A framework for deflated and augmented Krylov subspace methods

We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) "removes" certain parts from the operator making it singular, while augmentation adds a subspace to the Krylov subspace (often the one that is generated by the singular operator); in contrast, preconditioning changes the spectrum of the operator without making it singular. Deflation and augmentation have been used in a variety of methods and settings. Typically, deflation is combined with augmentation to compensate for the singularity of the operator, but both techniques can be applied separately. We introduce a framework of Krylov subspace methods that satisfy a Galerkin condition. It includes the families of orthogonal residual (OR) and minimal residual (MR) methods. We show that in this framework augmentation can be achieved either explicitly or, equivalently, implicitly by projecting the residuals appropriately and correcting the approximate solutions in a final step. We study conditions for a breakdown of the deflated methods, and we show several possibilities to avoid such breakdowns for the deflated MINRES method. Numerical experiments illustrate properties of different variants of deflated MINRES analyzed in this paper.Comment: 24 pages, 3 figure

Electron gas polarization effect induced by heavy H-like ions of moderate velocities channeled in a silicon crystal

We report on the observation of a strong perturbation of the electron gas induced by 20 MeV/u U$^{91+}$ ions and 13 MeV/u Pb$^{81+}$ ions channeled in silicon crystals. This collective response (wake effect) in-duces a shift of the continuum energy level by more than 100 eV, which is observed by means of Radiative Electron Capture into the K and L-shells of the projectiles. We also observe an increase of the REC probability by 20-50% relative to the probability in a non-perturbed electron gas. The energy shift is in agreement with calculations using the linear response theory, whereas the local electron density enhancement is much smaller than predicted by the same model. This shows that, for the small values of the adiabaticity parameter achieved in our experiments, the density fluctuations are not strongly localized at the vicinity of the heavy ions

Refined saddle-point preconditioners for discretized Stokes problems

This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying the preconditioned MINRES method. The codes used in the numerical studies are available online

How Preussag became TUI : kissing too many toads can make you a toad

In the period 1997-2004, Preussag, a diversified German conglomerate of old economy businesses, changed itself into TUI, a company focused almost entirely on tourism and logistics. This paper analyzes how this strategy was executed and how it contributed to Preussag’s underperformance of the stock market. We collect 417 announcements of acquisitions, financial disclosures and other news and disentangle the impact of different parts of the company’s strategy. We find that only the divestitures created value, that the strategy to invest in tourism destroyed value, and that the acquisition premiums Preussag paid were mostly unjustified. Bad luck like the events of September 11, 2001 cannot account for the poor performance of the stock. Poor management resulted from poor governance, combining a state-owned bank as the largest shareholder, board interlocks, and insufficient managerial incentives. The case shows how divestiture programs increase the liquid resources available to management beyond free operating cash flows and casts doubt on the positive governance role of institutional blockholders

Differential energy measurement between He- and Li-like uranium intra-shell transitions

We present the first clear identification and highly accurate measurement of the intra-shell transition 1s2p\, ^3P_2 \to 1s2s\, ^3S_1 of He-like uranium performed via X-ray spectroscopy. The present experiment has been conducted at the gas-jet target of the ESR storage ring in GSI (Darmstadt, Germany) where a Bragg spectrometer, with a bent germanium crystal, and a Ge(i) detector were mounted. Using the ESR deceleration capabilities, we performed a differential measurement between the 1s2p\, ^3P_2 \to 1s2s\, ^3S_1 He-like U transition energy, at 4510 eV, and the 1s^22p\ ^2P_{3/2} \to 1s^22s\, ^2S_{1/2} Li-like U transition energy, at 4460 eV. By a proper choice of the ion velocities, the X-ray energies from the He- and Li-like ions could be measured, in the laboratory frame, at the same photon energy. This allowed for a drastic reduction of the experimental systematic uncertainties, principally due to the Doppler effect, and for a comparison with the theory without the uncertainties arising from one-photon QED predictions and nuclear size corrections

Subshell-selective x-ray studies of radiative recombination of U92+{\mathrm{U}}^{92+} ions with electrons for very low relative energies

Radiative recombination (RR) into the K shell and L subshells of U92+ ions interacting with cooling electrons has been studied in an x-ray RR experiment at the electron cooler of the Experimental Storage Ring at GSI. The measured radiative recombination rate coefficients for electron-ion relative energies in the range 0–1000 meV demonstrate the importance of relativistic effects. The observed asymmetry of the measured K-RR x-ray emission with respect to the cooling energy, i.e., zero average relative velocity (⟨vrel⟩=0), are explained by fully relativistic RR calculations. With our new approach, we show that the study of the angular distribution of RR photons for different relative energies opens new perspectives for detailed understanding of the RR of ions with cooling electrons in cold magnetized plasma

Ion slowing down and charge exchange at small impact parameters selected by channeling: superdensity effects

CASInternational audienceIn two experiments performed with 20-30 MeV/u highly charged heavy ions (Pb56+, U91+) channeled through thin silicon crystals, we observed the original features of superdensity, associated to the glancing collisions with atomic rows undergone by part of the incident projectiles. In particular the very high collision rate yields a quite specific charge exchange regime, that leads to a higher ionization probability than in random conditions. X-ray measurements show that electrons captured in outershells are prevented from being stabilized, which enhances the lifetime of the projectile innershell vacancies. The charge state distributions and the energy loss spectra are compared to Monte-Carlo simulations. These simulations confirm, extend and illustrate the qualitative analysis of the experimental results