A TV-Gaussian prior for infinite-dimensional Bayesian inverse problems and its numerical implementations

Many scientific and engineering problems require to perform Bayesian inferences in function spaces, in which the unknowns are of infinite dimension. In such problems, choosing an appropriate prior distribution is an important task. In particular we consider problems where the function to infer is subject to sharp jumps which render the commonly used Gaussian measures unsuitable. On the other hand, the so-called total variation (TV) prior can only be defined in a finite dimensional setting, and does not lead to a well-defined posterior measure in function spaces. In this work we present a TV-Gaussian (TG) prior to address such problems, where the TV term is used to detect sharp jumps of the function, and the Gaussian distribution is used as a reference measure so that it results in a well-defined posterior measure in the function space. We also present an efficient Markov Chain Monte Carlo (MCMC) algorithm to draw samples from the posterior distribution of the TG prior. With numerical examples we demonstrate the performance of the TG prior and the efficiency of the proposed MCMC algorithm

Fast Gibbs sampling for high-dimensional Bayesian inversion

Solving ill-posed inverse problems by Bayesian inference has recently attracted considerable attention. Compared to deterministic approaches, the probabilistic representation of the solution by the posterior distribution can be exploited to explore and quantify its uncertainties. In applications where the inverse solution is subject to further analysis procedures, this can be a significant advantage. Alongside theoretical progress, various new computational techniques allow to sample very high dimensional posterior distributions: In [Lucka2012], a Markov chain Monte Carlo (MCMC) posterior sampler was developed for linear inverse problems with $\ell_1$-type priors. In this article, we extend this single component Gibbs-type sampler to a wide range of priors used in Bayesian inversion, such as general $\ell_p^q$ priors with additional hard constraints. Besides a fast computation of the conditional, single component densities in an explicit, parameterized form, a fast, robust and exact sampling from these one-dimensional densities is key to obtain an efficient algorithm. We demonstrate that a generalization of slice sampling can utilize their specific structure for this task and illustrate the performance of the resulting slice-within-Gibbs samplers by different computed examples. These new samplers allow us to perform sample-based Bayesian inference in high-dimensional scenarios with certain priors for the first time, including the inversion of computed tomography (CT) data with the popular isotropic total variation (TV) prior.Comment: submitted to "Inverse Problems

Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532-900 nm wavelength window fabricated within a CMOS pilot line

PECVD silicon nitride photonic wire waveguides have been fabricated in a CMOS pilot line. Both clad and unclad single mode wire waveguides were measured at lambda = 532, 780, and 900 nm, respectively. The dependence of loss on wire width, wavelength, and cladding is discussed in detail. Cladded multimode and singlemode waveguides show a loss well below 1 dB/cm in the 532-900 nm wavelength range. For singlemode unclad waveguides, losses < 1 dB/cm were achieved at lambda = 900 nm, whereas losses were measured in the range of 1-3 dB/cm for lambda = 780 and 532 nm, respectively

Perfluorodecaline residue in the anterior chamber of a patient with an intact crystalline lens: a case report

Abstract Background Perfluorocarbon liquids are frequently used as intraoperative tools in vitreoretinal surgery and may occasionally be retained in the vitreous cavity. We report a patient who underwent pars plana vitrectomy for a giant tear after receiving blunt trauma to his right eye and sustained postoperative perfluorocarbon liquid residue in the anterior chamber in spite of an intact crystalline lens. Case presentation Perfluorodecaline was used as a temporary retinal tamponade. Three weeks after the surgery, a residue of heavy liquid was observed in the anterior chamber, even though the patient had an intact crystalline lens without any tilt or dislocation. The remnant of the heavy liquid was taken out of the anterior chamber immediately to avoid secondary complications. Conclusion Presence of heavy liquids in the anterior chamber may be associated with zonular defects even though the patient has an intact crystalline lens.</p

Large-scale Graphitic Thin Films Synthesized on Ni and Transferred to Insulators: Structural and Electronic Properties

