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

Properties of mass-loading shocks, 2. Magnetohydrodynamics

The one-dimensional magnetohydrodynamics of shocked flows subjected to significant mass loading are considered. Recent observations at comets Giacobini-Zinner and Halley suggest that simple nonreacting MHD is an inappropriate description for active cometary bow shocks. The thickness of the observed cometary shock implies that mass loading represents an important dynamical process within the shock itself, thereby requiring that the Rankine-Hugoniot condition for the mass flux possess a source term. In a formal sense, this renders mass-loading shocks qualitatively similar to combustion shocks, except that mass loading induces the shocked flow to shear. Nevertheless, a large class of stable shocks exist, identified by means of the Lax conditions appropriate to MHD. Thus mass-loading shocks represent a new and interesting class of shocks, which, although found frequently in the solar system, both at the head of comets and, under suitable conditions, upsteam of weakly magnetized and nonmagnetized planets, has not been discussed in any detail. Owing to the shearing of the flow, mass-loading shocks can behave like switch-on shocks regardless of the magnitude of the plasma beta. Thus the behavior of the magnetic field in mass-loading shocks is significantly different from that occurring in nonreacting classical MHD shocks. It is demonstrated that there exist two types of mass-loading fronts for which no classical MHD analogue exists, these being the fast and slow compound mass-loading shocks. These shocks are characterized by an initial deceleration of the fluid flow to either the fast or the slow magnetosonic speed followed by an isentropic expansion to the final decelerated downstream state. Thus these transitions take the flow from a supersonic to a supersonic, although decelerated, downstream state, unlike shocks which occur in classical MHD or gasdynamics. It is possible that such structures have been observed during the Giotto-Halley encounter, and a brief discussion of the appropriate Halley parameters is therefore given, together with a short discussion of the determination of the shock normal from observations. A further interesting new form of mass-loading shock is the “slow-intermediate” shock, a stable shock which possesses many of the properties of intermediate MHD shocks yet which propagates like a slow mode MHD shock. An important property of mass-loading shocks is the large parameter regime (compared with classical MHD) which does not admit simple or stable transitions from a given upstream to a downstream state. This suggests that it is often necessary to construct compound structures consisting of shocks, slip waves, rarefactions, and fast and slow compound waves in order to connect given upstream and downstream states. Thus the Riemann problem is significantly different from that of classical MHD

Mass-loading and parallel magnetized shocks

Recent observations at comets Giacobini-Zinner and Halley suggest that simple non-reacting gas dynamics or MHD is an inappropriate description for the bow shock. The thickness of the observed (sub)shock implies that mass-loading is an important dynamical process within the shock itself, thereby requiring that the Rankine-Hugoniot conditions possess source terms. This leads to shocks with properties similar to those of combustion shocks. We consider parallel magnetized shocks subjected to mass-loading, describe some properties which distinguish them from classical MHD parallel shocks, and establish the existence of a new kind of MHD compound shock. These results will be of importance both to observations and numerical simulations of the comet-solar wind interaction

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

Simulator study of the low-speed handling qualities of a supersonic cruise arrow-wing transport configuration during approach and landing

A fixed-based simulator study was conducted to determine the low-speed flight characteristics of an advanced supersonic cruise transport having an arrow wing, a horizontal tail, and four dry turbojets with variable geometry turbines. The primary piloting task was the approach and landing. The statically unstable (longitudinally) subject configuration has unacceptable low-speed handling qualities with no augmentation. Therefore, a hardened stability augmentation system is required to achieve acceptable handling qualities, should the normal operational stability and control augmentation system fail. In order to achieve satisfactory handling qualities, considerable augmentation was required

Large-scale variations of the interplanetary magnetic field: Voyager 1 and 2 observations between 1-5 AU

Observations by the Voyager 1 and 2 spacecraft of the interplanetary magnetic field between 1 and 5 AU were used to investigate the large scale structure of the IMF in a period of increasing solar activity. The Voyager spacecraft found notable deviations from the Parker axial model. These deviations are attributed both to temporal variations associated with increasing solar activity, and to the effects of fluctuations of the field in the radial direction. The amplitude of the latter fluctuations were found to be large relative to the magnitude of the radial field component itself beyond approximately 3 AU. Both Voyager 1 and Voyager 2 observed decreases with increasing heliocentric distance in the amplitude of transverse fluctuations in the averaged field strength (B) which are consistent with the presence of predominantly undamped Alfven waves in the solar wind, although and necessarily implying the presence of them. Fluctuations in the strength of B (relative to mean field strength) were found to be small in amplitude, with a RMS which is approximately one third of that for the transverse fluctuations and they are essentially independent of distance from the Sun

Magnetic field experiment for Voyagers 1 and 2

The magnetic field experiment to be carried on the Voyager 1 and 2 missions consists of dual low field (LFM) and high field magnetometer (HFM) systems. The dual systems provide greater reliability and, in the case of the LFM's, permit the separation of spacecraft magnetic fields from the ambient fields. Additional reliability is achieved through electronics redundancy. The wide dynamic ranges of plus or minus 0.5G for the LFM's and plus or minus 20G for the HFM's, low quantization uncertainty of plus or minus 0.002 gamma in the most sensitive (plus or minus 8 gamma) LFM range, low sensor RMS noise level of 0.006 gamma, and use of data compaction schemes to optimize the experiment information rate all combine to permit the study of a broad spectrum of phenomena during the mission. Planetary fields at Jupiter, Saturn, and possibly Uranus; satellites of these planets; solar wind and satellite interactions with the planetary fields; and the large-scale structure and microscale characteristics of the interplanetary magnetic field are studied. The interstellar field may also be measured

Skyrmion Lattice in a Chiral Magnet

Skyrmions represent topologically stable field configurations with particle-like properties. We used neutron scattering to observe the spontaneous formation of a two-dimensional lattice of skyrmion lines, a type of magnetic vortices, in the chiral itinerant-electron magnet MnSi. The skyrmion lattice stabilizes at the border between paramagnetism and long-range helimagnetic order perpendicular to a small applied magnetic field regardless of the direction of the magnetic field relative to the atomic lattice. Our study experimentally establishes magnetic materials lacking inversion symmetry as an arena for new forms of crystalline order composed of topologically stable spin states