Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

In this paper, we prove optimal convergence rates results for regularisation methods for solving linear ill-posed operator equations in Hilbert spaces. The result generalises existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connection to approximative source conditions

Motion Detection in Diffraction Tomography by Common Circle Methods

The method of common lines is a well-established reconstruction technique in cryogenic electron microscopy (cryo-EM), which can be used to extract the relative orientations of an object in tomographic projection images from different directions. In this paper, we deal with an analogous problem in optical diffraction tomography. Based on the Fourier diffraction theorem, we show that rigid motions, i.e., a map composed of rotations and translations, can be determined by detecting common circles in the Fourier-transformed data. We introduce two methods based on the idea of identifying common circles to reconstruct the object motion: While the first one is motivated by the common line approach for projection images and detects the relative orientation by the shape of the common circles in the two images, the second one assumes a smooth motion over time and calculates the angular velocity of the rotational motion from an infinitesimal version of the common circle method. Interestingly, using the stereographic projection, both methods can be reformulated as common line methods, but these lines are, in contrast to those used in cryo-EM, not confined to pass through the origin and allow for a full reconstruction of the relative orientation. Numerical proof-of-the-concept examples demonstrate the performance of our reconstruction methods.Comment: 35 pages, 13 figure

Convergence Rates of First and Higher Order Dynamics for Solving Linear Ill-posed Problems

Recently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov's algorithm and the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), respectively. In this paper we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residuum of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. We prove that the proposed flows for minimising the residuum of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions. In particular we show that in comparison to convex analysis results the rates can be significantly higher, which is possible by constraining the investigations to the particular convex energy functional, which is the squared norm of the residuum

Carotid Intima-Media Thickness Progression in HIV-Infected Adults Occurs Preferentially at the Carotid Bifurcation and Is Predicted by Inflammation.

BackgroundShear stress gradients and inflammation have been causally associated with atherosclerosis development in carotid bifurcation regions. The mechanism underlying higher levels of carotid intima-media thickness observed among HIV-infected individuals remains unknown.Methods and resultsWe measured carotid intima-media thickness progression and development of plaque in the common carotid, bifurcation region, and internal carotid artery in 300 HIV-infected persons and 47 controls. The median duration of follow-up was 2.4 years. When all segments were included, the rate of intima-media thickness progression was greater in HIV-infected subjects compared with controls after adjustment for traditional risk factors (0.055 vs. 0.024 mm/year, P=0.016). Rate of progression was also greater in the bifurcation region (0.067 vs. 0.025 mm/year, P=0.042) whereas differences were smaller in the common and internal regions. HIV-infected individuals had a greater incidence of plaque compared with controls in the internal (23% vs. 6.4%, P=0.0037) and bifurcation regions (34% vs. 17%, P=0.014). Among HIV-infected individuals, the rate of progression in the bifurcation region was more rapid compared with the common carotid, internal, or mean intima-media thickness; in contrast, progression rates among controls were similar at all sites. Baseline hsCRP was elevated in HIV-infected persons and was a predictor of progression in the bifurcation region.ConclusionsAtherosclerosis progresses preferentially in the carotid bifurcation region in HIV-infected individuals. hsCRP, a marker of inflammation, is elevated in HIV and is associated with progression in the bifurcation region. These data are consistent with a model in which the interplay between hemodynamic shear stresses and HIV-associated inflammation contribute to accelerated atherosclerosis. (J Am Heart Assoc. 2012;1:jah3-e000422 doi: 10.1161/JAHA.111.000422.)Clinical trial registrationURL: http://clinicaltrials.gov. Unique identifier: NCT01519141