def __init__(self, space):
"""Initialize an instance.
Parameters
----------
space : `FnBase`
The domain of the operator.
"""
if not isinstance(space, LinearSpace):
raise TypeError('`space` {!r} not a `LinearSpace`'.format(space))
if _is_integer_only_ufunc(name) and not is_int_dtype(space.dtype):
raise ValueError("ufunc '{}' only defined with integral dtype"
"".format(name))
if nargin == 1:
domain = space
else:
domain = ProductSpace(space, nargin)
if nargout == 1:
range = space
else:
range = ProductSpace(space, nargout)
linear = name in LINEAR_UFUNCS
Operator.__init__(self, domain=domain, range=range, linear=linear)
def __init__(self, sigma):
"""Initialize a new instance.
Parameters
----------
sigma : positive float
Scaling parameter in the proximal operator.
"""
if isinstance(space, ProductSpace) and space.is_power_space:
self.exponent = space[0].element(exponent)
elif isinstance(space, DiscreteLp):
self.exponent = space.element(exponent)
else:
raise TypeError('space must be a `DiscreteLp` instance or '
'a power space of those, got {!r}'
''.format(space))
if g is not None:
self.g = self.domain.element(g)
else:
self.g = None
Operator.__init__(self, domain=space, range=space, linear=False)
self.sigma = float(sigma)
self.impl = impl
def __init__(self, space):
"""Initialize an instance.
Parameters
----------
space : `TensorSpace`
The domain of the operator.
"""
if not isinstance(space, LinearSpace):
raise TypeError('`space` {!r} not a `LinearSpace`'.format(space))
if nargin == 1:
domain = space0 = space
dtypes = [space.dtype]
elif nargin == len(space) == 2 and isinstance(space, ProductSpace):
domain = space
space0 = space[0]
dtypes = [space[0].dtype, space[1].dtype]
else:
domain = ProductSpace(space, nargin)
space0 = space
dtypes = [space.dtype, space.dtype]
dts_out = dtypes_out(name, dtypes)
if nargout == 1:
range = space0.astype(dts_out[0])
else:
range = ProductSpace(space0.astype(dts_out[0]),
space0.astype(dts_out[1]))
linear = name in LINEAR_UFUNCS
Operator.__init__(self, domain=domain, range=range, linear=linear)
def __init__(self, space):
"""Initialize an instance.
Parameters
----------
space : `FnBase`
The domain of the operator.
"""
if not isinstance(space, LinearSpace):
raise TypeError('`space` {!r} not a `LinearSpace`'.format(space))
if _is_integer_only_ufunc(name) and not is_int_dtype(space.dtype):
raise ValueError("ufunc '{}' only defined with integral dtype"
"".format(name))
if nargin == 1:
domain = space
else:
domain = ProductSpace(space, nargin)
if nargout == 1:
range = space
else:
range = ProductSpace(space, nargout)
linear = name in LINEAR_UFUNCS
Operator.__init__(self, domain=domain, range=range, linear=linear)
def __init__(self, map_type, fset, partition, dspace, linear=False,
**kwargs):
"""Initialize a new instance.
Parameters
----------
map_type : {'sampling', 'interpolation'}
The type of operator
fset : `FunctionSet`
The non-discretized (abstract) set of functions to be
discretized
partition : `RectPartition`
Partition of (a subset of) ``fset.domain`` based on a
`RectGrid`.
dspace : `NtuplesBase`
Data space providing containers for the values of a
discretized object. Its `NtuplesBase.size` must be equal
to the total number of grid points.
linear : bool, optional
Create a linear operator if ``True``, otherwise a non-linear
operator.
order : {'C', 'F'}, optional
Ordering of the axes in the data storage. 'C' means the
first axis varies slowest, the last axis fastest;
vice versa for 'F'.
