32` of
length 2. `shape[0]` is the dimension on the domain of the operator,
`shape[1]` is the dimension of the co-domain of the operator. On other
words, if operator represents an N x N matrix A, `shape` must contain
`[N, N]`.
- dtype: The datatype of input to and output from `apply`.
- apply: Callable object taking a vector `x` as input and returning a
vector with the result of applying the operator to `x`, i.e. if
`operator` represents matrix `A`, `apply` should return `A * x`.
rhs: A rank-1 `Tensor` of shape `[N]` containing the right-hand size vector.
preconditioner: An object representing a linear operator, see `operator`
for detail. The preconditioner should approximate the inverse of `A`.
An efficient preconditioner could dramatically improve the rate of
convergence. If `preconditioner` represents matrix `M`(`M` approximates
`A^{-1}`), the algorithm uses `preconditioner.apply(x)` to estimate
`A^{-1}x`. For this to be useful, the cost of applying `M` should be
much lower than computing `A^{-1}` directly.
x: A rank-1 `Tensor` of shape `[N]` containing the initial guess for the
solution.
tol: A float scalar convergence tolerance.
max_iter: An integer giving the maximum number of iterations.
name: A name scope for the operation.
Returns:
output: A namedtuple representing the final state with fields:
- i: A scalar `int32` `Tensor`. Number of iterations executed.
- x: A rank-1 `Tensor` of shape `[N]` containing the computed solution.
- r: A rank-1 `Tensor` of shape `[M]` containing the residual vector.
- p: A rank-1 `Tensor` of shape `[N]`. `A`-conjugate basis vector.
- gamma: \\(r \dot M \dot r\\), equivalent to \\(||r||_2^2\\) when
`preconditioner=None`.
"""
# ephemeral class holding CG state.
cg_state = collections.namedtuple("CGState", ["i", "x", "r", "p", "gamma"])
def stopping_criterion(i, state):
return math_ops.logical_and(i < max_iter, linalg_ops.norm(state.r) > tol)
def cg_step(i, state): # pylint: disable=missing-docstring
z = operator.apply(state.p)
alpha = state.gamma / util.dot(state.p, z)
x = state.x + alpha * state.p
r = state.r - alpha * z
if preconditioner is None:
gamma = util.dot(r, r)
beta = gamma / state.gamma
p = r + beta * state.p
else:
q = preconditioner.apply(r)
gamma = util.dot(r, q)
beta = gamma / state.gamma
p = q + beta * state.p
return i + 1, cg_state(i + 1, x, r, p, gamma)
with ops.name_scope(name):
n = operator.shape[1:]
rhs = array_ops.expand_dims(rhs, -1)
if x is None:
x = array_ops.expand_dims(
array_ops.zeros(n, dtype=rhs.dtype.base_dtype), -1)
r0 = rhs
else:
x = array_ops.expand_dims(x, -1)
r0 = rhs - operator.apply(x)
if preconditioner is None:
p0 = r0
else:
p0 = preconditioner.apply(r0)
gamma0 = util.dot(r0, p0)
tol *= linalg_ops.norm(r0)
i = constant_op.constant(0, dtype=dtypes.int32)
state = cg_state(i=i, x=x, r=r0, p=p0, gamma=gamma0)
_, state = control_flow_ops.while_loop(stopping_criterion, cg_step,
[i, state])
return cg_state(
state.i,
x=array_ops.squeeze(state.x),
r=array_ops.squeeze(state.r),
p=array_ops.squeeze(state.p),
gamma=state.gamma)