Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data

In this paper we study a Tikhonov-type method for ill-posed nonlinear operator equations \gdag = F( ag) where \gdag is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density t\gdag where $t>0$ may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations, and phase retrieval problems in optics. Our approach uses a Kullback-Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as $t\to\infty$ both for an a priori and for a Lepski{\u\i}-type parameter choice rule

Proton-induced noise in digicons

The Space Telescope, which carries four Digicons, will pass several times per day through a low-altitude portion of the radiation belt called the South Atlantic Anomaly. This is expected to create interference in what is otherwise anticipated to be a noise-free device. Two essential components of the Digicon, the semiconductor diode array and the UV transmitting window, generate noise when subjected to medium-energy proton radiation, a primary component of the belt. These trapped protons, having energies ranging from 2 to 400 Mev and fluences at the Digicon up to 4,000 P+/sec-sq cm, pass through both the window and the diode array, depositing energy in each. In order to evaluate the effect of these protons, engineering test models of Digicon tubes to be flown on the High Resolution Spectrograph were irradiated with low-flux monoenergetic proton beams at the University of Maryland cyclotron. Electron-hole pairs produced by the protons passing through the diodes or the surrounding bulk caused a background count rate. This is the result of holes diffusing over a distance of many diode spacings, causing counts to be triggered simultaneously in the output circuits of several adjacent diodes. Pulse-height spectra of these proton-induced counts indicate that most of the bulk-related counts overlap the single photoelectron peak. A geometrical model will be presented of the charge collection characteristics of the diode array that accounts for most of the observed effects

Evolution of precipitates, in particular cruciform and cuboid particles, during simulated direct charging of thin slab cast vanadium microalloyed steels

A study has been undertaken of four vanadium based steels which have been processed by a simulated direct charging route using processing parameters typical of thin slab casting, where the cast product has a thickness of 50 to 80mm ( in this study 50 mm) and is fed directly to a furnace to equalise the microstructure prior to rolling. In the direct charging process, cooling rates are faster, equalisation times shorter and the amount of deformation introduced during rolling less than in conventional practice. Samples in this study were quenched after casting, after equalisation, after 4th rolling pass and after coiling, to follow the evolution of microstructure. The mechanical and toughness properties and the microstructural features might be expected to differ from equivalent steels, which have undergone conventional processing. The four low carbon steels (~0.06wt%) which were studied contained 0.1wt%V (V-N), 0.1wt%V and 0.010wt%Ti (V-Ti), 0.1wt%V and 0.03wt%Nb (V-Nb), and 0.1wt%V, 0.03wt%Nb and 0.007wt%Ti (V-Nb-Ti). Steels V-N and V-Ti contained around 0.02wt% N, while the other two contained about 0.01wt%N. The as-cast steels were heated at three equalising temperatures of 1050C, 1100C or 1200C and held for 30-60 minutes prior to rolling. Optical microscopy and analytical electron microscopy, including parallel electron energy loss spectroscopy (PEELS), were used to characterise the precipitates. In the as-cast condition, dendrites and plates were found. Cuboid particles were seen at this stage in Steel V-Ti, but they appeared only in the other steels after equalization. In addition, in the final product of all the steels, fine particles were seen, but it was only in the two titanium steels that cruciform precipitates were present. PEELS analysis showed that the dendrites, plates, cuboids, cruciforms and fine precipitates were essentially nitrides. The two Ti steels had better toughness than the other steels but inferior lower yield stress values. This was thought to be, in part, due to the formation of cruciform precipitates in austenite, thereby removing nitrogen and the microalloying elements which would have been expected to precipitate in ferrite as dispersion hardening particles

Necessary conditions for variational regularization schemes

We study variational regularization methods in a general framework, more precisely those methods that use a discrepancy and a regularization functional. While several sets of sufficient conditions are known to obtain a regularization method, we start with an investigation of the converse question: How could necessary conditions for a variational method to provide a regularization method look like? To this end, we formalize the notion of a variational scheme and start with comparison of three different instances of variational methods. Then we focus on the data space model and investigate the role and interplay of the topological structure, the convergence notion and the discrepancy functional. Especially, we deduce necessary conditions for the discrepancy functional to fulfill usual continuity assumptions. The results are applied to discrepancy functionals given by Bregman distances and especially to the Kullback-Leibler divergence.Comment: To appear in Inverse Problem

In Situ Detection of Active Edge Sites in Single-Layer MoS2_2 Catalysts

MoS2 nanoparticles are proven catalysts for processes such as hydrodesulphurization and hydrogen evolution, but unravelling their atomic-scale structure under catalytic working conditions has remained significantly challenging. Ambient pressure X-ray Photoelectron Spectroscopy (AP-XPS) allows us to follow in-situ the formation of the catalytically relevant MoS2 edge sites in their active state. The XPS fingerprint is described by independent contributions to the Mo3d core level spectrum whose relative intensity is sensitive to the thermodynamic conditions. Density Functional Theory (DFT) is used to model the triangular MoS2 particles on Au(111) and identify the particular sulphidation state of the edge sites. A consistent picture emerges in which the core level shifts for the edge Mo atoms evolve counter-intuitively towards higher binding energies when the active edges are reduced. The shift is explained by a surprising alteration in the metallic character of the edge sites, which is a distinct spectroscopic signature of the MoS2 edges under working conditions

Efficient CSL Model Checking Using Stratification

For continuous-time Markov chains, the model-checking problem with respect to continuous-time stochastic logic (CSL) has been introduced and shown to be decidable by Aziz, Sanwal, Singhal and Brayton in 1996. Their proof can be turned into an approximation algorithm with worse than exponential complexity. In 2000, Baier, Haverkort, Hermanns and Katoen presented an efficient polynomial-time approximation algorithm for the sublogic in which only binary until is allowed. In this paper, we propose such an efficient polynomial-time approximation algorithm for full CSL. The key to our method is the notion of stratified CTMCs with respect to the CSL property to be checked. On a stratified CTMC, the probability to satisfy a CSL path formula can be approximated by a transient analysis in polynomial time (using uniformization). We present a measure-preserving, linear-time and -space transformation of any CTMC into an equivalent, stratified one. This makes the present work the centerpiece of a broadly applicable full CSL model checker. Recently, the decision algorithm by Aziz et al. was shown to work only for stratified CTMCs. As an additional contribution, our measure-preserving transformation can be used to ensure the decidability for general CTMCs.Comment: 18 pages, preprint for LMCS. An extended abstract appeared in ICALP 201

Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data

We study Newton type methods for inverse problems described by nonlinear operator equations $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss-Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback-Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performence of the proposed method for these problems is illustrated in numerical examples