mathematical methods

The need for mathematical tools in the project arises from the ill-posedness and computational difficulties of atmospheric phenomena. The field of inverse problems develops analytical and computational methods for this kind of context. The theoretical aspect aims make a mathematical formulation of an ill-posed or unstable problem, analyse its instability and uncertainty, and investigate what kind of information can be extracted in a stable way. The computational aspect aims to implement various stable solution methods i.e. regularization, but also to quantify the uncertainties involved. A central tool for uncertainty quatification is Bayesian sampling methods such as Markov Chain Monte Carlo (MCMC).

Atmospheric chemisty gives an example of an ill-posed problem. A module in such a model contains reaction schemes consisting of known or assumed reaction pathways and the dynamics of the chemical kinetics are modelled as differential equation systems. The module includes several unknown reaction rates which one wishes to determine by calibrating the models against experimental data. This is severely under-determined: the complex reaction schemes consist of thousands of reactions, while the experimental data for each module is limited to a handful of measured mass spectra. As a consequence, there may be large ranges of possible – and physically reasonable – reaction rates that fit the experimental data within the measurement uncertainties. Because there is no unique solution fitting the data, we must instead find out to what extent the available data enables model calibration: what are the most relevant reaction pathways and their reaction constants, and which are the less sensitive reactions and the possible ranges of their respective constants.