A proper choice of a proposal distribution for Markov chain Monte Carlo methods, for example for the Metropolis±Hastings algorithm, is well known to be a crucial factor for the convergence of the algorithm. In this paper we introduce an adaptive Metropolis (AM) algorithm, where the Gaussian proposal distribution is updated along the process using the full information cumulated so far. Due to the adaptive nature of the process, the AM algorithm is non-Markovian, but we establish here that it has the correct ergodic properties. We also include the results of our numerical tests, which indicate that the AM algorithm competes well with traditional Metropolis±Hastings algorithms, and demonstrate that the AM algorithm is easy to use in practical computation.

We propose to combine two quite powerful ideas that have recently appeared in the Markov chain Monte Carlo literature: adaptive Metropolis samplers and delayed rejection. The ergodicity of the resulting non-Markovian sampler is proved, and the efficiency of the combination is demonstrated with various examples. We present situations where the combination outperforms the original methods: adaptation clearly enhances efficiency of the delayed rejection algorithm in cases where good proposal distributions are not available. Similarly, delayed rejection provides a systematic remedy when the adaptation process has a slow start.

No abstract

No abstract

Abstract. High-dimensional inverse problems present a challenge for Markov chain Monte Carlo (MCMC)-type sampling schemes. Typically, they rely on finding an efficient proposal distribution, which can be difficult for large-scale problems, even with adaptive approaches. Moreover, the autocorrelations of the samples typically increase with dimension, which leads to the need for long sample chains. We present an alternative method for sampling from posterior distributions in nonlinear inverse problems, when the measurement error and prior are both Gaussian. The approach computes a candidate sample by solving a stochastic optimization problem. In the linear case, these samples are directly from the posterior density, but this is not so in the nonlinear case. We derive the form of the sample density in the nonlinear case, and then show how to use it within both a Metropolis-Hastings and importance sampling framework to obtain samples from the posterior distribution of the parameters. We demonstrate, with various small-and medium-scale problems, that randomize-then-optimize can be efficient compared to standard adaptive MCMC algorithms.

No abstract

Changes in bacterioplankton community composition were followed in mesocosms set up in the littoral of Lake Vesijärvi, southern Finland, over two summers. Increasing nitrogen and phosphorus concentrations in the mesocosms represented different trophic states, from mesotrophic to hypertrophic. In 1998, the mesocosms were in a turbid state with a high biomass of phytoplankton, whereas in 1999, macrophytes proliferated and a clear-water state prevailed. The bacterial communities in the mesocosms also developed differently, as shown by denaturing gradient gel electrophoresis profiling of partial 16S rRNA gene fragments and by nonmetric multidimensional scaling analysis. In 1998, nutrient treatments affected the diversity and clustering of bacterial communities strongly, but in 1999, the bacterial communities were less diversified and not clearly affected by treatments. Canonical correspondence analysis indicated that bacterioplankton communities in the mesocosms were influenced by environmental physicochemical variables linked to the increasing level of eutrophication. Nitrogen concentration correlated directly with the bacterioplankton composition. In addition, the high nutrient levels had indirect effects through changes in the biomass and composition of phyto- and zooplankton. Sequencing analysis showed that the dominant bacterial divisions remained the same, but the dominant phylotypes changed during the 2-year period. The occurrence of Verrucomicrobia correlated with more eutrophic conditions, whereas the occurrence of Actinobacteria correlated with less eutrophic conditions.

Electronic and spectral properties of small TiO2 particles have been studied in order to gain more knowledge on their dependence on the crystal- and particle-size distributions. Our goal is to extend the recently developed light scattering based method for determining submicrometer size particles to nanoparticles. For that, we need to know how the refractive index function depends on the cluster size. As a first step, we have used time-dependent density functional theory (TDDFT) calculations having a focus on the shape changes of the calculated spectra, which can be related to changes in the refractive index function. Starting from the structure of TiO2 molecule for the two smallest particles and truncated bulk anatase structure for larger particles, the structures for (TiO2) n clusters, n = {1, 2, 8, 18, 28, 38}, have been modeled. After the structure optimization using standard density functional theory (DFT) approach, the photoabsorption spectra for the optimized particle structures have been calculated by using TDDFT calculations. The results show slight evidence of the band gap broadening in the case of three out of the smallest particles and strong structural dependence of electronic and spectral properties, which can partly be related to the transformation of the electron structure, and breaking of the crystal symmetry as the size of the particle becomes smaller. These findings indicate that in the case of small particles their refractive index function can differ from the bulk values, and this has to be taken into account in the interpretation of light-scattering measurements.