We present a comprehensive study of the structural and electronic properties of ultrathin films containing graphene layers synthesized by chemical vapor deposition (CVD) based surface segregation on polycrystalline Ni foils then transferred onto insulating SiO2/Si substrates. Films of size up to several mm's have been synthesized. Structural characterizations by atomic force microscopy (AFM), scanning tunneling microscopy (STM), cross-sectional transmission electron microscopy (XTEM) and Raman spectroscopy confirm that such large scale graphitic thin films (GTF) contain both thick graphite regions and thin regions of few layer graphene. The films also contain many wrinkles, with sharply-bent tips and dislocations revealed by XTEM, yielding insights on the growth and buckling processes of the GTF. Measurements on mm-scale back-gated transistor devices fabricated from the transferred GTF show ambipolar field effect with resistance modulation ~50% and carrier mobilities reaching ~2000 cm^2/Vs. We also demonstrate quantum transport of carriers with phase coherence length over 0.2 $\mu$m from the observation of 2D weak localization in low temperature magneto-transport measurements. Our results show that despite the non-uniformity and surface roughness, such large-scale, flexible thin films can have electronic properties promising for device applications.Comment: This version (as published) contains additional data, such as cross sectional TEM image

Characterization of PECVD Silicon Nitride Photonic Components at 532 and 900 nm Wavelength

Low temperature PECVD silicon nitride photonic waveguides have been fabricated by both electron beam lithography and 200 mm DUV lithography. Propagation losses and bend losses were both measured at 532 and 900 nm wavelength, revealing sub 1dB/cm propagation losses for cladded waveguides at both wavelengths for single mode operation. Without cladding, propagation losses were measured to be in the 1-3 dB range for 532 nm and remain below 1 dB/cm for 900 nm for single mode waveguides. Bend losses were measured for 532 nm and were well below 0.1 dB per 90 degree bend for radii larger than 10 mu m

Besov priors for Bayesian inverse problems

We consider the inverse problem of estimating a function $u$ from noisy, possibly nonlinear, observations. We adopt a Bayesian approach to the problem. This approach has a long history for inversion, dating back to 1970, and has, over the last decade, gained importance as a practical tool. However most of the existing theory has been developed for Gaussian prior measures. Recently Lassas, Saksman and Siltanen (Inv. Prob. Imag. 2009) showed how to construct Besov prior measures, based on wavelet expansions with random coefficients, and used these prior measures to study linear inverse problems. In this paper we build on this development of Besov priors to include the case of nonlinear measurements. In doing so a key technical tool, established here, is a Fernique-like theorem for Besov measures. This theorem enables us to identify appropriate conditions on the forward solution operator which, when matched to properties of the prior Besov measure, imply the well-definedness and well-posedness of the posterior measure. We then consider the application of these results to the inverse problem of finding the diffusion coefficient of an elliptic partial differential equation, given noisy measurements of its solution.Comment: 18 page

Association of Early Beta-amyloid Accumulation and Neuroinflammation Measured with [11C]PBR28 in Elderly Individuals Without Dementia

OBJECTIVE: To examine whether early β-amyloid (Aβ) accumulation and metabolic risk factors are associated with neuroinflammation in elderly individuals without dementia. METHODS: We examined 54 volunteers (mean age 70.0, 56% women, 51% APOE ε4 carriers) with a TSPO-tracer [11C]PBR28 to assess neuroinflammation and with [11C]Pittsburgh compound B (PiB) to assess cerebral Aβ accumulation. [11C]PBR28 and [11C]PiB standardized uptake value ratios (SUVRs) were quantified in six regions of interests by using the cerebellar cortex as a pseudo-reference/reference region, respectively. Fasting venous glucose, insulin, and high sensitivity C-reactive protein (hs-CRP) values were determined. Homeostatic model assessment of insulin resistance (HOMA-IR) was calculated. A subset of individuals (n=11) underwent CSF sampling, and Aβ40, Aβ42, total-tau, phospho-tau, soluble TREM2 and YKL-40 levels were measured. RESULTS: Among the whole study group, no significant association was found between [11C]PiB and [11C]PBR28 SUVR composite scores (slope 0.02, p=0.30). However, higher [11C]PiB binding was associated with higher [11C]PBR28 binding among amyloid negative ([11C]PiB composite score ≤1.5) (TSPO-genotype, age and sex adjusted slope 0.26, p=0.008) but not among amyloid positive participants (slope: -0.004, p=0.88). Higher CSF sTREM2 (rs 0.72, p=0.01) and YKL-40 (rs=0.63, p=0.04) concentrations were associated with a higher [11C]PBR28 composite score. Higher body mass index, HOMA-IR, and hs-CRP were associated with higher [11C]PBR28 binding in brain regions where Aβ accumulation is first detected in Alzheimer's disease (AD). CONCLUSIONS: While there was no association between amyloid and neuroinflammation in the overall study group, neuroinflammation was associated with amyloid among the subgroup at early stages of amyloid pathology