Default: 'C'
"""
map_type_ = str(map_type).lower()
if map_type_ not in ('sampling', 'interpolation'):
raise ValueError('`map_type` {} not understood'
''.format(map_type))
if not isinstance(fset, FunctionSet):
raise TypeError('`fset` {!r} is not a `FunctionSet` '
'instance'.format(fset))
if not isinstance(partition, RectPartition):
raise TypeError('`partition` {!r} is not a `RectPartition` '
'instance'.format(partition))
if not isinstance(dspace, NtuplesBase):
raise TypeError('`dspace` {!r} is not an `NtuplesBase` instance'
''.format(dspace))
if not fset.domain.contains_set(partition):
raise ValueError('{} not contained in the domain {} '
'of the function set {}'
''.format(partition, fset.domain, fset))
if dspace.size != partition.size:
raise ValueError('size {} of the data space {} not equal '
'to the size {} of the partition'
''.format(dspace.size, dspace, partition.size))
domain = fset if map_type_ == 'sampling' else dspace
range = dspace if map_type_ == 'sampling' else fset
Operator.__init__(self, domain, range, linear=linear)
self.__partition = partition
if self.is_linear:
if not isinstance(fset, FunctionSpace):
raise TypeError('`fset` {!r} is not a `FunctionSpace` '
'instance'.format(fset))
if not isinstance(dspace, FnBase):
raise TypeError('`dspace` {!r} is not an `FnBase` instance'
''.format(dspace))
if fset.field != dspace.field:
raise ValueError('`field` {} of the function space and `field`'
' {} of the data space are not equal'
''.format(fset.field, dspace.field))
order = str(kwargs.pop('order', 'C'))
if str(order).upper() not in ('C', 'F'):
raise ValueError('`order` {!r} not recognized'.format(order))
else:
self.__order = str(order).upper()
def __init__(self, stepsize):
Operator.__init__(self, func.domain, func.domain, False)
self.stepsize = stepsize
def __init__(self, operator, point, method='forward', step=None):
"""Initialize a new instance.
Parameters
----------
operator : `Operator`
The operator whose derivative should be computed numerically. Its
domain and range must be `FnBase` spaces.
point : ``operator.domain`` `element-like`
The point to compute the derivative in.
method : {'backward', 'forward', 'central'}
The method to use to compute the derivative.
step : float
The step length used in the derivative computation.
Default: selects the step according to the dtype of the space.
Examples
--------
Compute a numerical estimate of the derivative (Hessian) of the squared
L2 norm:
>>> space = odl.rn(3)
>>> func = odl.solvers.L2NormSquared(space)
>>> hess = NumericalDerivative(func.gradient, [1, 1, 1])
>>> hess([0, 0, 1])
rn(3).element([0.0, 0.0, 2.0])
Find the Hessian matrix:
>>> odl.matrix_representation(hess)
array([[ 2., 0., 0.],
[ 0., 2., 0.],
[ 0., 0., 2.]])
Notes
-----
If the operator is :math:`A` and step size :math:`h` is used, the
derivative in the point :math:`x` and direction :math:`dx` is computed
as follows.
``method='backward'``:
.. math::
\\partial A(x)(dx) =
(A(x) - A(x - dx \\cdot h / \| dx \|))
\\cdot \\frac{\| dx \|}{h}
``method='forward'``:
.. math::
\\partial A(x)(dx) =
(A(x + dx \\cdot h / \| dx \|) - A(x))
\\cdot \\frac{\| dx \|}{h}
``method='central'``:
.. math::
\\partial A(x)(dx) =
(A(x + dx \\cdot h / (2 \| dx \|)) -
A(x - dx \\cdot h / (2 \| dx \|))
\\cdot \\frac{\| dx \|}{h}
The number of operator evaluations is ``2``, regardless of parameters.
"""
if not isinstance(operator, Operator):
raise TypeError('`operator` has to be an `Operator` instance')
if not isinstance(operator.domain, FnBase):
raise TypeError('`operator.domain` has to be an `FnBase` '
'instance')
if not isinstance(operator.range, FnBase):
raise TypeError('`operator.range` has to be an `FnBase` '
'instance')
self.operator = operator
self.point = operator.domain.element(point)
if step is None:
# Use half of the number of digits as machine epsilon, this
# "usually" gives a good balance between precision and numerical
# stability.
self.step = np.sqrt(np.finfo(operator.domain.dtype).eps)
else:
self.step = float(step)
self.method, method_in = str(method).lower(), method
if self.method not in ('backward', 'forward', 'central'):
raise ValueError("`method` '{}' not understood").format(method_in)
Operator.__init__(self, operator.domain, operator.range,
linear=True)
def __init__(self, functional, method='forward', step=None):
"""Initialize a new instance.
Parameters
----------
functional : `Functional`
The functional whose gradient should be computed. Its domain must
be an `FnBase` space.
method : {'backward', 'forward', 'central'}
The method to use to compute the gradient.
step : float
The step length used in the derivative computation.
Default: selects the step according to the dtype of the space.
Examples
--------
>>> space = odl.rn(3)
>>> func = odl.solvers.L2NormSquared(space)
>>> grad = NumericalGradient(func)
>>> grad([1, 1, 1])
rn(3).element([2.0, 2.0, 2.0])
The gradient gives the correct value with sufficiently small step size:
>>> grad([1, 1, 1]) == func.gradient([1, 1, 1])
True
If the step is too large the result is not correct:
>>> grad = NumericalGradient(func, step=0.5)
>>> grad([1, 1, 1])
rn(3).element([2.5, 2.5, 2.5])
But it can be improved by using the more accurate ``method='central'``:
>>> grad = NumericalGradient(func, method='central', step=0.5)
>>> grad([1, 1, 1])
rn(3).element([2.0, 2.0, 2.0])
Notes
-----
If the functional is :math:`f` and step size :math:`h` is used, the
gradient is computed as follows.
