Convergence rates of general regularization methods for statistical inverse problems and applications

During the past the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut-off and Tikhonov type estimators. Spectral cut-off estimators achieve minimax rates for a broad range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator. Tikhonov estimators are simpler to compute, but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes. In this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods (spectral methods) including the aforementioned estimators as well as many iterative methods, such as í-methods and the Landweber iteration. The latter estimators converge at the same rate as spectral cut-off, but only require matrixvector products. Our results are applied to various problems, in particular we obtain precise convergence rates for satellite gradiometry, L2-boosting, and errors in variable problems. --Statistical inverse problems,iterative regularization methods,Tikhonov regularization,nonparametric regression,minimax convergence rates,satellite gradiometry,Hilbert scales,boosting,errors in variable

M\"{o}bius deconvolution on the hyperbolic plane with application to impedance density estimation

In this paper we consider a novel statistical inverse problem on the Poincar\'{e}, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of $2\times2$ real matrices of determinant one via M\"{o}bius transformations. Our approach is based on a deconvolution technique which relies on the Helgason--Fourier calculus adapted to this hyperbolic space. This gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random M\"{o}bius transform. A motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the Poincar\'{e} plane exactly describes the physical system that is of statistical interest.Comment: Published in at http://dx.doi.org/10.1214/09-AOS783 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

PET kinetics of radiolabeled antidepressant, [N-methyl-11C]mirtazapine, in the human brain

BACKGROUND: We compared six kinetic models with and without the requirement of arterial cannulation for estimating the binding potential of [N-methyl-(11)C]mirtazapine in the living human brain. METHODS: Distribution volumes of [N-methyl-(11)C]mirtazapine in brain regions were estimated using single- and two-tissue compartment models as well as a graphical plasma input model. The two-tissue compartment model provided a direct estimate of the binding potentials of [N-methyl-(11)C]mirtazapine in brain regions, while binding potentials of the single-tissue compartment model and the graphical plasma input model were estimated indirectly from ratios of distribution volumes in brain regions. We obtained also direct estimates of binding potentials using a graphical reference tissue model and two nonlinear reference tissue models. RESULTS: The two-tissue compartment model required several fits with different initial guesses for avoiding negative values of parameters. Despite the extra fits, estimates of distribution volumes and binding potentials of [N-methyl-(11)C]mirtazapine obtained by the two-tissue compartment model were far more variable than those produced by the other methods. The graphical plasma input method and the graphical reference tissue method provided estimates of the binding potential that correlated closely, but differed in magnitude. The single-tissue compartment model provided relatively low estimates of binding potentials with curves that failed to fit the data as well as the three other methods that used the entire series of positron emission tomography data. The reference tissue method and the simplified reference tissue method provided similar, consistent estimates of binding potentials. However, certain assumptions of the simplified reference tissue method may not be fulfilled by the radioligand. CONCLUSION: The reference tissue method is appropriate for estimating the binding potential of [N-methyl-(11)C]mirtazapine in regions of the human brain so that the binding potential of [N-methyl-(11)C]mirtazapine can be estimated without arterial cannulation

Kantorovich-Rubinstein Distance and Barycenter for Finitely Supported Measures: Foundations and Algorithms

The purpose of this paper is to provide a systematic discussion of a generalized barycenter based on a variant of unbalanced optimal transport (UOT) that defines a distance between general non-negative, finitely supported measures by allowing for mass creation and destruction modeled by some cost parameter. They are denoted as Kantorovich–Rubinstein (KR) barycenter and distance. In particular, we detail the influence of the cost parameter to structural properties of the KR barycenter and the KR distance. For the latter we highlight a closed form solution on ultra-metric trees. The support of such KR barycenters of finitely supported measures turns out to be finite in general and its structure to be explicitly specified by the support of the input measures. Additionally, we prove the existence of sparse KR barycenters and discuss potential computational approaches. The performance of the KR barycenter is compared to the OT barycenter on a multitude of synthetic datasets. We also consider barycenters based on the recently introduced Gaussian Hellinger–Kantorovich and Wasserstein–Fisher–Rao distances

Convergence rates of general regularization methods for statistical inverse problems and applications

