Bernstein-von Mises theorems for statistical inverse problems I: Schrödinger equation

© European Mathematical Society 2020 We consider the inverse problem of determining the potential f > 0 in the partial differential equation 1 2 u − fu = 0 on O, u = g on ∂O, where O is a bounded C∞-domain in Rd and g > 0 is a given function prescribing boundary values. The data consist of the solution u corrupted by additive Gaussian noise. A nonparametric Bayesian prior for the function f is devised and a Bernstein-von Mises theorem is proved which entails that the posterior distribution given the observations is approximated in a suitable function space by an infinite-dimensional Gaussian measure that has a 'minimal' covariance structure in an information-theoretic sense. As a consequence the posterior distribution performs valid and optimal frequentist statistical inference on various aspects of f in the small noise limit

Consistency of Bayesian inference with Gaussian process priors in an elliptic inverse problem

For $\mathcal{O}$ a bounded domain in $\mathbb{R}^d$ and a given smooth function $g:\mathcal{O}\to\mathbb{R}$, we consider the statistical nonlinear inverse problem of recovering the conductivity $f>0$ in the divergence form equation $$\nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad u=0\ \textrm{on}\ \partial\mathcal{O},$$ from $N$ discrete noisy point evaluations of the solution $u=u_f$ on $\mathcal O$. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number $N$ of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate $N^{-\lambda}, \lambda>0,$ for the reconstruction error of the associated posterior means, in $L^2(\mathcal{O})$-distance

Efficient nonparametric bayesian inference for X-ray transforms

We consider the statistical inverse problem of recovering a function $f: M \to \mathbb R$, where $M$ is a smooth compact Riemannian manifold with boundary, from measurements of general $X$-ray transforms $I_a(f)$ of $f$, corrupted by additive Gaussian noise. For $M$ equal to the unit disk with `flat' geometry and $a=0$ this reduces to the standard Radon transform, but our general setting allows for anisotropic media $M$ and can further model local `attenuation' effects -- both highly relevant in practical imaging problems such as SPECT tomography. We propose a nonparametric Bayesian inference approach based on standard Gaussian process priors for $f$. The posterior reconstruction of $f$ corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty that does not require the calculation of the singular value decomposition of the forward operator $I_a$. We prove Bernstein-von Mises theorems that entail that posterior-based inferences such as credible sets are valid and optimal from a frequentist point of view for a large family of semi-parametric aspects of $f$. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which is shown to attain the semi-parametric Cram\'er-Rao information bound. The proofs rely on an invertibility result for the `Fisher information' operator $I_a^*I_a$ between suitable function spaces, a result of independent interest that relies on techniques from microlocal analysis. We illustrate the performance of the proposed method via simulations in various settings

Совершенствование конструкции плоскореза глубокорыхлителя КПГ-250 в условиях КФХ "Правда" Беловского района, Кемеровской области

Объектом исследования является плоскорежущая лапа глубокорыхлителя КПГ-250. Цель работы – повышение качества обработки почвы, с совершенствованием конструкции глубокорыхлителя КПГ-250. повышение эффективности уборки зерновых культур, с разработкой конструкции В процессе исследования проводились технологические и конструкторские расчеты В результате исследования предложены мероприятия по снижению уплотнения почвы, а также конструкторские решения по повышению эффективности обработки почвы культиватором КПГ-250.The object of the study is flat-cutting paw cultivator KPG-250. Purpose – to improve the quality of the soil, by improving the design of the chisel KPG-250. Improving the efficiency of cleaning of grain crops, with the development of the design. In the process of research was conducted technological and design calculations The study proposed measures to reduce soil compaction, and design solutions for improving the effectiveness of soil treatment with cultivator KPG-250

Functionalized Fullerene for Inhibition of SARS-CoV-2 Variants

As virus outbreaks continue to pose a challenge, a nonspecific viral inhibitor can provide significant benefits, especially against respiratory viruses. Polyglycerol sulfates recently emerge as promising agents that mediate interactions between cells and viruses through electrostatics, leading to virus inhibition. Similarly, hydrophobic C60 fullerene can prevent virus infection via interactions with hydrophobic cavities of surface proteins. Here, two strategies are combined to inhibit infection of SARS-CoV-2 variants in vitro. Effective inhibitory concentrations in the millimolar range highlight the significance of bare fullerene's hydrophobic moiety and electrostatic interactions of polysulfates with surface proteins of SARS-CoV-2. Furthermore, microscale thermophoresis measurements support that fullerene linear polyglycerol sulfates interact with the SARS-CoV-2 virus via its spike protein, and highlight importance of electrostatic interactions within it. All-atom molecular dynamics simulations reveal that the fullerene binding site is situated close to the receptor binding domain, within 4 nm of polyglycerol sulfate binding sites, feasibly allowing both portions of the material to interact simultaneously

The Impact of the Physical Environment on Intrapartum Maternity Care: Identification of Eight Crucial Building Spaces.