``method='backward'``:
.. math::
(\\nabla f(x))_i = \\frac{f(x) - f(x - h e_i)}{h}
``method='forward'``:
.. math::
(\\nabla f(x))_i = \\frac{f(x + h e_i) - f(x)}{h}
``method='central'``:
.. math::
(\\nabla f(x))_i = \\frac{f(x + (h/2) e_i) - f(x - (h/2) e_i)}{h}
The number of function evaluations is ``functional.domain.size + 1`` if
``'backward'`` or ``'forward'`` is used and
``2 * functional.domain.size`` if ``'central'`` is used.
On large domains this will be computationally infeasible.
"""
if not isinstance(functional, Functional):
raise TypeError('`functional` has to be a `Functional` instance')
if not isinstance(functional.domain, FnBase):
raise TypeError('`functional.domain` has to be an `FnBase` '
'instance')
self.functional = functional
if step is None:
# Use half of the number of digits as machine epsilon, this
# "usually" gives a good balance between precision and numerical
# stability.
self.step = np.sqrt(np.finfo(functional.domain.dtype).eps)
else:
self.step = float(step)
self.method, method_in = str(method).lower(), method
if self.method not in ('backward', 'forward', 'central'):
raise ValueError("`method` '{}' not understood").format(method_in)
Operator.__init__(self, functional.domain, functional.domain,
linear=functional.is_linear)
def __init__(self, map_type, fset, partition, dspace, linear=False,
**kwargs):
"""Initialize a new instance.
Parameters
----------
map_type : {'sampling', 'interpolation'}
The type of operator
fset : `FunctionSet`
The non-discretized (abstract) set of functions to be
discretized
partition : `RectPartition`
Partition of (a subset of) ``fset.domain`` based on a
`RectGrid`.
dspace : `NtuplesBase`
Data space providing containers for the values of a
discretized object. Its `NtuplesBase.size` must be equal
to the total number of grid points.
linear : bool, optional
Create a linear operator if ``True``, otherwise a non-linear
operator.
order : {'C', 'F'}, optional
Ordering of the axes in the data storage. 'C' means the
first axis varies slowest, the last axis fastest;
vice versa for 'F'.
Default: 'C'
"""
map_type_ = str(map_type).lower()
if map_type_ not in ('sampling', 'interpolation'):
raise ValueError('`map_type` {} not understood'
''.format(map_type))
if not isinstance(fset, FunctionSet):
raise TypeError('`fset` {!r} is not a `FunctionSet` '
'instance'.format(fset))
if not isinstance(partition, RectPartition):
raise TypeError('`partition` {!r} is not a `RectPartition` '
'instance'.format(partition))
if not isinstance(dspace, NtuplesBase):
raise TypeError('`dspace` {!r} is not an `NtuplesBase` instance'
''.format(dspace))
if not fset.domain.contains_set(partition):
raise ValueError('{} not contained in the domain {} '
'of the function set {}'
''.format(partition, fset.domain, fset))
if dspace.size != partition.size:
raise ValueError('size {} of the data space {} not equal '
'to the size {} of the partition'
''.format(dspace.size, dspace, partition.size))
domain = fset if map_type_ == 'sampling' else dspace
range = dspace if map_type_ == 'sampling' else fset
Operator.__init__(self, domain, range, linear=linear)
self.__partition = partition
if self.is_linear:
if not isinstance(fset, FunctionSpace):
raise TypeError('`fset` {!r} is not a `FunctionSpace` '
'instance'.format(fset))
if not isinstance(dspace, FnBase):
raise TypeError('`dspace` {!r} is not an `FnBase` instance'
''.format(dspace))
if fset.field != dspace.field:
raise ValueError('`field` {} of the function space and `field`'
' {} of the data space are not equal'
''.format(fset.field, dspace.field))
order = str(kwargs.pop('order', 'C'))
if str(order).upper() not in ('C', 'F'):
raise ValueError('`order` {!r} not recognized'.format(order))
else:
self.__order = str(order).upper()
def __init__(self, operator, point, method='forward', step=None):
"""Initialize a new instance.
Parameters
----------
operator : `Operator`
The operator whose derivative should be computed numerically. Its
domain and range must be `FnBase` spaces.
point : ``operator.domain`` `element-like`
The point to compute the derivative in.
method : {'backward', 'forward', 'central'}, optional
The method to use to compute the derivative.
step : float, optional
The step length used in the derivative computation.
Default: selects the step according to the dtype of the space.
