Daan Crommelin - Research
My research focuses on stochastic methods for multiscale dynamical systems, with an emphasis on data-driven methods such as estimation of stochastic-dynamical models from data. Applications in atmosphere-ocean science are a major source of motivation.
The climate system and its components (e.g. atmosphere and ocean) are nonlinear systems with a wide range of active scales, both in time and space. A (re-)emerging approach for the study of these systems makes use of stochastic methods. Put crudely, the dynamical processes at small space and time scales are interpreted as noise, so that what remains is a system of nonlinear interactions on macroscopic space and time scales, augmented with noise that represents the collective impact of microscopic processes on macrocsopic dynamics. For the climate system, matters are complicated by the fact that the dictinstion between macroscopic and microscopic is not so clear: there is a hierarchy of scales but not a scale gap.
Some current and recent projects:
Stochastic parameterization of subgrid scale processes
Probably the main obstacle right
now for further improvement of climate and weather forecasting models is the representation of
unresolved (subgrid scale) processes in these models. Cloud formation, small-scale turbulence,
interaction with boundaries, all these processes should be taken into account. However, they
take place on spatial scales that are too small to be resolved explicitly in numerical models.
Parameterizations are methods to represent the "bulk effect" of these subgrid scale processes
in models. The use of stochastic methods for parameterizations has gotten a lot of attention
in recent years. We have proposed a new approach to stochastic parameterization, one in which
the subgrid processes are modelled as conditional Markov chains. Currently, we are applying this
approach to the problem of atmospheric convection parameterization.
[in collaboration with J. Dorrestijn, P. Siebesma, H. Jonker, F. Selten, E. Vanden-Eijnden]
Estimation of Markov processes from discretely sampled data
Modeling phenomena by Markov
("random") processes can be difficult because derivation of such models from first principles
may be impossible. A way out is provided by estimating models from available data. We work
on the development of algorithms that are suitable for estimation from finite data that is both
discretely sampled (non-negligible time interval between datapoints) and not entirely
consistent with a Markov process. These two aspects are common to many datasets arising in
applications. Within the class of Markov processes, we focus mainly on diffusion processes
(which can be modeled with stochastic differential equations) and jump processes (which have
discrete states). Estimation from multiscale data, in such a way that the inferred Markov process
gives a correct coarse-grained description of the observed process, has our particular interest.
[in collaboration with E. Vanden-Eijnden]
Modeling of sea-surface winds with stochastic differential equations
The behavior of the
wind at sea surface level is important for the interaction between ocean and atmosphere.
In order to capture the correct mean and fluctuations of both direction and speed of sea-surface
winds, stochastic differential equations can be used. After providing a physically motivated
form for these equations, we estimate the free parameters by fitting the equations to datasets
from measurement stations at sea.
[in collaboration with A. Monahan]
Regime behavior and metastability in atmospheric datasets
The framework of so-called
Hidden Markov Models (HMMs) is very suitable to analyse whether there exist preferred states
("regimes") of the large-scale atmospheric circulation. The concept of atmospheric regimes has
been around for at least half a century, but the last word on their existence has not been said
yet. The limited amount of available measurement data and the absence of a clear time scale gap
make the study of regimes notoriously difficult. HMMs are a useful analysis tool that had
previously not been used for this purpose.
[in collaboration with C. Franzke, A. Majda, A. Fischer]