OBJECTIVES, PURPOSE, OR AIM: This article investigates whether the physical environment in which childbirth occurs impacts the intrapartum intervention rates and how this might happen. The study explores the spatial physical characteristics that can support the design of spaces to promote the health and well-being of women, their supporters, and maternity care professionals. BACKGROUND: Medical interventions during childbirth have consequences for the health of women and babies in the immediate and long term. The increase in interventions is multifactorial and may be influenced by the model of care adopted, the relationships between caregivers and the organizational culture, which is made up of many factors, including the built environment. In the field of birth architecture research, there is a gap in the description of the physical characteristics of birth environments that impact users' health. METHOD: A scoping review on the topic was performed to understand the direct and indirect impacts of the physical environment on birth intervention rates. RESULTS AND DISCUSSION: The findings are organized into three tables reporting the influence that the physical characteristics of a space might have on people's behaviors, experiences, practices and birth health outcomes. Eight building spaces that require further investigation and research were highlighted: unit layout configuration, midwives' hub/desk, social room, birth philosophy vectors, configuration of the birth room, size and shape of the birth room, filter, and sensory elements. CONCLUSIONS: The findings show the importance of considering the physical environment in maternity care and that further interdisciplinary studies focused on architectural design are needed to enrich the knowledge and evidence on this topic and to develop accurate recommendations for designers

The Many Virtues of Second Nature : Habitus in Latin Medieval Philosophy

This chapter consists of a systematic introduction to the nature and function of habitus in Latin medieval philosophy. Over the course of this introduction, several topics are treated: the theoretical necessity to posit habitus; their nature; their causal contribution to the production of internal and external acts; how and why habitus can grow and decay; what makes their unity when they can have multiple objects and work in clusters. Finally we examine two specific questions: why intellectual habitus represent a special case that triggered considerable debate; how human beings can be said to be free if their actions are determined by moral habitus

On strong causal binomial approximation for stochastic processes

This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can be approximated in mean square sense and in other strong metrics by binomial processes, i.e., by processes with fixed size binary increments at sampling points. Moreover, this approximation can be causal, i.e., at every time it requires only past historical values of the underlying process. In addition, possibility of approximation of solutions of stochastic differential equations by solutions of ordinary equations with binary noise is established. Some consequences for the financial modelling and options pricing models are discussed

High sensitive troponin T and heart fatty acid binding protein: Novel biomarker in heart failure with normal ejection fraction?: A cross-sectional study

Background: High sensitive troponin T (hsTnT) and heart fatty acid binding protein (hFABP) are both markers of myocardial injury and predict adverse outcome in patients with systolic heart failure (SHF). We tested whether hsTnT and hFABP plasma levels are elevated in patients with heart failure with normal ejection fraction (HFnEF). Methods: We analyzed hsTnT, hFABP and N-terminal brain natriuretic peptide in 130 patients comprising 49 HFnEF patients, 51 patients with asymptomatic left ventricular diastolic dysfunction (LVDD), and 30 controls with normal diastolic function. Patients were classified to have HFnEF when the diagnostic criteria as recommended by the European Society of Cardiology were met. Results: Levels of hs TnT and hFABP were significantly higher in patients with asymptomatic LVDD and HFnEF (both p < 0.001) compared to controls. The hsTnT levels were 5.6 [0.0-9.8] pg/ml in LVDD vs. 8.5 [3.9-17.5] pg/ml in HFnEF vs. < 0.03 [< 0.03-6.4] pg/ml in controls; hFABP levels were 3029 [2533-3761] pg/ml in LVDD vs. 3669 [2918-4839] pg/ml in HFnEF vs. 2361 [1860-3081] pg/ml in controls. Furthermore, hsTnT and hFABP levels were higher in subjects with HFnEF compared to LVDD (p = 0.015 and p = 0.022). Conclusion: In HFnEF patients, hsTnT and hFABP are elevated independent of coronary artery disease, suggesting that ongoing myocardial damage plays a critical role in the pathophysiology. A combination of biomarkers and echocardiographic parameters might improve diagnostic accuracy and risk stratification of patients with HFnEF