Examples
--------
Compute a numerical estimate of the derivative (Hessian) of the squared
L2 norm:
>>> space = odl.rn(3)
>>> func = odl.solvers.L2NormSquared(space)
>>> hess = NumericalDerivative(func.gradient, [1, 1, 1])
>>> hess([0, 0, 1])
rn(3).element([0.0, 0.0, 2.0])
Find the Hessian matrix:
>>> odl.matrix_representation(hess)
array([[ 2., 0., 0.],
[ 0., 2., 0.],
[ 0., 0., 2.]])
Notes
-----
If the operator is :math:`A` and step size :math:`h` is used, the
derivative in the point :math:`x` and direction :math:`dx` is computed
as follows.
``method='backward'``:
.. math::
\\partial A(x)(dx) =
(A(x) - A(x - dx \\cdot h / \| dx \|))
\\cdot \\frac{\| dx \|}{h}
``method='forward'``:
.. math::
\\partial A(x)(dx) =
(A(x + dx \\cdot h / \| dx \|) - A(x))
\\cdot \\frac{\| dx \|}{h}
``method='central'``:
.. math::
\\partial A(x)(dx) =
(A(x + dx \\cdot h / (2 \| dx \|)) -
A(x - dx \\cdot h / (2 \| dx \|))
\\cdot \\frac{\| dx \|}{h}
The number of operator evaluations is ``2``, regardless of parameters.
"""
if not isinstance(operator, Operator):
raise TypeError('`operator` has to be an `Operator` instance')
if not isinstance(operator.domain, FnBase):
raise TypeError('`operator.domain` has to be an `FnBase` '
'instance')
if not isinstance(operator.range, FnBase):
raise TypeError('`operator.range` has to be an `FnBase` '
'instance')
self.operator = operator
self.point = operator.domain.element(point)
if step is None:
# Use half of the number of digits as machine epsilon, this
# "usually" gives a good balance between precision and numerical
# stability.
self.step = np.sqrt(np.finfo(operator.domain.dtype).eps)
else:
self.step = float(step)
self.method, method_in = str(method).lower(), method
if self.method not in ('backward', 'forward', 'central'):
raise ValueError("`method` '{}' not understood").format(method_in)
Operator.__init__(self, operator.domain, operator.range,
linear=True)
def __init__(self, functional, method='forward', step=None):
"""Initialize a new instance.
Parameters
----------
functional : `Functional`
The functional whose gradient should be computed. Its domain must
be an `FnBase` space.
method : {'backward', 'forward', 'central'}, optional
The method to use to compute the gradient.
step : float, optional
The step length used in the derivative computation.
Default: selects the step according to the dtype of the space.
Examples
--------
>>> space = odl.rn(3)
>>> func = odl.solvers.L2NormSquared(space)
>>> grad = NumericalGradient(func)
>>> grad([1, 1, 1])
rn(3).element([2.0, 2.0, 2.0])
The gradient gives the correct value with sufficiently small step size:
>>> grad([1, 1, 1]) == func.gradient([1, 1, 1])
True
If the step is too large the result is not correct:
>>> grad = NumericalGradient(func, step=0.5)
>>> grad([1, 1, 1])
rn(3).element([2.5, 2.5, 2.5])
But it can be improved by using the more accurate ``method='central'``:
>>> grad = NumericalGradient(func, method='central', step=0.5)
>>> grad([1, 1, 1])
rn(3).element([2.0, 2.0, 2.0])
Notes
-----
If the functional is :math:`f` and step size :math:`h` is used, the
gradient is computed as follows.
``method='backward'``:
.. math::
(\\nabla f(x))_i = \\frac{f(x) - f(x - h e_i)}{h}
``method='forward'``:
.. math::
(\\nabla f(x))_i = \\frac{f(x + h e_i) - f(x)}{h}
``method='central'``:
.. math::
(\\nabla f(x))_i = \\frac{f(x + (h/2) e_i) - f(x - (h/2) e_i)}{h}
The number of function evaluations is ``functional.domain.size + 1`` if
``'backward'`` or ``'forward'`` is used and
``2 * functional.domain.size`` if ``'central'`` is used.
On large domains this will be computationally infeasible.
"""
if not isinstance(functional, Functional):
raise TypeError('`functional` has to be a `Functional` instance')
if not isinstance(functional.domain, FnBase):
raise TypeError('`functional.domain` has to be an `FnBase` '
'instance')
self.functional = functional
if step is None:
# Use half of the number of digits as machine epsilon, this
# "usually" gives a good balance between precision and numerical
# stability.
self.step = np.sqrt(np.finfo(functional.domain.dtype).eps)
else:
self.step = float(step)
self.method, method_in = str(method).lower(), method
if self.method not in ('backward', 'forward', 'central'):
raise ValueError("`method` '{}' not understood").format(method_in)
Operator.__init__(self, functional.domain, functional.domain,
linear=functional.is_linear)
