Efficient Krylov subspace methods for uncertainty quantification in large Bayesian linear inverse problems

Uncertainty quantification for linear inverse problems remains a challenging task, especially for problems with a very large number of unknown parameters (e.g., dynamic inverse problems) and for problems where computation of the square root and inverse of the prior covariance matrix are not feasible. This work exploits Krylov subspace methods to develop and analyze new techniques for large‐scale uncertainty quantification in inverse problems. In this work, we assume that generalized Golub‐Kahan‐based methods have been used to compute an estimate of the solution, and we describe efficient methods to explore the posterior distribution. In particular, we use the generalized Golub‐Kahan bidiagonalization to derive an approximation of the posterior covariance matrix, and we provide theoretical results that quantify the accuracy of the approximate posterior covariance matrix and of the resulting posterior distribution. Then, we describe efficient methods that use the approximation to compute measures of uncertainty, including the Kullback‐Liebler divergence. We present two methods that use the preconditioned Lanczos algorithm to efficiently generate samples from the posterior distribution. Numerical examples from dynamic photoacoustic tomography demonstrate the effectiveness of the described approaches.     

An economic¯X-chart approach to the joint control of the means of independent quality characteristics

Summary In controlling the mean of a multivariate normally distributed quality characteristic the main disadvantage of the globalT ²-control charts is that they don't indicate the component of the quality characteristic which gives rise to an alarm. To avoid this disadvantage the joint use of¯x-charts is proposed in this paper, where — for the sake of simplicity — the investigations are restricted to the case of independent components and to one-sided¯x-charts. On the basis of an economic objective function an approximation to the optimal design of the¯x-charts procedure is derived and discussed. It turns out that at least in the bivariate case the nearly optimal economic design is very close to the exact solution.Supported by the Deutsche Forschungsgemeinschaft (DFG).     

基于动态线性模型的中国狭义货币需求及缺口的贝叶斯估计

本文应用动态线性模型研究我国的狭义货币需求,利用贝叶斯吉布斯抽样方法估计模型的参数和方差,获得了潜在货币需求趋势和货币缺口,对我国的货币供给进行分析并获得一些有益结论,对于央行更好地实施货币管理,保持经济平稳发展有积极意义。     

Simulation from Wishart Distributions with Eigenvalue Constraints

Abstract

This article provides an efficient algorithm for generating a random matrix according to a Wishart distribution, but with eigenvalues constrained to be less than a given vector of positive values. The procedure of Odell and Feiveson provides a guide, but the modifications here ensure that the diagonal elements of a candidate matrix are less than the corresponding elements of the constraint vector, thus greatly improving the chances that the matrix will be acceptable. The Normal hierarchical model with vector outcomes and the multivariate random effects model provide motivating applications.     

Bayesian estimation of finite time ruin probabilities

In this paper, we consider Bayesian inference and estimation of finite time ruin probabilities for the Sparre Andersen risk model. The dense family of Coxian distributions is considered for the approximation of both the inter‐claim time and claim size distributions. We illustrate that the Coxian model can be well fitted to real, long‐tailed claims data and that this compares well with the generalized Pareto model. The main advantage of using the Coxian model for inter‐claim times and claim sizes is that it is possible to compute finite time ruin probabilities making use of recent results from queueing theory. In practice, finite time ruin probabilities are much more useful than infinite time ruin probabilities as insurance companies are usually interested in predictions for short periods of future time and not just in the limit. We show how to obtain predictive distributions of these finite time ruin probabilities, which are more informative than simple point estimations and take account of model and parameter uncertainty. We illustrate the procedure with simulated data and the well‐known Danish fire loss data set. Copyright © 2009 John Wiley & Sons, Ltd.     

Orbital deletion procedure and its applications

The orbital deletion procedure is introduced, which is suited to quantitatively investigating the electronic delocalization effiect in earboeations and boranes. While the routine, ab initio molecular orbital methods can generate wavefunetions for real systems where all electrons are delocalized, the present orbital deletion procedure can generate wavefunctions for hypothetical reference molecules where electronic delocalization effect is deactivated. The latter wavefunetion normlly corresponds In the most stable resonance structure in terms of the resonance theory. By comparing and analyzing the delocalized and the localized wavefunetions, one can obtain a quantitative and instinct pieture to show how electronic deloealizalion inside a molecule affects the molecular structure, energy as well as other physical properties. Two examples are detailedly discussed. The first is related to the hypercoujugation of alkyl groups in carbocations and a comparison of the order of stability of carbocations is made, T     

