We study Tikhonov regularization for possibly nonlinear inverse problems with weighted
ℓ
1
-penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically ...an
L
2
-space, is assumed to satisfy a two-sided Lipschitz condition with respect to a weighted
ℓ
2
-norm and the norm of the image space. We show that in this setting approximation rates of arbitrarily high Hölder-type order in the regularization parameter can be achieved, and we characterize maximal subspaces of sequences on which these rates are attained. On these subspaces the method also converges with optimal rates in terms of the noise level with the discrepancy principle as parameter choice rule. Our analysis includes the case that the penalty term is not finite at the exact solution (’oversmoothing’). As a standard example we discuss wavelet regularization in Besov spaces
B
1
,
1
r
. In this setting we demonstrate in numerical simulations for a parameter identification problem in a differential equation that our theoretical results correctly predict improved rates of convergence for piecewise smooth unknown coefficients.
Lorentz microscopy of optical fields Gaida, John H.; Lourenço-Martins, Hugo; Yalunin, Sergey V. ...
Nature communications,
10/2023, Volume:
14, Issue:
1
Journal Article
Peer reviewed
Open access
Abstract
In electron microscopy, detailed insights into nanoscale optical properties of materials are gained by spontaneous inelastic scattering leading to electron-energy loss and ...cathodoluminescence. Stimulated scattering in the presence of external sample excitation allows for mode- and polarization-selective photon-induced near-field electron microscopy (PINEM). This process imprints a spatial phase profile inherited from the optical fields onto the wave function of the probing electrons. Here, we introduce Lorentz-PINEM for the full-field, non-invasive imaging of complex optical near fields at high spatial resolution. We use energy-filtered defocus phase-contrast imaging and iterative phase retrieval to reconstruct the phase distribution of interfering surface-bound modes on a plasmonic nanotip. Our approach is universally applicable to retrieve the spatially varying phase of nanoscale fields and topological modes.
This paper introduces a new type of infinite element for scattering and resonance problems that is derived from a variant of the pole condition as radiation condition. This condition states that a ...certain transform of the exterior solution belongs to the Hardy space of L² boundary values of holomorphic functions on the unit disc if and only if the solution is outgoing. We obtain a symmetric variational formulation of the problem in this Hardy space. Our infinite elements correspond to a Galerkin discretization with respect to the standard monomial orthogonal basis of this Hardy space and lead to simple element matrices. Hardy space infinite elements are particularly well suited for solving resonance problems since they preserve the eigenvalue structure of the problem. We prove superalgebraic convergence for a separated problem. Numerical experiments exhibit fast convergence over a wide range of wave numbers.
We describe a general strategy for the verification of variational source condition by formulating two sufficient criteria describing the smoothness of the solution and the degree of illposedness of ...the forward operator in terms of a family of subspaces. For linear deterministic inverse problems we show that variational source conditions are necessary and sufficient for convergence rates of spectral regularization methods, which are slower than the square root of the noise level. A similar result is shown for linear inverse problems with white noise. In many cases variational source conditions can be characterized by Besov spaces. This is discussed for a number of prominent inverse problems.
We demonstrate the generation and optical control of ultrashort high-coherence electron pulses. The free-electron quantum state is phase-modulated in the longitudinal and transverse dimensions, and ...the formation of attosecond electron pulse trains is quantitatively probed.
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 measured, the main mathematical challenge consists in a phase-retrieval problem in the near-field regime. We treat an often used linearized version of this problem known as the contract transfer function model. Surprisingly, this inverse problem turns out to be well-posed assuming only a compact support of the imaged object. Moreover, we establish bounds on the Lipschitz stability constant. In general this constant grows exponentially with the Fresnel number of the imaging setup. However, both for homogeneous objects, characterized by a fixed ratio of the induced refractive phase shifts and attenuation, and in the case of measurements at two distances, a much more favorable algebraic dependence on the Fresnel number can be shown. In some cases we establish order optimality of our estimates.
Abstract
We study Tikhonov regularization for possibly nonlinear inverse problems with weighted
$$\ell ^1$$
ℓ
1
-penalization. The forward operator, mapping from a sequence space to an arbitrary ...Banach space, typically an
$$L^2$$
L
2
-space, is assumed to satisfy a two-sided Lipschitz condition with respect to a weighted
$$\ell ^2$$
ℓ
2
-norm and the norm of the image space. We show that in this setting approximation rates of arbitrarily high Hölder-type order in the regularization parameter can be achieved, and we characterize maximal subspaces of sequences on which these rates are attained. On these subspaces the method also converges with optimal rates in terms of the noise level with the discrepancy principle as parameter choice rule. Our analysis includes the case that the penalty term is not finite at the exact solution (’oversmoothing’). As a standard example we discuss wavelet regularization in Besov spaces
$$B^r_{1,1}$$
B
1
,
1
r
. In this setting we demonstrate in numerical simulations for a parameter identification problem in a differential equation that our theoretical results correctly predict improved rates of convergence for piecewise smooth unknown coefficients.
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