landweber¶

odl.solvers.iterative.iterative.landweber(op, x, rhs, niter, omega=None, projection=None, callback=None)[source]¶

Optimized implementation of Landweber’s method.

Solves the inverse problem:

A(x) = rhs

Parameters

opOperator: Operator in the inverse problem. op.derivative(x).adjoint must be well-defined for x in the operator domain.
xop.domain element: Element to which the result is written. Its initial value is used as starting point of the iteration, and its values are updated in each iteration step.
rhsop.range element: Right-hand side of the equation defining the inverse problem.
niterint: Number of iterations.
omegapositive float, optional: Relaxation parameter in the iteration. Default: 1 / op.norm(estimate=True) ** 2
projectioncallable, optional: Function that can be used to modify the iterates in each iteration, for example enforcing positivity. The function should take one argument and modify it in-place.
callbackcallable, optional: Object executing code per iteration, e.g. plotting each iterate.

Notes

This method calculates an approximate least-squares solution of the inverse problem of the first kind

$\mathcal{A} (x) = y,$

for a given $y\in \mathcal{Y}$ , i.e. an approximate solution $x^*$ to

$\min_{x\in \mathcal{X}} \| \mathcal{A}(x) - y \|_{\mathcal{Y}}^2$

for a (Frechet-) differentiable operator $\mathcal{A}: \mathcal{X} \to \mathcal{Y}$ between Hilbert spaces $\mathcal{X}$ and $\mathcal{Y}$ . The method starts from an initial guess $x_0$ and uses the iteration

$x_{k+1} = x_k - \omega \ \partial \mathcal{A}(x)^* (\mathcal{A}(x_k) - y),$

where $\partial \mathcal{A}(x)$ is the Frechet derivative of $\mathcal{A}$ at $x$ and $\omega$ is a relaxation parameter. For linear problems, a choice $0 < \omega < 2/\lVert \mathcal{A}^2\rVert$ guarantees convergence, where $\lVert\mathcal{A}\rVert$ stands for the operator norm of $\mathcal{A}$ .

Users may also optionally provide a projection to project each iterate onto some subset. For example enforcing positivity.

This implementation uses a minimum amount of memory copies by applying re-usable temporaries and in-place evaluation.

The method is also described in a Wikipedia article.