During the past the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut-off and Tikhonov type estimators. Spectral cut-off estimators achieve minimax rates for a broad range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator. Tikhonov estimators are simpler to compute, but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes. In this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods (spectral methods) including the aforementioned estimators as well as many iterative methods, such as ν-methods and the Landweber iteration. The latter estimators converge at the same rate as spectral cut-off, but only require matrixvector products. Our results are applied to various problems, in particular we obtain precise convergence rates for satellite gradiometry, L2-boosting, and errors in variable problems. AMS subject classifications: 62G05, 62J05, 62P35, 65J10, 35R3

Note on the dynamics of the Gulf Stream

The nonlinear inertial terms have been neglected in Stommel\u27s and in Munk\u27s theory for the wind-driven ocean circulation. Using a method of successive approximations, the effect of these terms on the mass transport in the Gulf Stream region has been computed under greatly simplifying assumptions. These assumptions involve Reid\u27s model of the vertical density structure, which consists of an exponential decrease in the density upward to the thermocline and a homogeneous upper layer...

In-Space Propulsion (ISP) Aerocapture Technology

A viewgraph presentation is shown to raise awareness of aerocapture technology through in-space propulsion. The topics include: 1) Purpose; 2) In-Space Propulsion Program; 3) Aerocapture Overview; 4) Aerocapture Technology Alternatives; 5) Aerocapture Technology Project Process; 6) Results from 2002 Aerocapture TAG; 7) Bounding Case Requirements; 8) ST9 Flight Demonstration Opportunity; 9) Aerocapture NRA Content: Cycles 1 and 2; 10) Ames Research Center TPS Development; 11) Applied Research Associates TPS Development; 12) LaRC Structures Development; 13) Lockheed Martin Astronautics Aeroshell Development; 14) ELORET/ARC Sensor Development; 15) Ball Aerospace Trailing Ballute Development; 16) Cycle 2 NRA Selections - Aerocapture; and 17) Summary

Entangled Dynamics of a Stiff Polymer

Entangled networks of stiff biopolymers exhibit complex dynamic response, emerging from the topological constraints that neighboring filaments impose upon each other. We propose a class of reference models for entanglement dynamics of stiff polymers and provide a quantitative foundation of the tube concept for stiff polymers. For an infinitely thin needle exploring a planar course of point obstacles, we have performed large-scale computer simulations proving the conjectured scaling relations from the fast transverse equilibration to the slowest process of orientational relaxation. We determine the rotational diffusion coefficient of the tracer, its angular confinement, the tube diameter and the orientational correlation functions

The influence of potassium on core and geodynamo evolution

We model the thermal evolution of the core and mantle using a parametrized convection scheme, and calculate the entropy available to drive the geodynamo as a function of time. The cooling of the core is controlled by the rate at which the mantle can remove heat. Rapid core cooling favours the operation of a geodynamo but creates an inner core that is too large; slower cooling reduces the inner core size but makes a geodynamo less likely to operate. Introducing potassium into the core retards inner core growth and provides an additional source of entropy. For our nominal model parameters, a core containing approximate to 400 ppm potassium satisfies the criteria of present-day inner core size, surface heat flux, mantle temperature and cooling rate, and positive core entropy production.We have identified three possibilities that may allow the criteria to be satisfied without potassium in the core. (1) The core thermal conductivity is less than half the generally accepted value of 50 W m(-1) K-1. (2) The core solidus and adiabat are significantly colder and shallower than results from shock experiments and ab initio simulations indicate. (3) The core heat flux has varied by no more than a factor of 2 over Earth history. All models we examined with the correct present-day inner core radius have an inner core age of < 1.5 Gyr; prior to this time the geodynamo was sustained by cooling and radioactive heat production within a completely liquid core

A Comparative Study of Aerocapture Missions with a Mars Destination

Conventional interplanetary spacecraft use propulsive systems to decelerate into orbit. Aerocapture is an alternative approach for orbit capture, in which the spacecraft makes a single pass through a target destination's atmosphere. Although this technique has never been performed, studies show there are substantial benefits of using aerocapture for reduction of propellant mass, spacecraft size, and mission cost. The In-Space Propulsion (ISP) Program, part of NASA's Science Mission Directorate, has invested in aerocapture technology development since 2002. Aerocapture investments within ISP are largely driven by mission systems analysis studies, The purpose of this NASA-funded report is to identify and document the fundamental parameters of aerocapture within previous human and robotic Mars mission studies which will assist the community in identifying technology research gaps in human and robotic missions, and provide insight for future technology investments. Upon examination of the final data set, some key attributes within the aerocapture disciplines are identified