A variational Bayesian method for inverse problems with impulsive noise

We propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers. The approach is of Bayesian type, and it exploits a heavy-tailed t distribution for data noise to achieve robustness with respect to outliers. A hierarchical model with all hyper-parameters automatically determined from the given data is described. An algorithm of variational type by minimizing the Kullback-Leibler divergence between the true posteriori distribution and a separable approximation is developed. The numerical method is illustrated on several one- and two-dimensional linear and nonlinear inverse problems arising from heat conduction, including estimating boundary temperature, heat flux and heat transfer coefficient. The results show its robustness to outliers and the fast and steady convergence of the algorithm.Comment: 20 pages, to appear in J. Comput. Phy

Parameter Identification in a Probabilistic Setting

Parameter identification problems are formulated in a probabilistic language, where the randomness reflects the uncertainty about the knowledge of the true values. This setting allows conceptually easily to incorporate new information, e.g. through a measurement, by connecting it to Bayes's theorem. The unknown quantity is modelled as a (may be high-dimensional) random variable. Such a description has two constituents, the measurable function and the measure. One group of methods is identified as updating the measure, the other group changes the measurable function. We connect both groups with the relatively recent methods of functional approximation of stochastic problems, and introduce especially in combination with the second group of methods a new procedure which does not need any sampling, hence works completely deterministically. It also seems to be the fastest and more reliable when compared with other methods. We show by example that it also works for highly nonlinear non-smooth problems with non-Gaussian measures.Comment: 29 pages, 16 figure

Aging of dissolved copper and copper-based nanoparticles in five different soils: short-term kinetics vs long-term fate

With the growing availability and use of copper based nanomaterials (Cu-NMs), there is increasing concern regarding their release and potential impact on the environment. In this study, the short term (≤ 5 days) ageing profile and the long term (4 months) speciation of dissolved Cu, copper oxide (CuO-) and copper sulfide nanoparticles (CuS-NPs) were investigated in five different soils using X-ray absorption spectroscopy (XAS). Soil pH was found to strongly influence the short term chemistry of the Cu-NMs added at 100 mg/kg above background. Low pH soils promoted rapid dissolution of CuO-NPs that effectively aligned their behaviour to that of dissolved Cu within 3 days. In higher pH soils, CuO-NPs persisted longer due to slower dissolution in the soil and resulted in contrasting short term speciation compared to dissolved Cu, which formed copper hydroxides and carbonates that were reflective of the soil chemistry. Organic matter appeared to slow the dissolution process but in the long term, the speciation of Cu added as dissolved Cu, CuO-NPs and CuS-NPs were found to be same for each soil. The results imply that in the short term Cu-NMs may exhibit unique behaviour in alkaline soils compared to their conventional forms (e.g. in the event of an adverse leaching event), but in the long term (≥ 4 months), their fates are dictated by the soil properties and are independent of the initial Cu form, and are likely to present minimal risk of nano-specific Cu-NM impact in the soil environment for the concentration studied here