Generalized Qualification and Qualification Levels for Spectral Regularization Methods

The concept of qualification for spectral regularization methods for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error. In this article, the definition of qualification is extended and three different levels are introduced: weak, strong and optimal. It is shown that the weak qualification extends the definition introduced by Mathe and Pereverzev in 2003, mainly in the sense that the functions associated to orders of convergence and source sets need not be the same. It is shown that certain methods possessing infinite classical qualification, e.g. truncated singular value decomposition (TSVD), Landweber's method and Showalter's method, also have generalized qualification leading to an optimal order of convergence of the regularization error. Sufficient conditions for a SRM to have weak qualification are provided and necessary and sufficient conditions for a given order of convergence to be strong or optimal qualification are found. Examples of all three qualification levels are provided and the relationships between them as well as with the classical concept of qualification and the qualification introduced by Mathe and Perevezev are shown. In particular, spectral regularization methods having extended qualification in each one of the three levels and having zero or infinite classical qualification are presented. Finally several implications of this theory in the context of orders of convergence, converse results and maximal source sets for inverse ill-posed problems, are shown.Comment: 20 pages, 1 figur

ATLAS monitored drift tube chambers for super-LHC

After the high-luminosity upgrade of the Large Hadron Collider (LHC) at CERN, the ATLAS muon spectrometer is expected to work at 10 times increased background rates of gammas and neutrons. This is challenging as the momentum resolution of the spectrometer is expected to be 10 %. This requires a single tube resolution of the muon drift tubes of 80 mum. At background rates around 1000 Hz/cm2 space charge effects will lead in the slow and non-linear AR:CO2 = 93:7 gas mixture to a degradation of the drift-tube spatial resolution. This was studied before experimentally for gammas and low energetic neutrons. Almost no information exists for fast neutrons. Therefore, we organized our studies under the following aspects: - We investigated the influence of 11 MeV neutrons on the position resolution of ATLAS MDT chambers. At flux densities between 4 and 16 kHz/cm2, almost no influence on the position resolution was found, it degrades by only 10 mum at a detection efficiency of only 4*10-4. - We investigated inert gas mixtures on fastness and linearity of their position-drifttime (r-t) relation. At a reduction of the maximum drift time by a factor of 2, the use of the present hardware and electronics might be possible. For our experimental studies we used our Munich cosmic ray facility. Two gas mixtures show almost identical position resolution as the standard gas. - For spectrometer regions of highest background rates we contributed to the investigation of newly developed 15 mm drift tubes. Position resolutions have been measured as a function of gamma background rates between 0 and 1400 Hz/cm2. - Garfield simulations have been performed to simulate space charge effects due to gamma irradiation. Results will be presented for the standard geometry as well as for the new 15 mm drift tubes.Comment: 3 pages, 7 figures, conferenc

Regularization independent of the noise level: an analysis of quasi-optimality

The quasi-optimality criterion chooses the regularization parameter in inverse problems without taking into account the noise level. This rule works remarkably well in practice, although Bakushinskii has shown that there are always counterexamples with very poor performance. We propose an average case analysis of quasi-optimality for spectral cut-off estimators and we prove that the quasi-optimality criterion determines estimators which are rate-optimal {\em on average}. Its practical performance is illustrated with a calibration problem from mathematical finance.Comment: 18 pages, 3 figure

Development of Muon Drift-Tube Detectors for High-Luminosity Upgrades of the Large Hadron Collider

The muon detectors of the experiments at the Large Hadron Collider (LHC) have to cope with unprecedentedly high neutron and gamma ray background rates. In the forward regions of the muon spectrometer of the ATLAS detector, for instance, counting rates of 1.7 kHz/square cm are reached at the LHC design luminosity. For high-luminosity upgrades of the LHC, up to 10 times higher background rates are expected which require replacement of the muon chambers in the critical detector regions. Tests at the CERN Gamma Irradiation Facility showed that drift-tube detectors with 15 mm diameter aluminum tubes operated with Ar:CO2 (93:7) gas at 3 bar and a maximum drift time of about 200 ns provide efficient and high-resolution muon tracking up to the highest expected rates. For 15 mm tube diameter, space charge effects deteriorating the spatial resolution at high rates are strongly suppressed. The sense wires have to be positioned in the chamber with an accuracy of better than 50 ?micons in order to achieve the desired spatial resolution of a chamber of 50 ?microns up to the highest rates. We report about the design, construction and test of prototype detectors which fulfill these requirements

Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

In this article the concept of saturation of an arbitrary regularization method is formalized based upon the original idea of saturation for spectral regularization methods introduced by A. Neubauer in 1994. Necessary and sufficient conditions for a regularization method to have global saturation are provided. It is shown that for a method to have global saturation the total error must be optimal in two senses, namely as optimal order of convergence over a certain set which at the same time, must be optimal (in a very precise sense) with respect to the error. Finally, two converse results are proved and the theory is applied to find sufficient conditions which ensure the existence of global saturation for spectral methods with classical qualification of finite positive order and for methods with maximal qualification. Finally, several examples of regularization methods possessing global saturation are shown.Comment: 29 page

The density of states of chaotic Andreev billiards

Quantum cavities or dots have markedly different properties depending on whether their classical counterparts are chaotic or not. Connecting a superconductor to such a cavity leads to notable proximity effects, particularly the appearance, predicted by random matrix theory, of a hard gap in the excitation spectrum of quantum chaotic systems. Andreev billiards are interesting examples of such structures built with superconductors connected to a ballistic normal metal billiard since each time an electron hits the superconducting part it is retroreflected as a hole (and vice-versa). Using a semiclassical framework for systems with chaotic dynamics, we show how this reflection, along with the interference due to subtle correlations between the classical paths of electrons and holes inside the system, are ultimately responsible for the gap formation. The treatment can be extended to include the effects of a symmetry breaking magnetic field in the normal part of the billiard or an Andreev billiard connected to two phase shifted superconductors. Therefore we are able to see how these effects can remold and eventually suppress the gap. Furthermore the semiclassical framework is able to cover the effect of a finite Ehrenfest time which also causes the gap to shrink. However for intermediate values this leads to the appearance of a second hard gap - a clear signature of the Ehrenfest time.Comment: Refereed version. 23 pages, 19 figure