Noisy Monte Carlo: convergence of Markov chains with approximate transition kernels

Monte Carlo algorithms often aim to draw from a distribution π by simulating a Markov chain with transition kernel P such that π is invariant under P. However, there are many situations for which it is impractical or impossible to draw from the transition kernel P. For instance, this is the case with massive datasets, where is it prohibitively expensive to calculate the likelihood and is also the case for intractable likelihood models arising from, for example, Gibbs random fields, such as those found in spatial statistics and network analysis. A natural approach in these cases is to replace P by an approximation Pˆ. Using theory from the stability of Markov chains we explore a variety of situations where it is possible to quantify how ’close’ the chain given by the transition kernel Pˆ is to the chain given by P . We apply these results to several examples from spatial statistics and network analysis.

Adaptive approximate Bayesian computation

Sequential techniques can enhance the efficiency of the approximate Bayesian computation algorithm, as in Sisson et al.'s (2007) partial rejection control version. While this method is based upon the theoretical works of Del Moral et al. (2006), the application to approximate Bayesian computation results in a bias in the approximation to the posterior. An alternative version based on genuine importance sampling arguments bypasses this difficulty, in connection with the population Monte Carlo method of Cappe et al. (2004), and it includes an automatic scaling of the forward kernel. When applied to a population genetics example, it compares favourably with two other versions of the approximate algorithm.

Femtosecond evolution of spatially inhomogeneous carrier excitations: part 2: stochastic approach and GRID implementation

We present a stochastic approach for solving the quantum-kinetic equation introduced in Part I. A Monte Carlo method based on backward time evolution of the numerical trajectories is developed. The computational complexity and the stochastic error are investigated numerically. Variance reduction techniques are applied, which demonstrate a clear advantage with respect to the approaches based on symmetry transformation. Parallel implementation is realized on a GRID infrastructure.

Efficient non-linear data assimilation in geophysical fluid dynamics

New ways of combining observations with numerical models are discussed in which the size of the state space can be very large, and the model can be highly nonlinear. Also the observations of the system can be related to the model variables in highly nonlinear ways, making this data-assimilation (or inverse) problem highly nonlinear. First we discuss the connection between data assimilation and inverse problems, including regularization. We explore the choice of proposal density in a Particle Filter and show how the ’curse of dimensionality’ might be beaten. In the standard Particle Filter ensembles of model runs are propagated forward in time until observations are encountered, rendering it a pure Monte-Carlo method. In large-dimensional systems this is very inefficient and very large numbers of model runs are needed to solve the data-assimilation problem realistically. In our approach we steer all model runs towards the observations resulting in a much more efficient method. By further ’ensuring almost equal weight’ we avoid performing model runs that are useless in the end. Results are shown for the 40 and 1000 dimensional Lorenz 1995 model.

A complete atomistic model of molten polyethylene from neutron scattering data: a new methodology for polymer structure

A new approach to the study of the local organization in amorphous polymer materials is presented. The method couples neutron diffraction experiments that explore the structure on the spatial scale 1–20 Å with the reverse Monte Carlo fitting procedure to predict structures that accurately represent the experimental scattering results over the whole momentum transfer range explored. Molecular mechanics and molecular dynamics techniques are also used to produce atomistic models independently from any experimental input, thereby providing a test of the viability of the reverse Monte Carlo method in generating realistic models for amorphous polymeric systems. An analysis of the obtained models in terms of single chain properties and of orientational correlations between chain segments is presented. We show the viability of the method with data from molten polyethylene. The analysis derives a model with average C-C and C-H bond lengths of 1.55 Å and 1.1 Å respectively, average backbone valence angle of 112, a torsional angle distribution characterized by a fraction of trans conformers of 0.67 and, finally, a weak interchain orientational correlation at around 4 Å.

Inverse modeling of the 14C bomb pulse in stalagmites to constrain the dynamics of soil carbon cycling at selected European cave sites

The decomposition of soil organic matter (SOM) is temperature dependent, but its response to a future warmer climate remains equivocal. Enhanced rates of decomposition of SOM under increased global temperatures might cause higher CO2 emissions to the atmosphere, and could therefore constitute a strong positive feedback. The magnitude of this feedback however remains poorly understood, primarily because of the difficulty in quantifying the temperature sensitivity of stored, recalcitrant carbon that comprises the bulk (>90%) of SOM in most soils. In this study we investigated the effects of climatic conditions on soil carbon dynamics using the attenuation of the 14C ‘bomb’ pulse as recorded in selected modern European speleothems. These new data were combined with published results to further examine soil carbon dynamics, and to explore the sensitivity of labile and recalcitrant organic matter decomposition to different climatic conditions. Temporal changes in 14C activity inferred from each speleothem was modelled using a three pool soil carbon inverse model (applying a Monte Carlo method) to constrain soil carbon turnover rates at each site. Speleothems from sites that are characterised by semi-arid conditions, sparse vegetation, thin soil cover and high mean annual air temperatures (MAATs), exhibit weak attenuation of atmospheric 14C ‘bomb’ peak (a low damping effect, D in the range: 55–77%) and low modelled mean respired carbon ages (MRCA), indicating that decomposition is dominated by young, recently fixed soil carbon. By contrast, humid and high MAAT sites that are characterised by a thick soil cover and dense, well developed vegetation, display the highest damping effect (D = c. 90%), and the highest MRCA values (in the range from 350 ± 126 years to 571 ± 128 years). This suggests that carbon incorporated into these stalagmites originates predominantly from decomposition of old, recalcitrant organic matter. SOM turnover rates cannot be ascribed to a single climate variable, e.g. (MAAT) but instead reflect a complex interplay of climate (e.g. MAAT and moisture budget) and vegetation development.

A new calibrated sunspot group series since 1749: statistics of active day fractions

Although the sunspot-number series have existed since the mid-19th century, they are still the subject of intense debate, with the largest uncertainty being related to the "calibration" of the visual acuity of individual observers in the past. Daisy-chain regression methods are applied to inter-calibrate the observers which may lead to significant bias and error accumulation. Here we present a novel method to calibrate the visual acuity of the key observers to the reference data set of Royal Greenwich Observatory sunspot groups for the period 1900-1976, using the statistics of the active-day fraction. For each observer we independently evaluate their observational thresholds [S_S] defined such that the observer is assumed to miss all of the groups with an area smaller than S_S and report all the groups larger than S_S. Next, using a Monte-Carlo method we construct, from the reference data set, a correction matrix for each observer. The correction matrices are significantly non-linear and cannot be approximated by a linear regression or proportionality. We emphasize that corrections based on a linear proportionality between annually averaged data lead to serious biases and distortions of the data. The correction matrices are applied to the original sunspot group records for each day, and finally the composite corrected series is produced for the period since 1748. The corrected series displays secular minima around 1800 (Dalton minimum) and 1900 (Gleissberg minimum), as well as the Modern grand maximum of activity in the second half of the 20th century. The uniqueness of the grand maximum is confirmed for the last 250 years. It is shown that the adoption of a linear relationship between the data of Wolf and Wolfer results in grossly inflated group numbers in the 18th and 19th centuries in some reconstructions.