Related papers: Phase retrieval in infinite-dimensional Hilbert sp…
The mutual intensity and its equivalent phase-space representations quantify an optical field's state of coherence and are important tools in the study of light propagation and dynamics, but they can only be estimated indirectly from…
The phase retrieval problem has a long history and is an important problem in many areas of optics. Theoretical understanding of phase retrieval is still limited and fundamental questions such as uniqueness and stability of the recovered…
This paper addresses the stability analysis of infinite-dimensional sampled-data systems under unbounded perturbations. We present two classes of unbounded perturbations preserving the exponential stability of sampled-data systems. To this…
The classical Kaczmarz iteration and its randomized variants are popular tools for fast inversion of linear overdetermined systems. This method extends naturally to the setting of the phase retrieval problem via substituting at each…
We study the stability issue for the inverse problem of determining a coefficient appearing in a Schr\"odinger equation defined on an infinite cylindrical waveguide. More precisely, we prove the stable recovery of some general class of…
The Fourier dimension is not in general stable under finite unions of sets. Moreover, the stability of the Fourier dimension on particular pairs of sets is independent from the stability of the compact Fourier dimension.
Solutions of bilevel optimization problems tend to suffer from instability under changes to problem data. In the optimistic setting, we construct a lifted formulation that exhibits desirable stability properties under mild assumptions that…
Conditional stability estimates require additional regularization for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this context, we consider ill-posed nonlinear inverse problems…
Phase retrieval deals with the estimation of complex-valued signals solely from the magnitudes of linear measurements. While there has been a recent explosion in the development of phase retrieval algorithms, the lack of a common interface…
We consider the recovery of square-integrable signals from discrete, equidistant samples of their Gabor transform magnitude and show that, in general, signals can not be recovered from such samples. In particular, we show that for any…
This paper addresses fundamental scaling issues that hinder phase retrieval (PR) in high dimensions. We show that, if the measurement matrix can be put into a generalized block-diagonal form, a large PR problem can be solved on separate…
Propagation-based X-ray phase contrast enables nanoscale imaging of biological tissue by probing not only the attenuation, but also the real part of the refractive index of the sample. Since only intensities of diffracted waves can be…
Instabilities and pattern formation is the rule in nonequilibrium systems. Selection of a persistent lengthscale, or coarsening (increase of the lengthscale with time) are the two major alternatives. When and under which conditions one…
In this work we shall review the (phased) inverse scattering problem and then pursue the phaseless reconstruction from far-field data with the help of the concept of scattering coefficients. We perform sensitivity, resolution and stability…
We study stability theory in Hilbert spaces quantitatively. We prove that the inner product on the unit ball is $(k,\epsilon)$-stable for all $k\ge \exp(\pi/\epsilon)$, and it is not $(k,\epsilon)$-stable for $k\le \exp(\log 2/\epsilon)$,…
In this paper, we study the success rate of the reconstruction of objects of finite extent given the magnitude of its Fourier transform and its geometrical shape. We demonstrate that the commonly used combination of the hybrid input output…
We introduce a method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we…
A Hilbert manifold structure is described for the ADM phase space of asymptotically flat initial data $(g,\pi)$ with local $H^2\times H^1$ Sobolev regularity. Solutions of the constraint equations form a Hilbert submanifold. A regularized…
Recovering an unknown complex signal from the magnitude of linear combinations of the signal is referred to as phase retrieval. We present an exact performance analysis of a recently proposed convex-optimization-formulation for this…
Stability and stabilization of linear port-Hamiltonian systems on infinite-dimensional spaces are investigated. This class is general enough to include models of beams and waves as well as transport and Schr\"odinger equations with boundary…