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.

Cambium non est mutuum: exchange and interest rates in medieval Europe

A major gap in our understanding of the medieval economy concerns interest rates, especially relating to commercial credit. Although direct evidence about interest rates is scattered and anecdotal, there is much more surviving information about exchange rates. Since both contemporaries and historians have suggested that exchange and rechange transactions could be used to disguise the charging of interest in order to circumvent the usury prohibition, it should be possible to back out the interest rates from exchange rates. The following analysis is based on a new dataset of medieval exchange rates collected from commercial correspondence in the archive of Francesco di Marco Datini of Prato, c.1383-1411. It demonstrates that the time value of money was consistently incorporated into market exchange rates. Moreover, these implicit interest rates are broadly comparable to those received from other types of commercial loan and investment. Although on average profitable, the return on any individual exchange and rechange transaction did involve a degree of uncertainty that may have justified their non-usurious nature. However, there were also practical reasons why medieval merchants may have used foreign exchange transactions as a means of extending credit.