IndicatorLpUnitBall¶

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

Bases: odl.solvers.functional.functional.Functional

The indicator function on the unit ball in given the Lp norm.

It does not implement gradient since it is not differentiable everywhere.

Notes

This functional is defined as

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where $||x||_{L_p}$ is the $L_p$ -norm, which for finite values of $p$ is defined as

$\| x \|_{L_p} = \left( \int_{\Omega} |x|^p dx \right)^{1/p},$

and for $p = \infty$ it is defined as

$||x||_{\infty} = \max_x (|x|).$

The functional also allows noninteger and nonpositive values of the exponent $p$ , however in this case $\| x \|_{L_p}$ is not a norm.

Attributes

adjoint: Adjoint of this operator (abstract).
convex_conj: The conjugate functional of IndicatorLpUnitBall.
domain: Set of objects on which this operator can be evaluated.
exponent: Exponent corresponding to the norm.
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)	Apply the functional to the given point.
`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, exponent)[source]¶

Initialize a new instance.

Parameters

spaceDiscretizedSpace or TensorSpace: Domain of the functional.
exponentint or infinity: Specifies wich norm to use.