L2NormSquared¶

class odl.solvers.functional.default_functionals.L2NormSquared(*args, **kwargs)[source]¶

Bases: odl.solvers.functional.functional.Functional

The functional corresponding to the squared L2-norm.

The squared L2-norm, ||x||_2^2, is defined as the integral/sum of x^2.

Notes

If the functional is defined on an $\mathbb{R}^n$ -like space, the $\| \cdot \|_2^2$ -functional is defined as

$\| x \|_2^2 = \sum_{i=1}^n |x_i|^2.$

If the functional is defined on an $L_2$ -like space, the $\| \cdot \|_2^2$ -functional is defined as

$\| x \|_2^2 = \int_\Omega |x(t)|^2 dt.$

The proximal factory allows using vector-valued stepsizes:

>>> space = odl.rn(3)
>>> f = odl.solvers.L2NormSquared(space)
>>> x = space.one()
>>> f.proximal([0.5, 1.5, 2.0])(x)
rn(3).element([ 0.5 ,  0.25,  0.2 ])

Attributes

adjoint: Adjoint of this operator (abstract).
convex_conj: The convex conjugate functional of the squared L2-norm.
domain: Set of objects on which this operator can be evaluated.
grad_lipschitz: Lipschitz constant for the gradient of the functional.
gradient: Gradient operator of the functional.
inverse: Return the operator inverse.
is_functional: True if this operator’s range is a Field.
is_linear: True if this operator is linear.
proximal: Return the proximal factory of the functional.
range: Set in which the result of an evaluation of this operator lies.

Methods

`_call`(self, x)	Return the squared L2-norm of `x`.
`bregman`(self, point, subgrad)	Return the Bregman distance functional.
`derivative`(self, point)	Return the derivative operator in the given point.
`norm`(self[, estimate])	Return the operator norm of this operator.
`translated`(self, shift)	Return a translation of the functional.

__init__(self, space)[source]¶

Initialize a new instance.

Parameters

spaceDiscretizedSpace or TensorSpace: Domain of the functional.