PointwiseNorm¶

class odl.operator.tensor_ops.PointwiseNorm(*args, **kwargs)[source]¶

Bases: PointwiseTensorFieldOperator

Take the point-wise norm of a vector field.

This operator computes the (weighted) p-norm in each point of a vector field, thus producing a scalar-valued function. It implements the formulas

||F(x)|| = [ sum_j( w_j * |F_j(x)|^p ) ]^(1/p)

for p finite and

||F(x)|| = max_j( w_j * |F_j(x)| )

for p = inf, where F is a vector field. This implies that the Operator.domain is a power space of a discretized function space. For example, if X is a DiscretizedSpace space, then ProductSpace(X, d) is a valid domain for any positive integer d.

Attributes:

adjoint: Adjoint of this operator (abstract).
base_space: Base space X of this operator's domain and range.
domain: Set of objects on which this operator can be evaluated.
exponent: Exponent p of this norm.
inverse: Return the operator inverse.
is_functional: True if this operator's range is a Field.
is_linear: True if this operator is linear.
is_weighted: True if weighting is not 1 or all ones.
range: Set in which the result of an evaluation of this operator lies.
weights: Weighting array of this operator.

Methods

`__call__`(x[, out])	Return `self(x[, out, **kwargs])`.
`derivative`(vf)	Derivative of the point-wise norm operator at `vf`.
`norm`([estimate])	Return the operator norm of this operator.

__init__(vfspace, exponent=None, weighting=None)[source]¶

Initialize a new instance.

Parameters:

vfspaceProductSpace: Space of vector fields on which the operator acts. It has to be a product space of identical spaces, i.e. a power space.
exponentnon-zero float, optional: Exponent of the norm in each point. Values between 0 and 1 are currently not supported due to numerical instability. Default: vfspace.exponent
weightingarray-like or positive float, optional: Weighting array or constant for the norm. If an array is given, its length must be equal to len(domain), and all entries must be positive. By default, the weights are is taken from domain.weighting. Note that this excludes unusual weightings with custom inner product, norm or dist.

Examples

We make a tiny vector field space in 2D and create the standard point-wise norm operator on that space. The operator maps a vector field to a scalar function:

>>> spc = odl.uniform_discr([-1, -1], [1, 1], (1, 2))
>>> vfspace = odl.ProductSpace(spc, 2)
>>> pw_norm = odl.PointwiseNorm(vfspace)
>>> pw_norm.range == spc
True

Now we can calculate the 2-norm in each point:

>>> x = vfspace.element([[[1, -4]],
...                      [[0, 3]]])
>>> print(pw_norm(x))
[[ 1.,  5.]]

We can change the exponent either in the vector field space or in the operator directly:

>>> vfspace = odl.ProductSpace(spc, 2, exponent=1)
>>> pw_norm = PointwiseNorm(vfspace)
>>> print(pw_norm(x))
[[ 1.,  7.]]
>>> vfspace = odl.ProductSpace(spc, 2)
>>> pw_norm = PointwiseNorm(vfspace, exponent=1)
>>> print(pw_norm(x))
[[ 1.,  7.]]