Data Assimilation of Nonlocal Observations in Complex Systems

Natural complex systems exhibit spatio-temporal dynamics on multiple scales that is difficult to predict and understand. Prominent examples for such systems are the earth atmosphere, the brain or even single biological cells. To gain deeper insights into the system's dynamics and to be able to predict the system evolution, typically observations are analyzed what allows to derive or motivate corresponding models. These days more and more observations of diverse types are available that mirror the systems dynamics. In general, one may distinguish two types of data. The so-called in-situ or local observations capture measures in the system itsself, such as temperature in the atmosphere at a certain vertical height or electric potentials in biological cells. Other observations are not measured at a certain location but represents the integral of system activity. Examples in meteorology for such nonlocal observations are satellite radiances, slant delays and radio occultation based on GPS data, or radar reflectivities. In biological systems, the non-invasive measurement techniques provide nonlocal observations, such as electro- and magnetoencephalogram or Magnetic Resonance Imaging. In addition to observations, realistic models are essential to improve the understanding of natural complex systems and to predict their dynamical evolution.

To merge both models and observations, it is essential to develop techniques that estimate optimally the system activity well-adapted to a model and observed data. Data assimilation is an important technique to match diverse experimental data with an underlying model. The technique combines optimally observations and a model to achieve a certain goal. This goal may represent optimal fitting of model parameters or providing optimal forecasts of the system's dynamics. Since the recent years have shown an increasing number of observation techniques capturing integrals of system activity, data assimilation of nonlocal observations becomes more and more important.

The present Research Topic aims to bring together recent theoretical work in data assimilation of nonlocal observations with a strong link to specific applications. This article collection reflects the state-of-the-art in the research field and permits to provide insight into different disciplines, such as fundamental mathematical research, earth science, computer science and the biological sciences. Examples of theoretical topics (as an unconstrained open list) are Kalman filters, variational assimilation techniques, regression techniques and stochastic optimization techniques. Applications may range from the parameter estimation in genetic regulatory networks over prediction of brain dynamics to weather forecast.