Bayesian Inversion of Stokes Profiles

[abridged] Inversion techniques are the most powerful methods to obtain information about the thermodynamical and magnetic properties of solar and stellar atmospheres. In the last years, we have witnessed the development of highly sophisticated inversion codes that are now widely applied to spectro-polarimetric observations. The majority of these inversion codes are based on the optimization of a complicated non-linear merit function. However, no reliable and statistically well-defined confidence intervals can be obtained for the parameters inferred from the inversions. A correct estimation of the confidence intervals for all the parameters that describe the model is mandatory. Additionally, it is fundamental to apply efficient techniques to assess the ability of models to reproduce the observations and to what extent the models have to be refined or can be simplified. Bayesian techniques are applied to analyze the performance of the model to fit a given observed Stokes vector. The posterior distribution, is efficiently sampled using a Markov Chain Monte Carlo method. For simplicity, we focus on the Milne-Eddington approximate solution of the radiative transfer equation and we only take into account the generation of polarization through the Zeeman effect. However, the method is extremely general and other more complex forward models can be applied. We illustrate the ability of the method with the aid of academic and realistic examples. We show that the information provided by the posterior distribution turns out to be fundamental to understand and determine the amount of information available in the Stokes profiles in these particular cases.Comment: 15 pages, 12 figures, accepted for publication in A&

Data-driven modelling of biological multi-scale processes

Biological processes involve a variety of spatial and temporal scales. A holistic understanding of many biological processes therefore requires multi-scale models which capture the relevant properties on all these scales. In this manuscript we review mathematical modelling approaches used to describe the individual spatial scales and how they are integrated into holistic models. We discuss the relation between spatial and temporal scales and the implication of that on multi-scale modelling. Based upon this overview over state-of-the-art modelling approaches, we formulate key challenges in mathematical and computational modelling of biological multi-scale and multi-physics processes. In particular, we considered the availability of analysis tools for multi-scale models and model-based multi-scale data integration. We provide a compact review of methods for model-based data integration and model-based hypothesis testing. Furthermore, novel approaches and recent trends are discussed, including computation time reduction using reduced order and surrogate models, which contribute to the solution of inference problems. We conclude the manuscript by providing a few ideas for the development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and Multiscale Dynamics (American Scientific Publishers

Finite-Size Effects on Traveling Wave Solutions to Neural Field Equations

Neural field equations are used to describe the spatiotemporal evolution of the activity in a network of synaptically coupled populations of neurons in the continuum limit. Their heuristic derivation involves two approximation steps. Under the assumption that each population in the network is large, the activity is described in terms of a population average. The discrete network is then approximated by a continuum. In this article we make the two approximation steps explicit. Extending a model by Bressloff and Newby, we describe the evolution of the activity in a discrete network of finite populations by a Markov chain. In order to determine finite-size effects - deviations from the mean field limit due to the finite size of the populations in the network - we analyze the fluctuations of this Markov chain and set up an approximating system of diffusion processes. We show that a well-posed stochastic neural field equation with a noise term accounting for finite-size effects on traveling wave solutions is obtained as the strong continuum limit

Rejoinder on: queueing models for the analysis of communication systems

In this rejoinder, we respond to the comments and questions of three discussants of our paper on queueing models for the analysis of communication systems. Our responses are structured around two main topics: discrete-time modeling and further extensions of the presented queueing analysis