A photo-oxidation procedure using UV radiation/H2O2 for decomposition of wine samples — Determination of iron and manganese content by flame atomic absorption spectrometry

This paper proposes the use of photo-oxidation with UV radiation/H₂O₂ as sample pretreatment for the determination of iron and manganese in wines by flame atomic absorption spectrometry (FAAS). The optimization involved the study of the following variables: pH and concentration of buffer solution, concentrated hydrogen peroxide volume and irradiation time. The evaluation of sample degradation was monitored by measuring the absorbance at the maximum wavelength of red wine (530 nm). Using the experimental conditions established during the optimization (irradiation time of 30 min, oxidant volume of 2.5 mL, pH 10, and a buffer concentration of 0.15 mol L^− 1), this procedure allows the determination of iron and manganese with limits of detection of 30 and 22 μg L^− 1, respectively, for a 5 mL volume of digested sample. The precision levels, expressed as relative standard deviation (RSD), were 2.8% and 0.65% for iron and 2.7% and 0.54% for manganese for concentrations of 0.5 and 2.0 mg L^− 1, respectively. Addition/recovery tests for evaluation of the accuracy were in the ranges of 90%–111% and 95%–107% for iron and manganese, respectively. This digestion procedure has been applied for the determination of iron and manganese in six wine samples. The concentrations varied from 1.58 to 2.77 mg L^− 1 for iron and from 1.30 to 1.91 mg L^− 1 for manganese. The results were compared with those obtained by an acid digestion procedure and determination of the elements by FAAS. There was no significant difference between the results obtained by the two methods based on a paired t-test (at 95% confidence level).     

The Influence of Synthesis Conditions on the Antioxidant Activity of Selenium Nanoparticles

Selenium nanoparticles (SeNPs) have attracted great attention in recent years due to their unique properties and potential bioactivities. While the production of SeNPs has been long reported, there is little news about the influence of reaction conditions and clean-up procedure on their physical properties (e.g., shape, size) as well as their antioxidant activity. This study takes up this issue. SeNPs were synthesized by two methods using cysteine and ascorbic acid as selenium reductants. The reactions were performed with and without the use of polyvinyl alcohol as a stabilizer. After the synthesis, SeNPs were cleaned using various procedures. The antioxidant properties of the obtained SeNPs were investigated using DPPH and hydroxyl radical scavenging assays. It was found that their antioxidant activity does not always depend only on the nanoparticles size but also on their homogeneity. Moreover, the size and morphology of selenium nanoparticles are controlled by the clean-up step.     

Estimating reaction parameters in mechanism-enabled population balance models of nanoparticle size distributions: A Bayesian inverse problem approach

In order to quantitatively predict nano- as well as other particle-size distributions, one needs to have both a mathematical model and estimates of the parameters that appear in these models. Here, we show how one can use Bayesian inversion to obtain statistical estimates for the parameters that appear in recently derived mechanism-enabled population balance models (ME-PBM) of nanoparticle growth. The Bayesian approach addresses the question of “how well do we know our parameters, along with their uncertainties?.” The results reveal that Bayesian inversion statistical analysis on an example, prototype nanoparticle formation system allows one to estimate not just the most likely rate constants and other parameter values, but also their SDs, confidence intervals, and other statistical information. Moreover, knowing the reliability of the mechanistic model's parameters in turn helps inform one about the reliability of the proposed mechanism, as well as the reliability of its predictions. The paper can also be seen as a tutorial with the additional goal of achieving a “Gold Standard” Bayesian inversion ME-PBM benchmark that others can use as a control to check their own use of this methodology for other systems of interest throughout nature. Overall, the results provide strong support for the hypothesis that there is substantial value in using a Bayesian inversion methodology for parameter estimation in particle formation systems.     

New method for calculating the adsorption of noble gases on amorphous surfaces

A new approach to calculating the equilibrium characteristics of the adsorption of noble gases on the amorphous surfaces of adsorbents was developed and applied to the Ar−TiO₂(rutile) system. Intermolecular adsorbate-adsorbate interactions are taken into account for the nearest neighbors in the quasi-chemical approximation. The lattice energy parameters of all interactions of the model are determined from the Lennard-Jones potential (12-6). The formation of amorphous TiO₂(rutile) surface includes completion of the surface layers and partial removal of the surface oxygen ions. The quality of the amorphization procedure was confirmed by the experimentally measured heats and isotherms of adsorption of the system under study. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1109–1118, June, 1997.